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<span class="a" style="left:1191px;top:1802px;word-spacing:5px;letter-spacing:-6px">APOSTILA DE ÁLGEBRA</span><span class="a" style="left:1917px;top:1964px;letter-spacing:-7px">LINEAR</span></div>
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<span class="a" style="left:1627px;top:2356px;word-spacing:5px">Prof. Dr Rogério de Aguiar</span></div>
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<span class="a" style="left:1506px;top:2474px;word-spacing:1px">Chefe do Departamento de Matemática</span><span class="a" style="left:1688px;top:2577px;word-spacing:3px">CCT - UDESC - JOINVILLE</span><span class="a" style="left:1620px;top:2680px;word-spacing:-7px">Email: <span class="w" style="width:21px"></span>dma2ra<span class="l6">@join<span class="l7">ville.udes<span class="l6">c.br</span></span></span></span><span class="a" style="left:1315px;top:2783px;word-spacing:16px;letter-spacing:-1px">Home Page: www.joinville.udesc.br/dmat/rogerio</span><span class="a" style="left:1352px;top:2957px;word-spacing:3px">Professores Integrantes do Projeto de Álgebra II</span><span class="a" style="left:1559px;top:3060px;word-spacing:1px">Graciela Moro - Coordenadora</span><span class="a" style="left:1610px;top:3163px;word-spacing:3px;letter-spacing:-1px">Ivanete Zucki</span><span class="a" style="left:1834px;top:3265px;word-spacing:2px">João de Azevedo</span><span class="a" style="left:1777px;top:3368px;word-spacing:1px">Jorge Mota</span><span class="a" style="left:1719px;top:3471px;word-spacing:1px">Marnei Luis Mandler</span><span class="a" style="left:1719px;top:3574px;word-spacing:3px">Milton Procópio de Borba</span><span class="a" style="left:1721px;top:3779px;word-spacing:2px">Rogério de Aguiar</span><span class="a" style="left:1860px;top:3992px;word-spacing:5px">25 de Março de 2008</span></div>
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<span class="a" style="left:1191px;top:1802px;word-spacing:5px;letter-spacing:-6px">APOSTILA DE ÁLGEBRA</span><span class="a" style="left:1917px;top:1964px;letter-spacing:-7px">LINEAR</span></div>
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<span class="a" style="left:1627px;top:2356px;word-spacing:5px">Prof. Dr Rogério de Aguiar</span></div>
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<span class="a" style="left:1506px;top:2474px;word-spacing:1px">Chefe do Departamento de Matemática</span><span class="a" style="left:1688px;top:2577px;word-spacing:3px">CCT - UDESC - JOINVILLE</span><span class="a" style="left:1620px;top:2680px;word-spacing:-7px">Email: <span class="w" style="width:21px"></span>dma2ra<span class="l6">@join<span class="l7">ville.udes<span class="l6">c.br</span></span></span></span><span class="a" style="left:1315px;top:2783px;word-spacing:16px;letter-spacing:-1px">Home Page: www.joinville.udesc.br/dmat/rogerio</span><span class="a" style="left:1352px;top:2957px;word-spacing:3px">Professores Integrantes do Projeto de Álgebra II</span><span class="a" style="left:1559px;top:3060px;word-spacing:1px">Graciela Moro - Coordenadora</span><span class="a" style="left:1610px;top:3163px;word-spacing:3px;letter-spacing:-1px">Ivanete Zucki</span><span class="a" style="left:1834px;top:3265px;word-spacing:2px">João de Azevedo</span><span class="a" style="left:1777px;top:3368px;word-spacing:1px">Jorge Mota</span><span class="a" style="left:1719px;top:3471px;word-spacing:1px">Marnei Luis Mandler</span><span class="a" style="left:1719px;top:3574px;word-spacing:3px">Milton Procópio de Borba</span><span class="a" style="left:1721px;top:3779px;word-spacing:2px">Rogério de Aguiar</span><span class="a" style="left:1860px;top:3992px;word-spacing:5px">25 de Março de 2008</span></div>
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<span class="a" style="left:985px;top:1415px;letter-spacing:-2px">Sumário</span></div>
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<span class="a" style="left:985px;top:1969px;word-spacing:139px;letter-spacing:40px">1 <span class="l" style="margin-left:-168px">M<span class="l" style="margin-left:-39px">A<span class="l" style="margin-left:-48px">T<span class="l" style="margin-left:-40px">R<span class="l" style="margin-left:-41px">I<span class="l" style="margin-left:-40px">Z<span class="l" style="margin-left:-40px">E<span class="l" style="margin-left:-41px">S <span class="l" style="margin-left:-180px">E <span class="l" style="margin-left:-208px">S<span class="l" style="margin-left:-40px">I<span class="l" style="margin-left:-40px">S<span class="l" style="margin-left:-40px">T<span class="l" style="margin-left:-40px">E<span class="l" style="margin-left:-41px">M<span class="l" style="margin-left:-40px">A<span class="l" style="margin-left:-40px">S <span class="w" style="width:1129px"></span>3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:2057px;word-spacing:4px;letter-spacing:7px">1<span class="l7">.<span class="l8">1 <span class="w" style="width:40px"></span>T<span class="l8">i<span class="l7">p<span class="l6">o<span class="l7">s <span class="l10">d<span class="l8">e <span class="l11">m<span class="l7">a<span class="l8">t<span class="l7">r<span class="l7">i<span class="l8">z<span class="l7">e<span class="l8">s <span class="l" style="margin-left:-13px">. . . . . <span class="w6"></span>. . . . . . <span class="w6"></span>. . . . . <span class="w6"></span>. . . . . . <span class="w6"></span>. . . . . <span class="w" style="width:85px"></span>3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2146px;word-spacing:3px;letter-spacing:7px">1<span class="l7">.<span class="l8">2 <span class="w" style="width:41px"></span>O<span class="l8">pe<span class="l12">r<span class="l8">a<span class="l7">ç<span class="l7">õ<span class="l8">e<span class="l7">s <span class="l9">c<span class="l8">o<span class="l7">m <span class="l11">m<span class="l7">a<span class="l8">t<span class="l7">r<span class="l7">i<span class="l8">z<span class="l7">e<span class="l8">s <span class="w12"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . <span class="w" style="width:84px"></span>6</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2234px;word-spacing:3px;letter-spacing:8px">1<span class="l8">.<span class="l9">3 <span class="w" style="width:39px"></span>M<span class="l9">a<span class="l8">t<span class="l8">r<span class="l9">i<span class="l8">z <span class="l12">e<span class="l8">s<span class="l8">c<span class="l9">a<span class="l8">l<span class="l9">o<span class="l8">n<span class="l8">a<span class="l9">d<span class="l8">a <span class="w" style="width:35px"></span>. . . . . . . . . . . . . . . . . . . . . . . . . . <span class="w" style="width:82px"></span>8</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2322px;word-spacing:3px;letter-spacing:7px">1<span class="l7">.<span class="l8">4 <span class="w" style="width:41px"></span>C<span class="l8">á<span class="l7">l<span class="l8">c<span class="l7">u<span class="l7">l<span class="l8">o <span class="l10">d<span class="l7">a <span class="l10">i<span class="l7">n<span class="l10">v<span class="l9">e<span class="l7">r<span class="l8">s<span class="l7">a <span class="w" style="width:29px"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . <span class="w" style="width:48px"></span>1<span class="l8">0</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2410px;word-spacing:5px;letter-spacing:7px">1<span class="l7">.<span class="l8">5 <span class="w" style="width:39px"></span>D<span class="l8">e<span class="l7">t<span class="l8">e<span class="l7">r<span class="l7">m<span class="l8">i<span class="l8">n<span class="l7">a<span class="l7">n<span class="l9">t<span class="l8">e<span class="l7">s <span class="w" style="width:36px"></span>. . . . . . . . . . . . . . . . . . . . . . . . . . . . <span class="w" style="width:46px"></span>1<span class="l8">1</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2498px">1.6</span></div>
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<span class="a" style="left:1264px;top:2497px;word-spacing:2px;letter-spacing:1px">Sexta lista de exercícios</span></div>
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<span class="a" style="left:2203px;top:2498px;letter-spacing:10px">. . . . . . <span class="l6">. . . . . . . <span class="l6">. . . . . . . <span class="l6">. <span class="w" style="width:42px"></span>1<span class="l11">4</span></span></span></span></span><span class="a" style="left:1095px;top:2586px;word-spacing:5px;letter-spacing:5px">1.<span class="l11">7 <span class="w" style="width:43px"></span>Si<span class="l11">st<span class="l10">e<span class="l6">ma <span class="l" style="margin-left:-13px">de <span class="l" style="margin-left:-13px">e<span class="l6">qu<span class="l10">a<span class="l6">çõ<span class="l10">e<span class="l6">s <span class="l7">l<span class="l6">in<span class="l11">ea<span class="l10">re<span class="l11">s <span class="l10">. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. <span class="w" style="width:47px"></span>1<span class="l6">6</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:2674px;word-spacing:3px;letter-spacing:8px">1<span class="l8">.<span class="l9">7<span class="l8">.<span class="l9">1 <span class="w" style="width:48px"></span>I<span class="l8">n<span class="l11">t<span class="l8">r<span class="l8">o<span class="l7">d<span class="l8">u<span class="l8">ç<span class="l8">ã<span class="l9">o <span class="w" style="width:33px"></span>. . . . . . . . . . . . . . . . . . . . . . . . . . <span class="w" style="width:46px"></span>1<span class="l9">6</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:2762px;word-spacing:5px;letter-spacing:6px">1<span class="l6">.<span class="l7">7<span class="l6">.<span class="l7">2 <span class="w" style="width:50px"></span>S<span class="l6">i<span class="l7">s<span class="l6">t<span class="l6">e<span class="l7">m<span class="l6">a<span class="l7">s <span class="l9">e <span class="l10">m<span class="l7">a<span class="l6">t<span class="l7">r<span class="l6">i<span class="l7">z<span class="l6">e<span class="l7">s<span class="l6">. <span class="w" style="width:20px"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w" style="width:50px"></span>1<span class="l7">7</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:2850px;word-spacing:8px;letter-spacing:3px">1.<span class="l7">7.<span class="l7">3 <span class="w" style="width:53px"></span>So<span class="l6">lu<span class="l7">çã<span class="l7">o <span class="l7">de <span class="l10">um <span class="l11">si<span class="l6">st<span class="l7">em<span class="l7">a <span class="l7">por <span class="l11">ma<span class="l7">tr<span class="l7">iz <span class="l10">in<span class="l9">v<span class="l6">er<span class="l6">sa <span class="w11"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. <span class="w6"></span>. . <span class="w" style="width:53px"></span>20</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2960px">1.8</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:1264px;top:2960px;word-spacing:2px;letter-spacing:1px">Sétima lista de exercícios</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2260px;top:2960px;letter-spacing:10px">. . . . . . <span class="l6">. . . . . . . <span class="l6">. . . . . . . <span class="w" style="width:37px"></span>2<span class="l11">1</span></span></span></span><span class="a" style="left:1095px;top:3049px;word-spacing:4px;letter-spacing:8px">1<span class="l8">.<span class="l9">9 <span class="w" style="width:38px"></span>A<span class="l9">p<span class="l6">ê<span class="l8">n<span class="l9">d<span class="l8">i<span class="l8">c<span class="l9">e <span class="w" style="width:33px"></span>. . . . . . <span class="l6">. . . . . . . <span class="l6">. . . . . . <span class="l6">. . . . . . . <span class="l6">. . . . . <span class="w" style="width:39px"></span>2<span class="l9">4</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:3137px;word-spacing:5px;letter-spacing:5px">1.<span class="l11">9.<span class="l11">1 <span class="w" style="width:52px"></span>C<span class="l6">ál<span class="l11">cu<span class="l10">l<span class="l6">o <span class="l8">da <span class="l" style="margin-left:-13px">i<span class="l6">n<span class="l7">v<span class="l7">er<span class="l11">sa <span class="l" style="margin-left:-13px">po<span class="l8">r <span class="l8">ad<span class="l10">j<span class="l6">un<span class="l12">t<span class="l6">a <span class="w" style="width:22px"></span>. . <span class="w8"></span>. . <span class="w9"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. <span class="w" style="width:47px"></span>2<span class="l6">4</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:3225px;word-spacing:3px;letter-spacing:7px">1<span class="l7">.<span class="l8">9<span class="l7">.<span class="l8">2 <span class="w" style="width:50px"></span>R<span class="l8">e<span class="l7">g<span class="l8">r<span class="l7">a <span class="l10">d<span class="l7">e <span class="l10">C<span class="l8">r<span class="l7">a<span class="l7">m<span class="l8">e<span class="l8">r . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. <span class="w" style="width:45px"></span>2<span class="l8">5</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:3386px;word-spacing:205px;letter-spacing:41px">2 <span class="l" style="margin-left:-236px">E<span class="l" style="margin-left:-41px">S<span class="l" style="margin-left:-41px">P<span class="l" style="margin-left:-49px">A<span class="l" style="margin-left:-41px">Ç<span class="l" style="margin-left:-41px">O<span class="l" style="margin-left:-41px">S <span class="l" style="margin-left:-276px">V<span class="l" style="margin-left:-42px">E<span class="l" style="margin-left:-41px">T<span class="l" style="margin-left:-41px">O<span class="l" style="margin-left:-42px">R<span class="l" style="margin-left:-42px">I<span class="l" style="margin-left:-41px">A<span class="l" style="margin-left:-41px">I<span class="l" style="margin-left:-41px">S <span class="w" style="width:1139px"></span>2<span class="l" style="margin-left:-41px">9</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:3474px;word-spacing:4px;letter-spacing:7px">2<span class="l7">.<span class="l8">1 <span class="w" style="width:40px"></span>I<span class="l7">n<span class="l10">t<span class="l7">r<span class="l7">o<span class="l6">d<span class="l7">u<span class="l7">ç<span class="l7">ã<span class="l8">o <span class="l" style="margin-left:-16px">. . . . . . <span class="w6"></span>. . . . . <span class="w6"></span>. . . . . . <span class="w6"></span>. . . . . <span class="w6"></span>. . . . . . <span class="w6"></span>. . . <span class="w" style="width:47px"></span>2<span class="l8">9</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:3562px;word-spacing:3px;letter-spacing:8px">2<span class="l8">.<span class="l9">2 <span class="w" style="width:39px"></span>S<span class="l8">u<span class="l8">b<span class="l6">e<span class="l9">s<span class="l8">p<span class="l8">a<span class="l8">ç<span class="l9">o<span class="l8">s <span class="w" style="width:27px"></span>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . <span class="w" style="width:46px"></span>3<span class="l9">6</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:3651px;word-spacing:5px;letter-spacing:4px">2.<span class="l9">3 <span class="w" style="width:45px"></span>In<span class="l11">te<span class="l8">rs<span class="l9">ec<span class="l8">çã<span class="l9">o <span class="l6">de <span class="l10">do<span class="l9">is <span class="l10">Su<span class="l8">be<span class="l7">sp<span class="l8">aç<span class="l9">os <span class="l10">V<span class="l11">et<span class="l9">or<span class="l8">ia<span class="l9">is <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w" style="width:49px"></span>38</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:3739px;word-spacing:5px;letter-spacing:6px">2<span class="l6">.<span class="l7">4 <span class="w" style="width:41px"></span>C<span class="l7">o<span class="l6">m<span class="l9">b<span class="l6">i<span class="l7">n<span class="l6">a<span class="l6">ç<span class="l7">ã<span class="l6">o <span class="l10">L<span class="l6">i<span class="l7">n<span class="l6">e<span class="l6">a<span class="l6">r . . . . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w" style="width:50px"></span>4<span class="l7">0</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:3827px;word-spacing:7px;letter-spacing:4px">2.<span class="l9">5 <span class="w" style="width:43px"></span>De<span class="l9">pe<span class="l7">nd<span class="l8">ên<span class="l9">ci<span class="l9">a <span class="l8">e <span class="l7">In<span class="l9">de<span class="l9">pe<span class="l6">nd<span class="l9">ên<span class="l8">ci<span class="l9">a <span class="l8">Li<span class="l9">ne<span class="l9">ar <span class="w7"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w" style="width:52px"></span>41</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:3915px;word-spacing:3px;letter-spacing:7px">2<span class="l7">.<span class="l8">6 <span class="w" style="width:41px"></span>S<span class="l7">u<span class="l7">be<span class="l" style="margin-left:-13px">s<span class="l7">p<span class="l7">a<span class="l7">ç<span class="l8">o<span class="l7">s <span class="l10">G<span class="l7">e<span class="l7">r<span class="l8">a<span class="l7">d<span class="l7">o<span class="l8">s <span class="w" style="width:28px"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. <span class="w" style="width:45px"></span>4<span class="l8">3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:4003px;word-spacing:3px;letter-spacing:7px">2<span class="l7">.<span class="l8">7 <span class="w" style="width:41px"></span>S<span class="l7">o<span class="l7">m<span class="l8">a <span class="l10">d<span class="l7">e <span class="l10">S<span class="l7">u<span class="l8">be<span class="l12">s<span class="l7">p<span class="l7">a<span class="l8">ç<span class="l7">o<span class="l7">s <span class="w" style="width:16px"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. <span class="w" style="width:45px"></span>4<span class="l8">5</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:4091px;word-spacing:5px;letter-spacing:5px">2.<span class="l11">8 <span class="w" style="width:43px"></span>Ba<span class="l10">se <span class="w11"></span>e <span class="l8">Di<span class="l11">m<span class="l6">en<span class="l11">sã<span class="l10">o <span class="l8">de <span class="l" style="margin-left:-13px">um <span class="l" style="margin-left:-13px">Es<span class="l10">p<span class="l6">aç<span class="l10">o <span class="l8">V<span class="l12">e<span class="l6">to<span class="l10">r<span class="l6">ia<span class="l11">l <span class="w" style="width:41px"></span>. . <span class="w9"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. <span class="w" style="width:47px"></span>4<span class="l6">6</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:4179px;word-spacing:1px;letter-spacing:9px">2<span class="l9">.<span class="l10">8<span class="l9">.<span class="l10">1 <span class="w" style="width:48px"></span>B<span class="l9">a<span class="l9">s<span class="l9">e . . . . . . . . . . . . . . . . . . . . . . . . . . . . <span class="w6"></span>. . <span class="w" style="width:43px"></span>4<span class="l10">6</span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:4267px;word-spacing:3px;letter-spacing:8px">2<span class="l8">.<span class="l9">8<span class="l8">.<span class="l9">2 <span class="w" style="width:48px"></span>D<span class="l9">i<span class="l9">m<span class="l8">e<span class="l9">n<span class="l8">s<span class="l8">ã<span class="l8">o <span class="w11"></span>. . . . . . . . . . . . . . . . . . . . . . . . . . . <span class="w" style="width:46px"></span>4<span class="l9">9</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:4355px;word-spacing:7px;letter-spacing:3px">2.<span class="l7">8.<span class="l7">3 <span class="w" style="width:54px"></span>Di<span class="l8">me<span class="l7">ns<span class="l6">ão <span class="l9">da <span class="l9">So<span class="l7">ma <span class="l9">de <span class="l9">Su<span class="l7">bes<span class="l8">pa<span class="l6">ço<span class="l7">s V<span class="l" style="margin-left:-15px">et<span class="l7">or<span class="l7">ia<span class="l7">is <span class="w7"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w6"></span>. <span class="w" style="width:49px"></span>50</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:4443px;word-spacing:5px;letter-spacing:7px">2<span class="l7">.<span class="l8">8<span class="l7">.<span class="l8">4 <span class="w" style="width:48px"></span>C<span class="l8">oo<span class="l12">r<span class="l8">d<span class="l7">e<span class="l7">n<span class="l8">a<span class="l7">d<span class="l7">a<span class="l7">s <span class="w" style="width:24px"></span>. . . . . . . . . . . . . . . . . . . . . . . . . <span class="w" style="width:46px"></span>5<span class="l8">0</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:4532px;word-spacing:3px;letter-spacing:7px">2<span class="l7">.<span class="l8">9 <span class="w" style="width:41px"></span>M<span class="l8">u<span class="l7">d<span class="l7">a<span class="l7">n<span class="l7">ç<span class="l8">a <span class="l10">d<span class="l7">e <span class="l10">B<span class="l7">a<span class="l7">s<span class="l7">e <span class="l7">. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w" style="width:50px"></span>5<span class="l8">1</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:4620px;word-spacing:4px;letter-spacing:4px">2.<span class="l9">10 A In<span class="l11">v<span class="l6">er<span class="l9">sa <span class="l9">da <span class="l9">Ma<span class="l9">tr<span class="l9">iz <span class="l9">de <span class="l9">Mu<span class="l9">da<span class="l9">nç<span class="l8">a d<span class="l9">e B<span class="l9">as<span class="l9">e <span class="w" style="width:26px"></span>. <span class="w7"></span>. <span class="w8"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w8"></span>. <span class="w" style="width:50px"></span>56</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:4708px">2.11</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:1264px;top:4707px;word-spacing:8px">Oitava lista de exercícios</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2203px;top:4708px;letter-spacing:10px">. . . . . . <span class="l6">. . . . . . . <span class="l6">. . . . . . . <span class="l6">. <span class="w" style="width:42px"></span>5<span class="l11">7</span></span></span></span></span><span class="a" style="left:2233px;top:5112px">1</span></div>
</div>
</div>
</div>
<div class="newpage" id="page2" style="width: 902px; height: 1167px; -webkit-transform: scale(0.9390243902439024); -webkit-transform-origin: 0% 0%; display: block;">
<div class="text_layer" style="z-index:2"><div class="ie_fix">
 
<div class="ff7" style="font-size:211px">
<span class="a" style="left:985px;top:1415px;letter-spacing:-2px">Sumário</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:1969px;word-spacing:139px;letter-spacing:40px">1 <span class="l" style="margin-left:-168px">M<span class="l" style="margin-left:-39px">A<span class="l" style="margin-left:-48px">T<span class="l" style="margin-left:-40px">R<span class="l" style="margin-left:-41px">I<span class="l" style="margin-left:-40px">Z<span class="l" style="margin-left:-40px">E<span class="l" style="margin-left:-41px">S <span class="l" style="margin-left:-180px">E <span class="l" style="margin-left:-208px">S<span class="l" style="margin-left:-40px">I<span class="l" style="margin-left:-40px">S<span class="l" style="margin-left:-40px">T<span class="l" style="margin-left:-40px">E<span class="l" style="margin-left:-41px">M<span class="l" style="margin-left:-40px">A<span class="l" style="margin-left:-40px">S <span class="w" style="width:1129px"></span>3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:2057px;word-spacing:4px;letter-spacing:7px">1<span class="l7">.<span class="l8">1 <span class="w" style="width:40px"></span>T<span class="l8">i<span class="l7">p<span class="l6">o<span class="l7">s <span class="l10">d<span class="l8">e <span class="l11">m<span class="l7">a<span class="l8">t<span class="l7">r<span class="l7">i<span class="l8">z<span class="l7">e<span class="l8">s <span class="l" style="margin-left:-13px">. . . . . <span class="w6"></span>. . . . . . <span class="w6"></span>. . . . . <span class="w6"></span>. . . . . . <span class="w6"></span>. . . . . <span class="w" style="width:85px"></span>3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2146px;word-spacing:3px;letter-spacing:7px">1<span class="l7">.<span class="l8">2 <span class="w" style="width:41px"></span>O<span class="l8">pe<span class="l12">r<span class="l8">a<span class="l7">ç<span class="l7">õ<span class="l8">e<span class="l7">s <span class="l9">c<span class="l8">o<span class="l7">m <span class="l11">m<span class="l7">a<span class="l8">t<span class="l7">r<span class="l7">i<span class="l8">z<span class="l7">e<span class="l8">s <span class="w12"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . <span class="w" style="width:84px"></span>6</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2234px;word-spacing:3px;letter-spacing:8px">1<span class="l8">.<span class="l9">3 <span class="w" style="width:39px"></span>M<span class="l9">a<span class="l8">t<span class="l8">r<span class="l9">i<span class="l8">z <span class="l12">e<span class="l8">s<span class="l8">c<span class="l9">a<span class="l8">l<span class="l9">o<span class="l8">n<span class="l8">a<span class="l9">d<span class="l8">a <span class="w" style="width:35px"></span>. . . . . . . . . . . . . . . . . . . . . . . . . . <span class="w" style="width:82px"></span>8</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2322px;word-spacing:3px;letter-spacing:7px">1<span class="l7">.<span class="l8">4 <span class="w" style="width:41px"></span>C<span class="l8">á<span class="l7">l<span class="l8">c<span class="l7">u<span class="l7">l<span class="l8">o <span class="l10">d<span class="l7">a <span class="l10">i<span class="l7">n<span class="l10">v<span class="l9">e<span class="l7">r<span class="l8">s<span class="l7">a <span class="w" style="width:29px"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . <span class="w" style="width:48px"></span>1<span class="l8">0</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2410px;word-spacing:5px;letter-spacing:7px">1<span class="l7">.<span class="l8">5 <span class="w" style="width:39px"></span>D<span class="l8">e<span class="l7">t<span class="l8">e<span class="l7">r<span class="l7">m<span class="l8">i<span class="l8">n<span class="l7">a<span class="l7">n<span class="l9">t<span class="l8">e<span class="l7">s <span class="w" style="width:36px"></span>. . . . . . . . . . . . . . . . . . . . . . . . . . . . <span class="w" style="width:46px"></span>1<span class="l8">1</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2498px">1.6</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:1264px;top:2497px;word-spacing:2px;letter-spacing:1px">Sexta lista de exercícios</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2203px;top:2498px;letter-spacing:10px">. . . . . . <span class="l6">. . . . . . . <span class="l6">. . . . . . . <span class="l6">. <span class="w" style="width:42px"></span>1<span class="l11">4</span></span></span></span></span><span class="a" style="left:1095px;top:2586px;word-spacing:5px;letter-spacing:5px">1.<span class="l11">7 <span class="w" style="width:43px"></span>Si<span class="l11">st<span class="l10">e<span class="l6">ma <span class="l" style="margin-left:-13px">de <span class="l" style="margin-left:-13px">e<span class="l6">qu<span class="l10">a<span class="l6">çõ<span class="l10">e<span class="l6">s <span class="l7">l<span class="l6">in<span class="l11">ea<span class="l10">re<span class="l11">s <span class="l10">. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. <span class="w" style="width:47px"></span>1<span class="l6">6</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:2674px;word-spacing:3px;letter-spacing:8px">1<span class="l8">.<span class="l9">7<span class="l8">.<span class="l9">1 <span class="w" style="width:48px"></span>I<span class="l8">n<span class="l11">t<span class="l8">r<span class="l8">o<span class="l7">d<span class="l8">u<span class="l8">ç<span class="l8">ã<span class="l9">o <span class="w" style="width:33px"></span>. . . . . . . . . . . . . . . . . . . . . . . . . . <span class="w" style="width:46px"></span>1<span class="l9">6</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:2762px;word-spacing:5px;letter-spacing:6px">1<span class="l6">.<span class="l7">7<span class="l6">.<span class="l7">2 <span class="w" style="width:50px"></span>S<span class="l6">i<span class="l7">s<span class="l6">t<span class="l6">e<span class="l7">m<span class="l6">a<span class="l7">s <span class="l9">e <span class="l10">m<span class="l7">a<span class="l6">t<span class="l7">r<span class="l6">i<span class="l7">z<span class="l6">e<span class="l7">s<span class="l6">. <span class="w" style="width:20px"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w" style="width:50px"></span>1<span class="l7">7</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:2850px;word-spacing:8px;letter-spacing:3px">1.<span class="l7">7.<span class="l7">3 <span class="w" style="width:53px"></span>So<span class="l6">lu<span class="l7">çã<span class="l7">o <span class="l7">de <span class="l10">um <span class="l11">si<span class="l6">st<span class="l7">em<span class="l7">a <span class="l7">por <span class="l11">ma<span class="l7">tr<span class="l7">iz <span class="l10">in<span class="l9">v<span class="l6">er<span class="l6">sa <span class="w11"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. <span class="w6"></span>. . <span class="w" style="width:53px"></span>20</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2960px">1.8</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:1264px;top:2960px;word-spacing:2px;letter-spacing:1px">Sétima lista de exercícios</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2260px;top:2960px;letter-spacing:10px">. . . . . . <span class="l6">. . . . . . . <span class="l6">. . . . . . . <span class="w" style="width:37px"></span>2<span class="l11">1</span></span></span></span><span class="a" style="left:1095px;top:3049px;word-spacing:4px;letter-spacing:8px">1<span class="l8">.<span class="l9">9 <span class="w" style="width:38px"></span>A<span class="l9">p<span class="l6">ê<span class="l8">n<span class="l9">d<span class="l8">i<span class="l8">c<span class="l9">e <span class="w" style="width:33px"></span>. . . . . . <span class="l6">. . . . . . . <span class="l6">. . . . . . <span class="l6">. . . . . . . <span class="l6">. . . . . <span class="w" style="width:39px"></span>2<span class="l9">4</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:3137px;word-spacing:5px;letter-spacing:5px">1.<span class="l11">9.<span class="l11">1 <span class="w" style="width:52px"></span>C<span class="l6">ál<span class="l11">cu<span class="l10">l<span class="l6">o <span class="l8">da <span class="l" style="margin-left:-13px">i<span class="l6">n<span class="l7">v<span class="l7">er<span class="l11">sa <span class="l" style="margin-left:-13px">po<span class="l8">r <span class="l8">ad<span class="l10">j<span class="l6">un<span class="l12">t<span class="l6">a <span class="w" style="width:22px"></span>. . <span class="w8"></span>. . <span class="w9"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. <span class="w" style="width:47px"></span>2<span class="l6">4</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:3225px;word-spacing:3px;letter-spacing:7px">1<span class="l7">.<span class="l8">9<span class="l7">.<span class="l8">2 <span class="w" style="width:50px"></span>R<span class="l8">e<span class="l7">g<span class="l8">r<span class="l7">a <span class="l10">d<span class="l7">e <span class="l10">C<span class="l8">r<span class="l7">a<span class="l7">m<span class="l8">e<span class="l8">r . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. <span class="w" style="width:45px"></span>2<span class="l8">5</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:3386px;word-spacing:205px;letter-spacing:41px">2 <span class="l" style="margin-left:-236px">E<span class="l" style="margin-left:-41px">S<span class="l" style="margin-left:-41px">P<span class="l" style="margin-left:-49px">A<span class="l" style="margin-left:-41px">Ç<span class="l" style="margin-left:-41px">O<span class="l" style="margin-left:-41px">S <span class="l" style="margin-left:-276px">V<span class="l" style="margin-left:-42px">E<span class="l" style="margin-left:-41px">T<span class="l" style="margin-left:-41px">O<span class="l" style="margin-left:-42px">R<span class="l" style="margin-left:-42px">I<span class="l" style="margin-left:-41px">A<span class="l" style="margin-left:-41px">I<span class="l" style="margin-left:-41px">S <span class="w" style="width:1139px"></span>2<span class="l" style="margin-left:-41px">9</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:3474px;word-spacing:4px;letter-spacing:7px">2<span class="l7">.<span class="l8">1 <span class="w" style="width:40px"></span>I<span class="l7">n<span class="l10">t<span class="l7">r<span class="l7">o<span class="l6">d<span class="l7">u<span class="l7">ç<span class="l7">ã<span class="l8">o <span class="l" style="margin-left:-16px">. . . . . . <span class="w6"></span>. . . . . <span class="w6"></span>. . . . . . <span class="w6"></span>. . . . . <span class="w6"></span>. . . . . . <span class="w6"></span>. . . <span class="w" style="width:47px"></span>2<span class="l8">9</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:3562px;word-spacing:3px;letter-spacing:8px">2<span class="l8">.<span class="l9">2 <span class="w" style="width:39px"></span>S<span class="l8">u<span class="l8">b<span class="l6">e<span class="l9">s<span class="l8">p<span class="l8">a<span class="l8">ç<span class="l9">o<span class="l8">s <span class="w" style="width:27px"></span>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . <span class="w" style="width:46px"></span>3<span class="l9">6</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:3651px;word-spacing:5px;letter-spacing:4px">2.<span class="l9">3 <span class="w" style="width:45px"></span>In<span class="l11">te<span class="l8">rs<span class="l9">ec<span class="l8">çã<span class="l9">o <span class="l6">de <span class="l10">do<span class="l9">is <span class="l10">Su<span class="l8">be<span class="l7">sp<span class="l8">aç<span class="l9">os <span class="l10">V<span class="l11">et<span class="l9">or<span class="l8">ia<span class="l9">is <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w" style="width:49px"></span>38</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:3739px;word-spacing:5px;letter-spacing:6px">2<span class="l6">.<span class="l7">4 <span class="w" style="width:41px"></span>C<span class="l7">o<span class="l6">m<span class="l9">b<span class="l6">i<span class="l7">n<span class="l6">a<span class="l6">ç<span class="l7">ã<span class="l6">o <span class="l10">L<span class="l6">i<span class="l7">n<span class="l6">e<span class="l6">a<span class="l6">r . . . . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w" style="width:50px"></span>4<span class="l7">0</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:3827px;word-spacing:7px;letter-spacing:4px">2.<span class="l9">5 <span class="w" style="width:43px"></span>De<span class="l9">pe<span class="l7">nd<span class="l8">ên<span class="l9">ci<span class="l9">a <span class="l8">e <span class="l7">In<span class="l9">de<span class="l9">pe<span class="l6">nd<span class="l9">ên<span class="l8">ci<span class="l9">a <span class="l8">Li<span class="l9">ne<span class="l9">ar <span class="w7"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w" style="width:52px"></span>41</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:3915px;word-spacing:3px;letter-spacing:7px">2<span class="l7">.<span class="l8">6 <span class="w" style="width:41px"></span>S<span class="l7">u<span class="l7">be<span class="l" style="margin-left:-13px">s<span class="l7">p<span class="l7">a<span class="l7">ç<span class="l8">o<span class="l7">s <span class="l10">G<span class="l7">e<span class="l7">r<span class="l8">a<span class="l7">d<span class="l7">o<span class="l8">s <span class="w" style="width:28px"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. <span class="w" style="width:45px"></span>4<span class="l8">3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:4003px;word-spacing:3px;letter-spacing:7px">2<span class="l7">.<span class="l8">7 <span class="w" style="width:41px"></span>S<span class="l7">o<span class="l7">m<span class="l8">a <span class="l10">d<span class="l7">e <span class="l10">S<span class="l7">u<span class="l8">be<span class="l12">s<span class="l7">p<span class="l7">a<span class="l8">ç<span class="l7">o<span class="l7">s <span class="w" style="width:16px"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. <span class="w" style="width:45px"></span>4<span class="l8">5</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:4091px;word-spacing:5px;letter-spacing:5px">2.<span class="l11">8 <span class="w" style="width:43px"></span>Ba<span class="l10">se <span class="w11"></span>e <span class="l8">Di<span class="l11">m<span class="l6">en<span class="l11">sã<span class="l10">o <span class="l8">de <span class="l" style="margin-left:-13px">um <span class="l" style="margin-left:-13px">Es<span class="l10">p<span class="l6">aç<span class="l10">o <span class="l8">V<span class="l12">e<span class="l6">to<span class="l10">r<span class="l6">ia<span class="l11">l <span class="w" style="width:41px"></span>. . <span class="w9"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. <span class="w" style="width:47px"></span>4<span class="l6">6</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:4179px;word-spacing:1px;letter-spacing:9px">2<span class="l9">.<span class="l10">8<span class="l9">.<span class="l10">1 <span class="w" style="width:48px"></span>B<span class="l9">a<span class="l9">s<span class="l9">e . . . . . . . . . . . . . . . . . . . . . . . . . . . . <span class="w6"></span>. . <span class="w" style="width:43px"></span>4<span class="l10">6</span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:4267px;word-spacing:3px;letter-spacing:8px">2<span class="l8">.<span class="l9">8<span class="l8">.<span class="l9">2 <span class="w" style="width:48px"></span>D<span class="l9">i<span class="l9">m<span class="l8">e<span class="l9">n<span class="l8">s<span class="l8">ã<span class="l8">o <span class="w11"></span>. . . . . . . . . . . . . . . . . . . . . . . . . . . <span class="w" style="width:46px"></span>4<span class="l9">9</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:4355px;word-spacing:7px;letter-spacing:3px">2.<span class="l7">8.<span class="l7">3 <span class="w" style="width:54px"></span>Di<span class="l8">me<span class="l7">ns<span class="l6">ão <span class="l9">da <span class="l9">So<span class="l7">ma <span class="l9">de <span class="l9">Su<span class="l7">bes<span class="l8">pa<span class="l6">ço<span class="l7">s V<span class="l" style="margin-left:-15px">et<span class="l7">or<span class="l7">ia<span class="l7">is <span class="w7"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w6"></span>. <span class="w" style="width:49px"></span>50</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:4443px;word-spacing:5px;letter-spacing:7px">2<span class="l7">.<span class="l8">8<span class="l7">.<span class="l8">4 <span class="w" style="width:48px"></span>C<span class="l8">oo<span class="l12">r<span class="l8">d<span class="l7">e<span class="l7">n<span class="l8">a<span class="l7">d<span class="l7">a<span class="l7">s <span class="w" style="width:24px"></span>. . . . . . . . . . . . . . . . . . . . . . . . . <span class="w" style="width:46px"></span>5<span class="l8">0</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:4532px;word-spacing:3px;letter-spacing:7px">2<span class="l7">.<span class="l8">9 <span class="w" style="width:41px"></span>M<span class="l8">u<span class="l7">d<span class="l7">a<span class="l7">n<span class="l7">ç<span class="l8">a <span class="l10">d<span class="l7">e <span class="l10">B<span class="l7">a<span class="l7">s<span class="l7">e <span class="l7">. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w" style="width:50px"></span>5<span class="l8">1</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:4620px;word-spacing:4px;letter-spacing:4px">2.<span class="l9">10 A In<span class="l11">v<span class="l6">er<span class="l9">sa <span class="l9">da <span class="l9">Ma<span class="l9">tr<span class="l9">iz <span class="l9">de <span class="l9">Mu<span class="l9">da<span class="l9">nç<span class="l8">a d<span class="l9">e B<span class="l9">as<span class="l9">e <span class="w" style="width:26px"></span>. <span class="w7"></span>. <span class="w8"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w8"></span>. <span class="w" style="width:50px"></span>56</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:4708px">2.11</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:1264px;top:4707px;word-spacing:8px">Oitava lista de exercícios</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2203px;top:4708px;letter-spacing:10px">. . . . . . <span class="l6">. . . . . . . <span class="l6">. . . . . . . <span class="l6">. <span class="w" style="width:42px"></span>5<span class="l11">7</span></span></span></span></span><span class="a" style="left:2233px;top:5112px">1</span></div>
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<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:926px;word-spacing:157px;letter-spacing:22px">3 <span class="l" style="margin-left:-150px">T<span class="l" style="margin-left:-22px">R<span class="l" style="margin-left:-23px">A<span class="l" style="margin-left:-22px">N<span class="l" style="margin-left:-22px">S<span class="l" style="margin-left:-22px">F<span class="l" style="margin-left:-24px">O<span class="l" style="margin-left:-23px">R<span class="l" style="margin-left:-22px">M<span class="l" style="margin-left:-22px">A<span class="l" style="margin-left:-22px">Ç<span class="l" style="margin-left:-22px">Õ<span class="l" style="margin-left:-23px">E<span class="l" style="margin-left:-23px">S <span class="l" style="margin-left:-190px">L<span class="l" style="margin-left:-22px">I<span class="l" style="margin-left:-22px">N<span class="l" style="margin-left:-22px">E<span class="l" style="margin-left:-22px">A<span class="l" style="margin-left:-22px">R<span class="l" style="margin-left:-23px">E<span class="l" style="margin-left:-23px">S <span class="w" style="width:812px"></span>6<span class="l" style="margin-left:-22px">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:1015px;word-spacing:8px;letter-spacing:3px">3.<span class="l7">1 <span class="w" style="width:44px"></span>Pr<span class="l6">op<span class="l7">ri<span class="l6">ed<span class="l7">ad<span class="l7">es <span class="l10">da<span class="l6">s <span class="l7">T<span class="l10">ra<span class="l6">ns<span class="l7">fo<span class="l6">rm<span class="l8">aç<span class="l6">õe<span class="l7">s <span class="l7">Li<span class="l6">ne<span class="l7">ar<span class="l6">es . . <span class="w8"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w11"></span>. <span class="w" style="width:48px"></span>66</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:1103px;word-spacing:6px;letter-spacing:4px">3.<span class="l9">2 <span class="w" style="width:44px"></span>T<span class="l11">ra<span class="l8">ns<span class="l9">fo<span class="l9">rm<span class="l9">aç<span class="l8">õe<span class="l9">s <span class="l7">Li<span class="l8">ne<span class="l9">ar<span class="l9">es <span class="l10">e <span class="l7">Ma<span class="l9">tr<span class="l9">iz<span class="l9">es <span class="l" style="margin-left:-16px">. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w11"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w11"></span>. <span class="w" style="width:48px"></span>72</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:1191px;word-spacing:6px;letter-spacing:3px">3.<span class="l7">2.<span class="l7">1 <span class="w" style="width:55px"></span>T<span class="l10">ra<span class="l7">ns<span class="l6">fo<span class="l7">rm<span class="l7">aç<span class="l7">ão <span class="l8">li<span class="l7">ne<span class="l6">ar <span class="l9">as<span class="l6">soc<span class="l8">ia<span class="l6">da <span class="l9">a u<span class="l8">ma <span class="l8">ma<span class="l7">tr<span class="l7">iz <span class="w10"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w8"></span>. <span class="w7"></span>. <span class="w" style="width:50px"></span>72</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:1279px;word-spacing:7px;letter-spacing:4px">3.<span class="l9">2.<span class="l9">2 <span class="w" style="width:52px"></span>Ma<span class="l9">tr<span class="l9">iz <span class="l12">de <span class="l12">um<span class="l9">a <span class="l8">tr<span class="l9">an<span class="l8">sf<span class="l9">or<span class="l9">ma<span class="l9">çã<span class="l8">o <span class="l8">li<span class="l9">ne<span class="l9">ar <span class="w" style="width:22px"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. . <span class="w" style="width:51px"></span>74</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:1367px;word-spacing:8px;letter-spacing:4px">3.<span class="l9">3 <span class="w" style="width:67px"></span>Co<span class="l10">mp<span class="l7">os<span class="l8">iç<span class="l9">ão <span class="l" style="margin-left:-13px">de <span class="l" style="margin-left:-13px">tr<span class="l9">an<span class="l9">sf<span class="l8">or<span class="l9">ma<span class="l9">çõ<span class="l8">es <span class="l" style="margin-left:-13px">li<span class="l10">ne<span class="l8">ar<span class="l9">es <span class="w" style="width:22px"></span>. . <span class="w6"></span>. . <span class="w7"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w" style="width:50px"></span>79</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:1455px;word-spacing:5px;letter-spacing:4px">3.<span class="l9">4 <span class="w" style="width:45px"></span>A <span class="l6">In<span class="l11">v<span class="l6">er<span class="l9">sa <span class="l10">de <span class="l10">um<span class="l9">a <span class="l6">tr<span class="l9">an<span class="l8">sf<span class="l9">or<span class="l8">ma<span class="l9">çã<span class="l9">o <span class="l6">li<span class="l9">ne<span class="l9">ar <span class="l7">. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w" style="width:49px"></span>79</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:1544px;word-spacing:5px;letter-spacing:6px">3<span class="l6">.<span class="l7">5 <span class="w" style="width:41px"></span>N<span class="l7">o<span class="l6">n<span class="l6">a <span class="l10">l<span class="l7">i<span class="l6">s<span class="l6">t<span class="l7">a <span class="l10">d<span class="l6">e <span class="l10">e<span class="l6">x<span class="l6">e<span class="l7">r<span class="l6">c<span class="l6">í<span class="l7">c<span class="l6">i<span class="l7">o<span class="l6">s <span class="w" style="width:34px"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . <span class="w" style="width:48px"></span>8<span class="l7">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:1704px;word-spacing:186px;letter-spacing:33px">4 <span class="l" style="margin-left:-201px">O<span class="l" style="margin-left:-33px">P<span class="l" style="margin-left:-34px">E<span class="l" style="margin-left:-33px">R<span class="l" style="margin-left:-34px">A<span class="l" style="margin-left:-33px">D<span class="l" style="margin-left:-33px">O<span class="l" style="margin-left:-34px">R<span class="l" style="margin-left:-33px">E<span class="l" style="margin-left:-34px">S <span class="l" style="margin-left:-241px">L<span class="l" style="margin-left:-33px">I<span class="l" style="margin-left:-33px">N<span class="l" style="margin-left:-33px">E<span class="l" style="margin-left:-34px">A<span class="l" style="margin-left:-33px">R<span class="l" style="margin-left:-34px">E<span class="l" style="margin-left:-33px">S <span class="w" style="width:1010px"></span>8<span class="l" style="margin-left:-33px">7</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:1881px;word-spacing:9px;letter-spacing:3px">4.<span class="l7">1 <span class="w" style="width:43px"></span>T<span class="l10">ra<span class="l6">ns<span class="l7">fo<span class="l7">rm<span class="l7">aç<span class="l6">õe<span class="l7">s <span class="l8">es<span class="l6">pec<span class="l8">ia<span class="l7">is <span class="l11">no <span class="l11">pl<span class="l7">an<span class="l7">o <span class="l7">e <span class="l8">no <span class="l11">es<span class="l7">pa<span class="l6">ço <span class="w" style="width:38px"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w" style="width:52px"></span>87</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:1969px;word-spacing:8px;letter-spacing:3px">4.<span class="l7">2 <span class="w" style="width:44px"></span>Pr<span class="l6">op<span class="l7">ri<span class="l6">ed<span class="l7">ad<span class="l7">es <span class="l10">do<span class="l6">s <span class="l7">ope<span class="l8">ra<span class="l6">do<span class="l7">re<span class="l6">s <span class="l7">in<span class="l9">ve<span class="l8">rs<span class="l7">ív<span class="l9">ei<span class="l7">s <span class="w" style="width:39px"></span>. . <span class="w10"></span>. <span class="w6"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w11"></span>. <span class="w11"></span>10<span class="l7">9</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:2057px;word-spacing:6px;letter-spacing:5px">4.<span class="l11">2.<span class="l11">1 <span class="w" style="width:51px"></span>M<span class="l6">at<span class="l10">r<span class="l6">iz<span class="l11">es <span class="l" style="margin-left:-14px">Se<span class="l10">m<span class="l6">el<span class="l11">ha<span class="l11">n<span class="l7">te<span class="l11">s <span class="w" style="width:34px"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w7"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w7"></span>. . <span class="w12"></span>11<span class="l11">1</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2146px;word-spacing:5px;letter-spacing:4px">4.<span class="l9">3 <span class="w" style="width:45px"></span>Op<span class="l7">er<span class="l9">ad<span class="l8">or<span class="l9">es <span class="l10">au<span class="l8">to<span class="l9">ad<span class="l8">ju<span class="l9">n<span class="l6">to<span class="l9">s <span class="l6">e <span class="l6">or<span class="l8">to<span class="l9">go<span class="l9">na<span class="l8">is <span class="w" style="width:27px"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w12"></span>11<span class="l9">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2234px;word-spacing:5px;letter-spacing:5px">4.<span class="l11">4 <span class="w" style="width:43px"></span>D<span class="l6">éc<span class="l10">i<span class="l6">m<span class="l6">a <span class="l8">li<span class="l11">st<span class="l11">a <span class="l7">d<span class="l6">e <span class="l7">e<span class="l6">xe<span class="l11">rc<span class="l10">í<span class="l6">ci<span class="l11">os <span class="w" style="width:19px"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w" style="width:15px"></span>11<span class="l11">3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2410px;word-spacing:4px;letter-spacing:5px">4.<span class="l11">5 <span class="w" style="width:44px"></span>A<span class="l6">ut<span class="l10">o<span class="l8">v<span class="l9">al<span class="l11">or<span class="l11">es <span class="l12">e <span class="l6">A<span class="l6">u<span class="l6">to<span class="l12">v<span class="l8">et<span class="l11">or<span class="l10">es . <span class="w7"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w11"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w11"></span>. <span class="w11"></span>11<span class="l11">5</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:2498px;word-spacing:7px;letter-spacing:3px">4.<span class="l7">5.<span class="l7">1 <span class="w" style="width:54px"></span>Au<span class="l7">to<span class="l9">v<span class="l7">al<span class="l7">or<span class="l7">es <span class="l9">e <span class="l6">au<span class="l6">to<span class="l9">ve<span class="l9">to<span class="l6">re<span class="l7">s <span class="l6">de <span class="l9">um<span class="l7">a <span class="l6">ma<span class="l7">tr<span class="l6">iz <span class="w" style="width:36px"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w6"></span>. <span class="w12"></span>11<span class="l7">6</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:2586px;word-spacing:8px;letter-spacing:4px">4.<span class="l9">5.<span class="l9">2 <span class="w" style="width:51px"></span>P<span class="l6">ol<span class="l9">in<span class="l9">ôm<span class="l9">io <span class="l" style="margin-left:-13px">Ca<span class="l10">ra<span class="l8">ct<span class="l9">er<span class="l9">ís<span class="l8">ti<span class="l9">co<span class="l9">. <span class="w" style="width:30px"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w7"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w7"></span>. . <span class="w12"></span>11<span class="l9">7</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2674px;word-spacing:5px;letter-spacing:4px">4.<span class="l9">6 <span class="w" style="width:45px"></span>Dé<span class="l9">ci<span class="l9">ma <span class="l11">pr<span class="l8">im<span class="l10">ei<span class="l9">ra <span class="l10">li<span class="l9">st<span class="l9">a d<span class="l10">e e<span class="l10">xe<span class="l8">rc<span class="l9">íc<span class="l9">io<span class="l9">s <span class="w" style="width:17px"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w12"></span>12<span class="l9">3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:2835px;word-spacing:376px;letter-spacing:68px">5 <span class="l" style="margin-left:-461px">A<span class="l" style="margin-left:-68px">P<span class="l" style="margin-left:-68px">L<span class="l" style="margin-left:-68px">I<span class="l" style="margin-left:-68px">C<span class="l" style="margin-left:-68px">A<span class="l" style="margin-left:-69px">Ç<span class="l" style="margin-left:-68px">Õ<span class="l" style="margin-left:-68px">E<span class="l" style="margin-left:-69px">S <span class="w" style="width:1212px"></span>1<span class="l" style="margin-left:-68px">2<span class="l" style="margin-left:-68px">6</span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:3012px;word-spacing:5px;letter-spacing:2px">5.1 <span class="w" style="width:44px"></span>Apl<span class="l8">ica<span class="l7">çõe<span class="l7">s da <span class="l6">Álg<span class="l7">ebr<span class="l7">a Li<span class="l7">nea<span class="l7">r na <span class="l6">Eng<span class="l6">enh<span class="l7">ari<span class="l7">a Ca<span class="l7">rto<span class="l7">grá<span class="l7">…ca <span class="w" style="width:23px"></span>. <span class="w10"></span>. <span class="w10"></span>. <span class="w11"></span>. <span class="w" style="width:16px"></span>126</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:3100px;word-spacing:7px;letter-spacing:2px">5.2 <span class="w" style="width:42px"></span>Apl<span class="l8">ica<span class="l7">çõe<span class="l7">s d<span class="l6">e e<span class="l6">spa<span class="l7">ços <span class="l8">ve<span class="l7">tor<span class="l7">iai<span class="l7">s n<span class="l6">a c<span class="l6">omp<span class="l7">uta<span class="l7">ção <span class="l9">grá<span class="l7">…ca <span class="w" style="width:25px"></span>. <span class="w8"></span>. <span class="w8"></span>. <span class="w8"></span>. <span class="w8"></span>. <span class="w9"></span>. <span class="w" style="width:14px"></span>127</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:3188px;word-spacing:10px;letter-spacing:1px">5.3 <span class="w" style="width:43px"></span>Apli<span class="l6">caçõ<span class="l6">es <span class="l6">de <span class="l6">auto<span class="l7">va<span class="l7">lores <span class="l10">e a<span class="l6">uto<span class="l6">vet<span class="l6">ores <span class="l9">na <span class="l6">engen<span class="l6">hari<span class="l6">a c<span class="l6">ivil <span class="w" style="width:36px"></span>. <span class="w7"></span>. <span class="w8"></span>. <span class="w" style="width:13px"></span>133</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:3276px;word-spacing:6px;letter-spacing:1px">5.3.<span class="l6">1 <span class="w" style="width:59px"></span>O Probl<span class="l6">ema de aut<span class="l6">ov<span class="l8">alor na a<span class="l6">v<span class="l6">alia<span class="l6">ção de model<span class="l6">os estru<span class="l6">-</span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1499px;top:3364px;word-spacing:5px;letter-spacing:5px">tu<span class="l11">ra<span class="l10">i<span class="l6">s <span class="l7">d<span class="l6">e <span class="l7">e<span class="l6">di<span class="l11">…c<span class="l10">a<span class="l6">çõ<span class="l10">e<span class="l6">s <span class="w" style="width:31px"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. <span class="w10"></span>13<span class="l11">3</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:2233px;top:5112px">2</span></div>
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<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:926px;word-spacing:157px;letter-spacing:22px">3 <span class="l" style="margin-left:-150px">T<span class="l" style="margin-left:-22px">R<span class="l" style="margin-left:-23px">A<span class="l" style="margin-left:-22px">N<span class="l" style="margin-left:-22px">S<span class="l" style="margin-left:-22px">F<span class="l" style="margin-left:-24px">O<span class="l" style="margin-left:-23px">R<span class="l" style="margin-left:-22px">M<span class="l" style="margin-left:-22px">A<span class="l" style="margin-left:-22px">Ç<span class="l" style="margin-left:-22px">Õ<span class="l" style="margin-left:-23px">E<span class="l" style="margin-left:-23px">S <span class="l" style="margin-left:-190px">L<span class="l" style="margin-left:-22px">I<span class="l" style="margin-left:-22px">N<span class="l" style="margin-left:-22px">E<span class="l" style="margin-left:-22px">A<span class="l" style="margin-left:-22px">R<span class="l" style="margin-left:-23px">E<span class="l" style="margin-left:-23px">S <span class="w" style="width:812px"></span>6<span class="l" style="margin-left:-22px">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:1015px;word-spacing:8px;letter-spacing:3px">3.<span class="l7">1 <span class="w" style="width:44px"></span>Pr<span class="l6">op<span class="l7">ri<span class="l6">ed<span class="l7">ad<span class="l7">es <span class="l10">da<span class="l6">s <span class="l7">T<span class="l10">ra<span class="l6">ns<span class="l7">fo<span class="l6">rm<span class="l8">aç<span class="l6">õe<span class="l7">s <span class="l7">Li<span class="l6">ne<span class="l7">ar<span class="l6">es . . <span class="w8"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w11"></span>. <span class="w" style="width:48px"></span>66</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:1103px;word-spacing:6px;letter-spacing:4px">3.<span class="l9">2 <span class="w" style="width:44px"></span>T<span class="l11">ra<span class="l8">ns<span class="l9">fo<span class="l9">rm<span class="l9">aç<span class="l8">õe<span class="l9">s <span class="l7">Li<span class="l8">ne<span class="l9">ar<span class="l9">es <span class="l10">e <span class="l7">Ma<span class="l9">tr<span class="l9">iz<span class="l9">es <span class="l" style="margin-left:-16px">. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w11"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w11"></span>. <span class="w" style="width:48px"></span>72</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:1191px;word-spacing:6px;letter-spacing:3px">3.<span class="l7">2.<span class="l7">1 <span class="w" style="width:55px"></span>T<span class="l10">ra<span class="l7">ns<span class="l6">fo<span class="l7">rm<span class="l7">aç<span class="l7">ão <span class="l8">li<span class="l7">ne<span class="l6">ar <span class="l9">as<span class="l6">soc<span class="l8">ia<span class="l6">da <span class="l9">a u<span class="l8">ma <span class="l8">ma<span class="l7">tr<span class="l7">iz <span class="w10"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w7"></span>. <span class="w8"></span>. <span class="w7"></span>. <span class="w" style="width:50px"></span>72</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:1279px;word-spacing:7px;letter-spacing:4px">3.<span class="l9">2.<span class="l9">2 <span class="w" style="width:52px"></span>Ma<span class="l9">tr<span class="l9">iz <span class="l12">de <span class="l12">um<span class="l9">a <span class="l8">tr<span class="l9">an<span class="l8">sf<span class="l9">or<span class="l9">ma<span class="l9">çã<span class="l8">o <span class="l8">li<span class="l9">ne<span class="l9">ar <span class="w" style="width:22px"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. . <span class="w" style="width:51px"></span>74</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:1367px;word-spacing:8px;letter-spacing:4px">3.<span class="l9">3 <span class="w" style="width:67px"></span>Co<span class="l10">mp<span class="l7">os<span class="l8">iç<span class="l9">ão <span class="l" style="margin-left:-13px">de <span class="l" style="margin-left:-13px">tr<span class="l9">an<span class="l9">sf<span class="l8">or<span class="l9">ma<span class="l9">çõ<span class="l8">es <span class="l" style="margin-left:-13px">li<span class="l10">ne<span class="l8">ar<span class="l9">es <span class="w" style="width:22px"></span>. . <span class="w6"></span>. . <span class="w7"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w" style="width:50px"></span>79</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:1455px;word-spacing:5px;letter-spacing:4px">3.<span class="l9">4 <span class="w" style="width:45px"></span>A <span class="l6">In<span class="l11">v<span class="l6">er<span class="l9">sa <span class="l10">de <span class="l10">um<span class="l9">a <span class="l6">tr<span class="l9">an<span class="l8">sf<span class="l9">or<span class="l8">ma<span class="l9">çã<span class="l9">o <span class="l6">li<span class="l9">ne<span class="l9">ar <span class="l7">. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w" style="width:49px"></span>79</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:1544px;word-spacing:5px;letter-spacing:6px">3<span class="l6">.<span class="l7">5 <span class="w" style="width:41px"></span>N<span class="l7">o<span class="l6">n<span class="l6">a <span class="l10">l<span class="l7">i<span class="l6">s<span class="l6">t<span class="l7">a <span class="l10">d<span class="l6">e <span class="l10">e<span class="l6">x<span class="l6">e<span class="l7">r<span class="l6">c<span class="l6">í<span class="l7">c<span class="l6">i<span class="l7">o<span class="l6">s <span class="w" style="width:34px"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w7"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . . <span class="w6"></span>. . <span class="w" style="width:48px"></span>8<span class="l7">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
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<span class="a" style="left:985px;top:1704px;word-spacing:186px;letter-spacing:33px">4 <span class="l" style="margin-left:-201px">O<span class="l" style="margin-left:-33px">P<span class="l" style="margin-left:-34px">E<span class="l" style="margin-left:-33px">R<span class="l" style="margin-left:-34px">A<span class="l" style="margin-left:-33px">D<span class="l" style="margin-left:-33px">O<span class="l" style="margin-left:-34px">R<span class="l" style="margin-left:-33px">E<span class="l" style="margin-left:-34px">S <span class="l" style="margin-left:-241px">L<span class="l" style="margin-left:-33px">I<span class="l" style="margin-left:-33px">N<span class="l" style="margin-left:-33px">E<span class="l" style="margin-left:-34px">A<span class="l" style="margin-left:-33px">R<span class="l" style="margin-left:-34px">E<span class="l" style="margin-left:-33px">S <span class="w" style="width:1010px"></span>8<span class="l" style="margin-left:-33px">7</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:1881px;word-spacing:9px;letter-spacing:3px">4.<span class="l7">1 <span class="w" style="width:43px"></span>T<span class="l10">ra<span class="l6">ns<span class="l7">fo<span class="l7">rm<span class="l7">aç<span class="l6">õe<span class="l7">s <span class="l8">es<span class="l6">pec<span class="l8">ia<span class="l7">is <span class="l11">no <span class="l11">pl<span class="l7">an<span class="l7">o <span class="l7">e <span class="l8">no <span class="l11">es<span class="l7">pa<span class="l6">ço <span class="w" style="width:38px"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w" style="width:52px"></span>87</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:1969px;word-spacing:8px;letter-spacing:3px">4.<span class="l7">2 <span class="w" style="width:44px"></span>Pr<span class="l6">op<span class="l7">ri<span class="l6">ed<span class="l7">ad<span class="l7">es <span class="l10">do<span class="l6">s <span class="l7">ope<span class="l8">ra<span class="l6">do<span class="l7">re<span class="l6">s <span class="l7">in<span class="l9">ve<span class="l8">rs<span class="l7">ív<span class="l9">ei<span class="l7">s <span class="w" style="width:39px"></span>. . <span class="w10"></span>. <span class="w6"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w11"></span>. <span class="w11"></span>10<span class="l7">9</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:2057px;word-spacing:6px;letter-spacing:5px">4.<span class="l11">2.<span class="l11">1 <span class="w" style="width:51px"></span>M<span class="l6">at<span class="l10">r<span class="l6">iz<span class="l11">es <span class="l" style="margin-left:-14px">Se<span class="l10">m<span class="l6">el<span class="l11">ha<span class="l11">n<span class="l7">te<span class="l11">s <span class="w" style="width:34px"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w7"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w7"></span>. . <span class="w12"></span>11<span class="l11">1</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2146px;word-spacing:5px;letter-spacing:4px">4.<span class="l9">3 <span class="w" style="width:45px"></span>Op<span class="l7">er<span class="l9">ad<span class="l8">or<span class="l9">es <span class="l10">au<span class="l8">to<span class="l9">ad<span class="l8">ju<span class="l9">n<span class="l6">to<span class="l9">s <span class="l6">e <span class="l6">or<span class="l8">to<span class="l9">go<span class="l9">na<span class="l8">is <span class="w" style="width:27px"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w12"></span>11<span class="l9">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2234px;word-spacing:5px;letter-spacing:5px">4.<span class="l11">4 <span class="w" style="width:43px"></span>D<span class="l6">éc<span class="l10">i<span class="l6">m<span class="l6">a <span class="l8">li<span class="l11">st<span class="l11">a <span class="l7">d<span class="l6">e <span class="l7">e<span class="l6">xe<span class="l11">rc<span class="l10">í<span class="l6">ci<span class="l11">os <span class="w" style="width:19px"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w" style="width:15px"></span>11<span class="l11">3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2410px;word-spacing:4px;letter-spacing:5px">4.<span class="l11">5 <span class="w" style="width:44px"></span>A<span class="l6">ut<span class="l10">o<span class="l8">v<span class="l9">al<span class="l11">or<span class="l11">es <span class="l12">e <span class="l6">A<span class="l6">u<span class="l6">to<span class="l12">v<span class="l8">et<span class="l11">or<span class="l10">es . <span class="w7"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w11"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w10"></span>. . <span class="w11"></span>. <span class="w11"></span>11<span class="l11">5</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:2498px;word-spacing:7px;letter-spacing:3px">4.<span class="l7">5.<span class="l7">1 <span class="w" style="width:54px"></span>Au<span class="l7">to<span class="l9">v<span class="l7">al<span class="l7">or<span class="l7">es <span class="l9">e <span class="l6">au<span class="l6">to<span class="l9">ve<span class="l9">to<span class="l6">re<span class="l7">s <span class="l6">de <span class="l9">um<span class="l7">a <span class="l6">ma<span class="l7">tr<span class="l6">iz <span class="w" style="width:36px"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w6"></span>. <span class="w12"></span>11<span class="l7">6</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:2586px;word-spacing:8px;letter-spacing:4px">4.<span class="l9">5.<span class="l9">2 <span class="w" style="width:51px"></span>P<span class="l6">ol<span class="l9">in<span class="l9">ôm<span class="l9">io <span class="l" style="margin-left:-13px">Ca<span class="l10">ra<span class="l8">ct<span class="l9">er<span class="l9">ís<span class="l8">ti<span class="l9">co<span class="l9">. <span class="w" style="width:30px"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w7"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w6"></span>. . <span class="w7"></span>. . <span class="w12"></span>11<span class="l9">7</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:2674px;word-spacing:5px;letter-spacing:4px">4.<span class="l9">6 <span class="w" style="width:45px"></span>Dé<span class="l9">ci<span class="l9">ma <span class="l11">pr<span class="l8">im<span class="l10">ei<span class="l9">ra <span class="l10">li<span class="l9">st<span class="l9">a d<span class="l10">e e<span class="l10">xe<span class="l8">rc<span class="l9">íc<span class="l9">io<span class="l9">s <span class="w" style="width:17px"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w6"></span>. <span class="w7"></span>. <span class="w12"></span>12<span class="l9">3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:2835px;word-spacing:376px;letter-spacing:68px">5 <span class="l" style="margin-left:-461px">A<span class="l" style="margin-left:-68px">P<span class="l" style="margin-left:-68px">L<span class="l" style="margin-left:-68px">I<span class="l" style="margin-left:-68px">C<span class="l" style="margin-left:-68px">A<span class="l" style="margin-left:-69px">Ç<span class="l" style="margin-left:-68px">Õ<span class="l" style="margin-left:-68px">E<span class="l" style="margin-left:-69px">S <span class="w" style="width:1212px"></span>1<span class="l" style="margin-left:-68px">2<span class="l" style="margin-left:-68px">6</span></span></span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:3012px;word-spacing:5px;letter-spacing:2px">5.1 <span class="w" style="width:44px"></span>Apl<span class="l8">ica<span class="l7">çõe<span class="l7">s da <span class="l6">Álg<span class="l7">ebr<span class="l7">a Li<span class="l7">nea<span class="l7">r na <span class="l6">Eng<span class="l6">enh<span class="l7">ari<span class="l7">a Ca<span class="l7">rto<span class="l7">grá<span class="l7">…ca <span class="w" style="width:23px"></span>. <span class="w10"></span>. <span class="w10"></span>. <span class="w11"></span>. <span class="w" style="width:16px"></span>126</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:3100px;word-spacing:7px;letter-spacing:2px">5.2 <span class="w" style="width:42px"></span>Apl<span class="l8">ica<span class="l7">çõe<span class="l7">s d<span class="l6">e e<span class="l6">spa<span class="l7">ços <span class="l8">ve<span class="l7">tor<span class="l7">iai<span class="l7">s n<span class="l6">a c<span class="l6">omp<span class="l7">uta<span class="l7">ção <span class="l9">grá<span class="l7">…ca <span class="w" style="width:25px"></span>. <span class="w8"></span>. <span class="w8"></span>. <span class="w8"></span>. <span class="w8"></span>. <span class="w9"></span>. <span class="w" style="width:14px"></span>127</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1095px;top:3188px;word-spacing:10px;letter-spacing:1px">5.3 <span class="w" style="width:43px"></span>Apli<span class="l6">caçõ<span class="l6">es <span class="l6">de <span class="l6">auto<span class="l7">va<span class="l7">lores <span class="l10">e a<span class="l6">uto<span class="l6">vet<span class="l6">ores <span class="l9">na <span class="l6">engen<span class="l6">hari<span class="l6">a c<span class="l6">ivil <span class="w" style="width:36px"></span>. <span class="w7"></span>. <span class="w8"></span>. <span class="w" style="width:13px"></span>133</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1264px;top:3276px;word-spacing:6px;letter-spacing:1px">5.3.<span class="l6">1 <span class="w" style="width:59px"></span>O Probl<span class="l6">ema de aut<span class="l6">ov<span class="l8">alor na a<span class="l6">v<span class="l6">alia<span class="l6">ção de model<span class="l6">os estru<span class="l6">-</span></span></span></span></span></span></span></span></span></span><span class="a" style="left:1499px;top:3364px;word-spacing:5px;letter-spacing:5px">tu<span class="l11">ra<span class="l10">i<span class="l6">s <span class="l7">d<span class="l6">e <span class="l7">e<span class="l6">di<span class="l11">…c<span class="l10">a<span class="l6">çõ<span class="l10">e<span class="l6">s <span class="w" style="width:31px"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w8"></span>. . <span class="w9"></span>. <span class="w10"></span>13<span class="l11">3</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:2233px;top:5112px">2</span></div>
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<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:927px">Resolução:</span><span class="a" style="left:1095px;top:1015px;word-spacing:3px">Precisamos encontrar números reais,</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2290px;top:1026px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2329px;top:1056px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2362px;top:1026px">;<span class="w11"></span>a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2433px;top:1056px">2</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2490px;top:1015px">e</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2547px;top:1026px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2586px;top:1056px">3</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2644px;top:1015px;word-spacing:5px">tais que:</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1169px"><span class="a" style="left:1734px"> p</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1169px"><span class="a" style="left:1786px">(</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1814px;top:1169px">x</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1856px;top:1169px">) =</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1982px;top:1169px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2021px;top:1199px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1169px"><span class="a" style="left:2054px">q </span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1169px"><span class="a" style="left:2090px">(</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2118px;top:1169px">x</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2160px;top:1169px">) +</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2278px;top:1169px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2317px;top:1199px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1169px"><span class="a" style="left:2350px">r</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1169px"><span class="a" style="left:2385px">(</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2414px;top:1169px">x</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2456px;top:1169px">) +</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2574px;top:1169px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2613px;top:1199px">3</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1169px"><span class="a" style="left:2646px">s</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2680px;top:1169px">(</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2709px;top:1169px">x</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2751px;top:1169px">)</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:985px;top:1302px;word-spacing:4px">Ou seja, precisamos encontrar</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1974px;top:1312px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2013px;top:1343px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2046px;top:1312px">;<span class="w11"></span>a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2117px;top:1343px">2</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2175px;top:1302px">e</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2232px;top:1312px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2270px;top:1343px">3</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2352px;top:1302px">satisfazendo:</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1611px;top:1456px">1 +</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1737px;top:1456px">x</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1779px;top:1445px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1886px;top:1456px">=</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2016px;top:1456px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2055px;top:1486px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2104px;top:1456px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2178px;top:1456px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2216px;top:1486px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1456px"><span class="a" style="left:2249px">(1 +</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2404px;top:1456px">x</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2446px;top:1456px">) +</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2565px;top:1456px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2603px;top:1486px">3</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1456px"><span class="a" style="left:2636px">(1 +</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2791px;top:1456px">x</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2850px;top:1456px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2923px;top:1456px">x</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2965px;top:1445px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2998px;top:1456px">)</span><span class="g" style="top:1566px"><span class="a" style="left:1328px">1</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1566px"><span class="a" style="left:1365px">:</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1566px"><span class="a" style="left:1385px">1 + 0</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1548px;top:1566px">x</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1607px;top:1566px">+ 1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1717px;top:1566px">:x</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1779px;top:1555px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1886px;top:1566px">= <span class="w" style="width:57px"></span>(</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2045px;top:1566px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2083px;top:1596px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2133px;top:1566px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2206px;top:1566px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2245px;top:1596px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2294px;top:1566px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2368px;top:1566px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2406px;top:1596px">3</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2439px;top:1566px">) <span class="w" style="width:38px"></span>+ (</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2623px;top:1566px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2662px;top:1596px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2711px;top:1566px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2784px;top:1566px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2823px;top:1596px">3</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1566px"><span class="a" style="left:2856px">)</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2897px;top:1566px">x</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2955px;top:1566px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3029px;top:1566px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:3067px;top:1596px">3</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3100px;top:1566px">x</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:3142px;top:1555px">3</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:985px;top:1699px;word-spacing:3px">que é equivalente ao sistema:</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:1440px;top:1838px">8</span><span class="a" style="left:1440px;top:1904px"><</span><span class="a" style="left:1440px;top:2036px">:</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1542px;top:1848px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1581px;top:1878px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1630px;top:1848px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1703px;top:1848px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1742px;top:1878px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1791px;top:1848px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1865px;top:1848px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1904px;top:1878px">3</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1957px;top:1848px">= 1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1623px;top:1936px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1661px;top:1966px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1711px;top:1936px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1784px;top:1936px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1823px;top:1966px">3</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1876px;top:1936px">= 0</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1703px;top:2024px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1742px;top:2054px">3</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1795px;top:2024px">= 1</span><span class="a" style="left:2137px;top:1937px">:</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2157px;top:1937px">,</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2251px;top:1937px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2290px;top:1967px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2343px;top:1937px">= 1;</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2490px;top:1937px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2529px;top:1967px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1937px"><span class="a" style="left:2582px">=</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2660px;top:1937px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2717px;top:1937px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2778px;top:1937px">e <span class="w10"></span>a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2876px;top:1967px">3</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2929px;top:1937px">= 1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3043px;top:1937px">:</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:2162px;word-spacing:1px;letter-spacing:1px">Exemplo 58</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1456px;top:2162px;word-spacing:5px;letter-spacing:1px">Consideremos , no</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:2106px;top:2177px">R</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2159px;top:2165px">3</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2192px;top:2162px;word-spacing:6px;letter-spacing:1px">, os seguintes vetores:</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2965px;top:2173px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:3000px;top:2203px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2173px"><span class="a" style="left:3068px">= <span class="w" style="width:19px"></span>(1</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2173px"><span class="a" style="left:3226px">;</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="g" style="top:2173px"><span class="a" style="left:3258px"></span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:3315px;top:2173px">3</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3352px;top:2173px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:3385px;top:2173px">2)</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:3484px;top:2162px">e </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:985px;top:2261px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1021px;top:2291px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2261px"><span class="a" style="left:1081px">= <span class="w12"></span>(2</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2261px"><span class="a" style="left:1231px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2261px"><span class="a" style="left:1264px">4</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2261px"><span class="a" style="left:1301px">;</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1333px;top:2261px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1390px;top:2261px">1)</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1456px;top:2261px">:</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1506px;top:2250px;word-spacing:3px;letter-spacing:1px">Escreva o vetor </span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2261px"><span class="a" style="left:2041px">v</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2261px"><span class="a" style="left:2107px">= <span class="w11"></span>(</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="g" style="top:2261px"><span class="a" style="left:2220px"></span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2261px"><span class="a" style="left:2277px">4</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2261px"><span class="a" style="left:2314px">;</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="g" style="top:2261px"><span class="a" style="left:2346px"></span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2403px;top:2261px">18</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2477px;top:2261px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2509px;top:2261px">7)</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2605px;top:2250px;word-spacing:6px">como combinação linear dos </span><span class="a" style="left:985px;top:2339px;letter-spacing:-1px">vetores </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1232px;top:2349px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1268px;top:2379px">1</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1327px;top:2339px">e </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1387px;top:2349px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1422px;top:2379px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1455px;top:2349px">:</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:2482px">Resolução:</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1999px;top:2669px">v</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2057px;top:2669px">=</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2135px;top:2669px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2173px;top:2699px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2206px;top:2669px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2242px;top:2699px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2291px;top:2669px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2365px;top:2669px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2403px;top:2699px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2436px;top:2669px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2472px;top:2699px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2790px"><span class="a" style="left:1095px">(</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="g" style="top:2790px"><span class="a" style="left:1124px"></span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2790px"><span class="a" style="left:1181px">4</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2790px"><span class="a" style="left:1218px">;</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="g" style="top:2790px"><span class="a" style="left:1250px"></span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2790px"><span class="a" style="left:1308px">18</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1381px;top:2790px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1414px;top:2790px">7) =</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1577px;top:2790px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1616px;top:2820px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2790px"><span class="a" style="left:1648px">(1</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2790px"><span class="a" style="left:1714px">;</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="g" style="top:2790px"><span class="a" style="left:1746px"></span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2790px"><span class="a" style="left:1803px">3</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1840px;top:2790px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1873px;top:2790px">2)+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1995px;top:2790px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2034px;top:2820px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2790px"><span class="a" style="left:2067px">(2</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2790px"><span class="a" style="left:2132px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2790px"><span class="a" style="left:2165px">4</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2790px"><span class="a" style="left:2201px">;</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2234px;top:2790px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2291px;top:2790px;word-spacing:1px;letter-spacing:1px">1) = (1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2520px;top:2790px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2558px;top:2820px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2790px"><span class="a" style="left:2591px">;</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2624px;top:2790px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2681px;top:2790px">3</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2718px;top:2790px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2757px;top:2820px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2790px;top:2790px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2822px;top:2790px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2859px;top:2790px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2898px;top:2820px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2931px;top:2790px;letter-spacing:-1px">)+(2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3082px;top:2790px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:3120px;top:2820px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3153px;top:2790px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:3186px;top:2790px">4</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3223px;top:2790px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:3261px;top:2820px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2790px"><span class="a" style="left:3294px">;</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:3327px;top:2790px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:3384px;top:2790px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3421px;top:2790px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:3460px;top:2820px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:3493px;top:2790px">) =</span><span class="a" style="left:1095px;top:2878px">= (</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1201px;top:2878px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1240px;top:2908px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1290px;top:2878px">+ 2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1400px;top:2878px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1438px;top:2908px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2878px"><span class="a" style="left:1471px">;</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1504px;top:2878px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1561px;top:2878px">3</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1598px;top:2878px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1637px;top:2908px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1686px;top:2878px">+ 4</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1796px;top:2878px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1835px;top:2908px">2</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:1864px;top:2908px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1885px;top:2878px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1922px;top:2878px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1961px;top:2908px">1</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2010px;top:2878px"></span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2083px;top:2878px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2122px;top:2908px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2155px;top:2878px">)</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2208px;top:2867px;word-spacing:3px">que é equivalente ao sistema:</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:1551px;top:3025px">8</span><span class="a" style="left:1551px;top:3091px"><</span><span class="a" style="left:1551px;top:3223px">:</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1718px;top:3035px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1757px;top:3065px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1806px;top:3035px">+ 2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1917px;top:3035px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1955px;top:3065px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2009px;top:3035px">=</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2086px;top:3035px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2143px;top:3035px">4</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1653px;top:3123px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1710px;top:3123px">3</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1747px;top:3123px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1786px;top:3154px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1835px;top:3123px">+ 4</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1945px;top:3123px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1984px;top:3154px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2037px;top:3123px">=</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2115px;top:3123px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2172px;top:3123px">18</span><span class="a" style="left:1747px;top:3211px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1784px;top:3211px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1822px;top:3242px">1</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1872px;top:3211px"></span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1945px;top:3211px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1984px;top:3242px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2037px;top:3211px">= 7</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2311px;top:3124px">,</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2405px;top:3124px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2444px;top:3154px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2497px;top:3124px">= 2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2611px;top:3124px">; <span class="w" style="width:21px"></span>a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2707px;top:3154px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3124px"><span class="a" style="left:2761px">=</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2838px;top:3124px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2895px;top:3124px">3</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2932px;top:3124px">:</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:3323px;letter-spacing:-1px">Portanto,</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1434px;top:3333px">v</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1506px;top:3333px">= <span class="w" style="width:19px"></span>2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1635px;top:3333px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1670px;top:3364px">1</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1725px;top:3333px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1805px;top:3333px">3</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1841px;top:3333px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1877px;top:3364px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1910px;top:3333px">:</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1963px;top:3323px;word-spacing:8px;letter-spacing:1px">Agora mostre que o vetor</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3333px"><span class="a" style="left:2851px">v</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3333px"><span class="a" style="left:2924px">= <span class="w" style="width:19px"></span>(4</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3333px"><span class="a" style="left:3081px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3333px"><span class="a" style="left:3114px">3</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3150px;top:3333px">;</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:3183px;top:3333px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:3240px;top:3333px">6)</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:3338px;top:3323px;word-spacing:4px;letter-spacing:3px">não é</span><span class="a" style="left:985px;top:3411px;word-spacing:3px">combinação linear dos vetores</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1967px;top:3422px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2003px;top:3452px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3422px"><span class="a" style="left:2056px">= (1</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3422px"><span class="a" style="left:2199px">;</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="g" style="top:3422px"><span class="a" style="left:2232px"></span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2289px;top:3422px">3</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2325px;top:3422px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2358px;top:3422px">2)</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2448px;top:3411px">e</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2505px;top:3422px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2540px;top:3452px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3422px"><span class="a" style="left:2594px">= (2</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3422px"><span class="a" style="left:2736px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3422px"><span class="a" style="left:2769px">4</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3422px"><span class="a" style="left:2806px">;</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2838px;top:3422px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2896px;top:3422px">1)</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2961px;top:3422px">:</span></div>
<div class="ff7" style="font-size:122px">
<span class="a" style="left:985px;top:3621px;word-spacing:14px;letter-spacing:1px">2.5 <span class="w" style="width:73px"></span>De<span class="l6">pend<span class="l8">ênc<span class="l8">ia e I<span class="l8">nd<span class="l6">epen<span class="l8">dê<span class="l6">nci<span class="l7">a Li<span class="l6">ne<span class="l6">ar</span></span></span></span></span></span></span></span></span></span></span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:3811px;word-spacing:1px;letter-spacing:1px">De…nição 59</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1476px;top:3811px">Seja </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1637px;top:3822px">V </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1723px;top:3811px;word-spacing:8px;letter-spacing:1px">um esp<span class="l10">aço ve<span class="l7">toria<span class="l7">l <span class="w" style="width:26px"></span>e </span></span></span></span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2434px;top:3822px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2470px;top:3852px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2503px;top:3822px">;<span class="w11"></span>v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2571px;top:3852px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2604px;top:3822px;letter-spacing:3px">;<span class="w9"></span>::<span class="l8">::<span class="l7">::<span class="l7">;<span class="w9"></span>v</span></span></span></span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2827px;top:3851px">n</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2888px;top:3822px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2958px;top:3822px">V:</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:3052px;top:3811px;word-spacing:1px;letter-spacing:1px">Dizemos que o</span><span class="a" style="left:985px;top:3899px;letter-spacing:-1px">conjunto</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1285px;top:3910px">f</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1322px;top:3910px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1358px;top:3940px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1391px;top:3910px">;<span class="w11"></span>v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1459px;top:3940px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1492px;top:3910px;letter-spacing:2px">;::::::;v</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:1715px;top:3940px">n</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1755px;top:3910px">g</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1818px;top:3899px">é </span></div>
<div class="ff8" style="font-size:74px">
<span class="a" style="left:1878px;top:3910px;word-spacing:8px">linearmente independente (LI),</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:3064px;top:3899px;word-spacing:5px">se a equação:</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1792px;top:4054px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1830px;top:4084px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1863px;top:4054px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1899px;top:4084px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1948px;top:4054px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2022px;top:4054px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2060px;top:4084px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2093px;top:4054px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2129px;top:4084px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4054px"><span class="a" style="left:2178px">+</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2252px;top:4054px;letter-spacing:-1px">::::</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2350px;top:4054px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2423px;top:4054px">a</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2462px;top:4083px">n</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2502px;top:4054px">v</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2537px;top:4083px">n</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2598px;top:4054px">= 0</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:985px;top:4186px;word-spacing:2px">implica que </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1886px;top:4285px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1924px;top:4315px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1978px;top:4285px">=</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2055px;top:4285px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2094px;top:4315px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4285px"><span class="a" style="left:2147px">=</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2225px;top:4285px">:::</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2307px;top:4285px">=</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2384px;top:4285px">a</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2423px;top:4314px">n</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2483px;top:4285px">= 0</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2597px;top:4285px">:</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:985px;top:4396px;word-spacing:7px;letter-spacing:1px">No caso, em que exista algum </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1998px;top:4406px">a</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2037px;top:4436px">i</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2092px;top:4406px">6</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2092px;top:4406px">= <span class="w" style="width:14px"></span>0</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2247px;top:4396px;word-spacing:2px;letter-spacing:1px">dizemos que </span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2669px;top:4406px">f</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2706px;top:4406px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2741px;top:4436px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2774px;top:4406px">;<span class="w12"></span>v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2842px;top:4436px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2875px;top:4406px;letter-spacing:3px">;<span class="w9"></span>::<span class="l7">::<span class="l7">::<span class="l8">;<span class="w9"></span>v</span></span></span></span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:3099px;top:4436px">n</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:3139px;top:4406px">g</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:3207px;top:4396px">é </span></div>
<div class="ff8" style="font-size:74px">
<span class="a" style="left:3277px;top:4406px;letter-spacing:-1px">linear-</span><span class="a" style="left:985px;top:4494px;word-spacing:1px;letter-spacing:1px">mente dependente (LD)</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1860px;top:4484px">.</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1201px;top:4627px;word-spacing:7px;letter-spacing:2px">Pa<span class="l6">ra deter<span class="l7">mi<span class="l6">nar<span class="l6">mo<span class="l6">s <span class="w8"></span>se um <span class="w9"></span>con<span class="l7">jun<span class="l9">to é <span class="w11"></span>L.I<span class="l7">. <span class="w6"></span>ou L.D. deve<span class="l7">mos <span class="w" style="width:37px"></span>faz<span class="l7">er a</span></span></span></span></span></span></span></span></span></span></span><span class="a" style="left:985px;top:4715px;word-spacing:2px;letter-spacing:1px">combinação linear do conjunto de vetores e igualar esta combinação linear ao</span><span class="a" style="left:985px;top:4803px;word-spacing:7px">vetor nulo do espaço. <span class="w" style="width:14px"></span>Portan<span class="l6">to é muito importante ter conhecimento do vetor</span></span><span class="a" style="left:985px;top:4891px;word-spacing:5px">nulo do espaço em que estamos trabalhando.</span><span class="a" style="left:2215px;top:5112px">41</span></div>
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<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:926px;word-spacing:1px;letter-spacing:1px">De…nição 60</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1476px;top:927px;word-spacing:4px">Considere o espaço vetorial </span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:2380px;top:942px">R</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2433px;top:930px">3</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2492px;top:927px;word-spacing:5px">e os conjunto de vetores:</span></div>
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<span class="g" style="top:1226px"><span class="a" style="left:1696px"></span></span>
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<span class="g" style="top:1226px"><span class="a" style="left:1816px">=</span></span>
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<div class="ff9" style="font-size:126px">
<span class="g" style="top:1226px"><span class="a" style="left:1947px">f</span></span>
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<div class="ff17" style="font-size:74px">
<span class="g" style="top:1226px"><span class="a" style="left:1984px">(1</span></span>
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<span class="g" style="top:1226px"><span class="a" style="left:2049px">;</span></span>
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<div class="ff17" style="font-size:74px">
<span class="g" style="top:1226px"><span class="a" style="left:2082px">2</span></span>
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<span class="g" style="top:1226px"><span class="a" style="left:2118px">;</span></span>
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<span class="g" style="top:1226px"><span class="a" style="left:2151px">3)</span></span>
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<span class="g" style="top:1226px"><span class="a" style="left:2228px">;</span></span>
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<div class="ff17" style="font-size:74px">
<span class="g" style="top:1226px"><span class="a" style="left:2261px">(1</span></span>
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<span class="g" style="top:1226px"><span class="a" style="left:2326px">;</span></span>
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<div class="ff17" style="font-size:74px">
<span class="g" style="top:1226px"><span class="a" style="left:2359px">1</span></span>
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<span class="g" style="top:1226px"><span class="a" style="left:2396px">;</span></span>
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<span class="g" style="top:1226px"><span class="a" style="left:2428px">1)</span></span>
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<div class="ff17" style="font-size:74px">
<span class="g" style="top:1226px"><span class="a" style="left:2526px">(1</span></span>
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<span class="g" style="top:1226px"><span class="a" style="left:2591px">;</span></span>
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<span class="g" style="top:1226px"><span class="a" style="left:2624px">0</span></span>
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<span class="a" style="left:2693px;top:1226px">0)</span></div>
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<span class="g" style="top:1336px"><span class="a" style="left:1698px"> </span></span>
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<span class="g" style="top:1336px"><span class="a" style="left:1816px">=</span></span>
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<span class="g" style="top:1336px"><span class="a" style="left:1947px">f</span></span>
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<div class="ff17" style="font-size:74px">
<span class="g" style="top:1336px"><span class="a" style="left:1984px">(1</span></span>
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<span class="g" style="top:1336px"><span class="a" style="left:2049px">;</span></span>
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<span class="g" style="top:1336px"><span class="a" style="left:2082px">2</span></span>
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<span class="g" style="top:1336px"><span class="a" style="left:2151px">3)</span></span>
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<span class="g" style="top:1336px"><span class="a" style="left:2228px">;</span></span>
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<div class="ff17" style="font-size:74px">
<span class="g" style="top:1336px"><span class="a" style="left:2261px">(1</span></span>
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<span class="g" style="top:1336px"><span class="a" style="left:2326px">;</span></span>
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<div class="ff17" style="font-size:74px">
<span class="g" style="top:1336px"><span class="a" style="left:2359px">1</span></span>
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<span class="g" style="top:1336px"><span class="a" style="left:2396px">;</span></span>
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<div class="ff17" style="font-size:74px">
<span class="g" style="top:1336px"><span class="a" style="left:2428px">1)</span></span>
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<span class="g" style="top:1336px"><span class="a" style="left:2494px">;</span></span>
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<div class="ff17" style="font-size:74px">
<span class="g" style="top:1336px"><span class="a" style="left:2526px">(3</span></span>
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<span class="g" style="top:1336px"><span class="a" style="left:2591px">;</span></span>
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<span class="g" style="top:1336px"><span class="a" style="left:2624px">5</span></span>
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<span class="a" style="left:2661px;top:1336px">;</span></div>
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<span class="a" style="left:2693px;top:1336px">7)</span></div>
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<span class="a" style="left:2759px;top:1336px">g</span></div>
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<span class="a" style="left:985px;top:1469px;word-spacing:2px">Os conjuntos</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1426px;top:1480px"></span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1498px;top:1469px">e</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1555px;top:1480px"> </span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1625px;top:1469px;word-spacing:7px;letter-spacing:1px">acim<span class="l6">a são <span class="w" style="width:19px"></span>L.I o<span class="l7">u L.D<span class="l6">?</span></span></span></span><span class="a" style="left:1095px;top:1558px">Solução:</span><span class="a" style="left:1095px;top:1646px;word-spacing:2px">Fazendo a combinação linear</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:1591px">a</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:1642px">(1</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:1707px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:1740px">2</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:1777px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:1809px">3) +</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:1964px">b</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:1996px">(1</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:2061px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:2094px">1</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:2130px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:2163px">1) +</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:2318px">c</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:2350px">(1</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:2415px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:2448px">0</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:2484px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:2517px;word-spacing:1px;letter-spacing:1px">0) = (0</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1801px"><span class="a" style="left:2745px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2778px;top:1801px">0</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2815px;top:1801px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2847px;top:1801px">0)</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:985px;top:1934px;word-spacing:4px">temos o sistema homogêneo:</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:1954px;top:2092px">8</span><span class="a" style="left:1954px;top:2158px"><</span><span class="a" style="left:1954px;top:2290px">:</span><span class="a" style="left:2019px;top:2095px">2</span><span class="a" style="left:2019px;top:2227px">4</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2102px"><span class="a" style="left:2068px">a</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2102px"><span class="a" style="left:2123px">+</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2102px"><span class="a" style="left:2196px">b</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2244px;top:2102px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2318px;top:2102px">c</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2423px;top:2102px">= <span class="w" style="width:57px"></span>0</span><span class="g" style="top:2190px"><span class="a" style="left:2110px">2</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2190px"><span class="a" style="left:2147px">a</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2202px;top:2190px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2275px;top:2190px">b</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2423px;top:2190px">= <span class="w" style="width:57px"></span>0</span><span class="g" style="top:2278px"><span class="a" style="left:2110px">3</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2278px"><span class="a" style="left:2147px">a</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2202px;top:2278px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2275px;top:2278px">b</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2423px;top:2278px">= <span class="w" style="width:57px"></span>0</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:2590px;top:2095px">3</span><span class="a" style="left:2590px;top:2227px">5</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:985px;top:2386px;word-spacing:5px">cuja única solução é</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2396px"><span class="a" style="left:1652px">a</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2396px"><span class="a" style="left:1712px">=</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2396px"><span class="a" style="left:1789px">b</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1841px;top:2396px">=</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1918px;top:2396px">c</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1971px;top:2396px">= 0</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2085px;top:2386px;word-spacing:6px">. Portanto o conjunto</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2808px;top:2396px"></span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2880px;top:2386px;word-spacing:2px;letter-spacing:1px">é L.I</span><span class="a" style="left:1095px;top:2474px;word-spacing:2px">Fazendo a combinação linear</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:1591px">a</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:1642px">(1</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:1707px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:1740px">2</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:1777px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:1809px">3) +</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:1964px">b</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:1996px">(1</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:2061px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:2094px">1</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:2130px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:2163px">1) +</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:2318px">c</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:2350px">(3</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:2415px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:2448px">5</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:2484px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:2517px;word-spacing:1px;letter-spacing:1px">7) = (0</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2629px"><span class="a" style="left:2745px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2778px;top:2629px">0</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2815px;top:2629px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2847px;top:2629px">0)</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:985px;top:2762px;word-spacing:4px">temos o sistema homogêneo:</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:1917px;top:2920px">8</span><span class="a" style="left:1917px;top:2986px"><</span><span class="a" style="left:1917px;top:3118px">:</span><span class="a" style="left:1982px;top:2923px">2</span><span class="a" style="left:1982px;top:3055px">4</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2930px"><span class="a" style="left:2049px">a</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2930px"><span class="a" style="left:2105px">+</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2930px"><span class="a" style="left:2178px">b</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2226px;top:2930px">+ 3</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2336px;top:2930px">c</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2459px;top:2930px">= <span class="w" style="width:58px"></span>0</span><span class="g" style="top:3018px"><span class="a" style="left:2031px">2</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3018px"><span class="a" style="left:2068px">a</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3018px"><span class="a" style="left:2123px">+</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3018px"><span class="a" style="left:2196px">b</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2244px;top:3018px">+ 5</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2354px;top:3018px">c</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2459px;top:3018px">= <span class="w" style="width:58px"></span>0</span><span class="g" style="top:3106px"><span class="a" style="left:2031px">3</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3106px"><span class="a" style="left:2068px">a</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3106px"><span class="a" style="left:2123px">+</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3106px"><span class="a" style="left:2196px">b</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2244px;top:3106px">+ 7</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2354px;top:3106px">c</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2459px;top:3106px">= <span class="w" style="width:58px"></span>0</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:2627px;top:2923px">3</span><span class="a" style="left:2627px;top:3055px">5</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:985px;top:3218px;word-spacing:3px;letter-spacing:1px">que possui in…nitas soluções ( grau de liberdade 1). Portanto além da solução</span><span class="a" style="left:985px;top:3306px;word-spacing:6px;letter-spacing:1px">nula ( que todo sistema homogêneo tem) este sistema possui outras soluções</span><span class="a" style="left:985px;top:3394px;word-spacing:4px">diferentes da solução nula, logo o conjunto</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2375px;top:3405px"> </span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2445px;top:3394px;word-spacing:5px">é L.D.</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:3538px;word-spacing:3px">Teorema 61</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1452px;top:3538px;word-spacing:4px;letter-spacing:-1px">O conjunto</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1823px;top:3549px">f</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1860px;top:3549px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1895px;top:3579px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1928px;top:3549px">;<span class="w12"></span>v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1996px;top:3579px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2029px;top:3549px;letter-spacing:3px">;<span class="w9"></span>::<span class="l7">::<span class="l8">::<span class="l7">;<span class="w9"></span>v</span></span></span></span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2252px;top:3579px">n</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2292px;top:3549px">g</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2349px;top:3538px;word-spacing:1px">é LD se, e somente se um dos vetores </span><span class="a" style="left:985px;top:3626px;word-spacing:4px">do conjunto for uma combinação linear dos outros.</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:3770px;word-spacing:1px;letter-spacing:1px">Exemplo 62</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1456px;top:3771px;word-spacing:7px">a) Seja </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1710px;top:3781px">V </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1790px;top:3781px">=</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:1867px;top:3785px">R</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1920px;top:3774px">3</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1953px;top:3771px;word-spacing:9px;letter-spacing:1px">. Sejam </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2230px;top:3781px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2266px;top:3811px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2299px;top:3781px">;<span class="w12"></span>v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2367px;top:3811px">2</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2421px;top:3781px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2490px;top:3781px">V:</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2557px;top:3771px;word-spacing:1px">O conjunto</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2939px;top:3781px">f</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2976px;top:3781px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:3011px;top:3811px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3044px;top:3781px">;<span class="w12"></span>v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:3112px;top:3811px">2</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:3145px;top:3781px">g</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:3208px;top:3771px;word-spacing:3px;letter-spacing:2px">é LD se e </span><span class="a" style="left:985px;top:3859px;word-spacing:7px">somente se </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1364px;top:3869px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1400px;top:3900px">1</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1460px;top:3859px">e </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1521px;top:3869px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1556px;top:3900px">2</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1644px;top:3859px;word-spacing:6px">estiverem na mesma reta que passa pela origem (um vetor </span><span class="a" style="left:985px;top:3947px;word-spacing:5px">é múltiplo do outro),</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1676px;top:3958px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1712px;top:3988px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1765px;top:3958px">=</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1843px;top:3958px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1921px;top:3988px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1954px;top:3958px">:</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:4091px;word-spacing:3px;letter-spacing:1px">b) Em</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1325px;top:4102px">V </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1404px;top:4102px">=</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:1482px;top:4106px">R</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1535px;top:4094px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1568px;top:4102px">; <span class="w" style="width:45px"></span>e</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1684px;top:4132px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1737px;top:4102px">= (1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1880px;top:4102px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1912px;top:4102px">0)</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2002px;top:4091px">e</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2059px;top:4102px">e</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2093px;top:4132px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2147px;top:4102px">= (0</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2289px;top:4102px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2322px;top:4102px">1)</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2412px;top:4091px;word-spacing:1px;letter-spacing:1px">são LI, pois:</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1192px;top:4246px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1231px;top:4276px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1264px;top:4246px">e</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1298px;top:4276px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1347px;top:4246px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1420px;top:4246px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1459px;top:4276px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1492px;top:4246px">e</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1526px;top:4276px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1580px;top:4246px">= 0 <span class="w9"></span>=</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1759px;top:4246px">)</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1853px;top:4246px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1892px;top:4276px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4246px"><span class="a" style="left:1925px">(1</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1990px;top:4246px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2023px;top:4246px">0) +</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2178px;top:4246px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2216px;top:4276px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4246px"><span class="a" style="left:2249px">(0</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:4246px"><span class="a" style="left:2315px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4246px"><span class="a" style="left:2347px;word-spacing:1px;letter-spacing:1px">1) = (0</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:4246px"><span class="a" style="left:2576px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4246px"><span class="a" style="left:2608px">0) =</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2739px;top:4246px">)</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2833px;top:4246px">(</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2861px;top:4246px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2900px;top:4276px">1</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2929px;top:4275px">;</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2950px;top:4246px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2989px;top:4276px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:3022px;top:4246px;word-spacing:2px;letter-spacing:1px">) = (0</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3214px;top:4246px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:3246px;top:4246px">0)</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:985px;top:4379px">logo</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1140px;top:4390px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1179px;top:4420px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1232px;top:4390px">= 0</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1371px;top:4379px">e</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1428px;top:4390px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1467px;top:4420px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1520px;top:4390px">= 0</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1659px;top:4379px">portanto,</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1981px;top:4390px">e</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2015px;top:4420px">1</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2048px;top:4379px">e</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2105px;top:4390px">e</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2140px;top:4420px">2</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2197px;top:4379px;word-spacing:6px">são LI.</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:4523px;word-spacing:1px;letter-spacing:1px">Exemplo 63</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1456px;top:4524px;word-spacing:2px">No espaço Vetorial </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2088px;top:4534px">M </span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2159px;top:4564px">2</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2218px;top:4524px;word-spacing:1px">o conjunto:</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1543px;top:4718px">A</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1619px;top:4718px">=</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:1696px;top:4666px"></span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1827px;top:4674px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1884px;top:4674px">1 <span class="w" style="width:58px"></span>2</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1827px;top:4762px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1884px;top:4762px">3 <span class="w" style="width:58px"></span>1</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:2067px;top:4666px"></span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2118px;top:4718px">;</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:2151px;top:4666px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2226px;top:4674px">2</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2336px;top:4674px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2393px;top:4674px">3</span><span class="a" style="left:2226px;top:4762px">3 <span class="w" style="width:87px"></span>0</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:2467px;top:4666px"></span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2518px;top:4718px">;</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:2551px;top:4666px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2626px;top:4674px">3</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2736px;top:4674px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2793px;top:4674px">4</span><span class="a" style="left:2626px;top:4762px">3 <span class="w" style="width:87px"></span>1</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:2867px;top:4666px"></span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:985px;top:4891px;word-spacing:6px">é LD. Examinemos a equação:</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2129px;top:4902px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2168px;top:4932px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2201px;top:4902px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2236px;top:4932px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2286px;top:4902px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2359px;top:4902px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2398px;top:4932px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2431px;top:4902px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2466px;top:4932px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2516px;top:4902px">+</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2589px;top:4902px">a</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2628px;top:4932px">3</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2661px;top:4902px">v</span></div>
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<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:1491px;word-spacing:2px;letter-spacing:1px">Propriedades da Dependência e da Independência Linear</span></div>
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<span class="a" style="left:985px;top:1627px;word-spacing:2px;letter-spacing:1px">Seja V um E.V</span><span class="a" style="left:1075px;top:1796px;word-spacing:7px;letter-spacing:3px">1. Se</span></div>
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<span class="a" style="left:1979px;top:1796px;word-spacing:4px">, então</span></div>
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<span class="a" style="left:2302px;top:1796px;word-spacing:2px;letter-spacing:1px">é LI.</span><span class="a" style="left:1075px;top:1943px;word-spacing:4px;letter-spacing:1px">2. Se um conjunto</span></div>
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<span class="a" style="left:985px;top:2303px;word-spacing:21px;letter-spacing:2px">2.<span class="l6">6 <span class="w" style="width:68px"></span>Su<span class="l8">bes<span class="l7">pa<span class="l7">ço<span class="l7">s <span class="l11">Ge<span class="l8">ra<span class="l7">do<span class="l7">s</span></span></span></span></span></span></span></span></span></span></div>
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<span class="a" style="left:985px;top:2493px;word-spacing:1px;letter-spacing:1px">De…nição 64</span></div>
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<span class="a" style="left:1724px;top:2494px;word-spacing:5px">um espaço vetorial. Consideramos um subconjunto</span></div>
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<span class="a" style="left:2436px;top:2582px;word-spacing:5px;letter-spacing:1px">de todos os vetores de </span></div>
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<div class="ff16" style="font-size:85px">
<span class="a" style="left:3275px;top:2582px;word-spacing:7px;letter-spacing:1px">que são</span><span class="a" style="left:985px;top:2670px;word-spacing:4px">combinações lineares dos vetores de A é um subespaço de </span></div>
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<span class="a" style="left:2939px;top:2670px;word-spacing:10px;letter-spacing:-1px">Simbolicamente, o</span><span class="a" style="left:985px;top:2758px;letter-spacing:-1px">subespaço</span></div>
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<span class="a" style="left:1519px;top:3092px">W </span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1621px;top:3081px;word-spacing:3px">diz-se gerado pelos vetores</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2491px;top:3092px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2526px;top:3122px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2559px;top:3092px">;<span class="w12"></span>v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2627px;top:3122px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2660px;top:3092px;letter-spacing:3px">;<span class="w9"></span>::<span class="l7">:v</span></span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2790px;top:3121px">n</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2830px;top:3092px">:;</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2893px;top:3081px;word-spacing:-1px;letter-spacing:1px">ou gerado pelo con-</span><span class="a" style="left:968px;top:3169px;word-spacing:5px"> junto A, e representa-se por:</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1702px;top:3341px">W </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1801px;top:3341px">= [</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1899px;top:3341px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1935px;top:3371px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1968px;top:3341px">;<span class="w11"></span>v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2036px;top:3371px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2069px;top:3341px;letter-spacing:3px">;<span class="w9"></span>::<span class="l8">:v</span></span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2198px;top:3371px">n</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3341px"><span class="a" style="left:2238px">:</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3341px"><span class="a" style="left:2259px">]</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3341px"><span class="a" style="left:2328px">ou <span class="w" style="width:34px"></span>W </span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3341px"><span class="a" style="left:2555px">=</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3341px"><span class="a" style="left:2632px">G</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2690px;top:3341px">(</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2718px;top:3341px">A</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2773px;top:3341px">)</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1719px;top:3492px;word-spacing:6px">Os vetores</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2085px;top:3503px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2121px;top:3533px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2153px;top:3503px">;<span class="w12"></span>v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2222px;top:3533px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2255px;top:3503px;letter-spacing:2px">;:::v</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2384px;top:3532px">n</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2424px;top:3503px">:</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2444px;top:3492px;word-spacing:1px;letter-spacing:1px">são chamados geradores do sube-</span><span class="a" style="left:985px;top:3580px">spaço</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1185px;top:3591px">W;</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1298px;top:3580px;letter-spacing:-1px">enquanto</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1616px;top:3591px">A</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1695px;top:3580px;letter-spacing:1px">é o conjunto gerador de</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2479px;top:3591px">W:</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:3676px;word-spacing:5px">Para o caso particular de</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1927px;top:3687px">A</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2003px;top:3687px">=</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:2080px;top:3691px">?</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2137px;top:3687px">;</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2182px;top:3676px">de…ne-se</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2480px;top:3687px">[</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:2501px;top:3691px">?</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2558px;top:3687px">] =</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2676px;top:3687px">f</span><span class="a" style="left:2713px;top:3640px"></span><span class="a" style="left:2713px;top:3640px">!</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2731px;top:3687px">0</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2786px;top:3687px">g</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3775px"><span class="a" style="left:1095px">A</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="g" style="top:3775px"><span class="a" style="left:1171px"></span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3775px"><span class="a" style="left:1248px">G</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1306px;top:3775px">(</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1335px;top:3775px">A</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1390px;top:3775px">)</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1418px;top:3764px;word-spacing:2px;letter-spacing:1px">, ou seja,</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1731px;top:3775px">f</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1792px;top:3775px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1827px;top:3805px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1860px;top:3775px">;<span class="w12"></span>v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1929px;top:3805px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1961px;top:3775px;letter-spacing:3px">;<span class="w9"></span>::<span class="l7">:v</span></span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2091px;top:3804px">n</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2131px;top:3775px">g<span class="w" style="width:20px"></span></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2265px;top:3775px">[</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2286px;top:3775px">v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2321px;top:3805px">1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2354px;top:3775px">;<span class="w12"></span>v</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2423px;top:3805px">2</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2455px;top:3775px;letter-spacing:3px">;<span class="w9"></span>::<span class="l7">:v</span></span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2585px;top:3804px">n</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2625px;top:3775px">]</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:3852px;word-spacing:3px">Todo conjunto</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1596px;top:3863px">A</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1680px;top:3863px"></span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1767px;top:3863px">V </span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1856px;top:3852px;word-spacing:4px;letter-spacing:1px">gera um subespaço vetorial de</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2877px;top:3863px">V </span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2936px;top:3852px;word-spacing:6px;letter-spacing:1px">, podendo ocorrer</span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3951px"><span class="a" style="left:985px">G</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3951px"><span class="a" style="left:1043px">(</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1072px;top:3951px">A</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1127px;top:3951px">) =</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1253px;top:3951px">V </span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1312px;top:3940px;word-spacing:4px;letter-spacing:1px">. Nesse caso,</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1748px;top:3951px">A</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1827px;top:3940px;word-spacing:5px">é um conjunto gerador de</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2676px;top:3951px">V:</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:4101px;word-spacing:1px;letter-spacing:1px">Exemplo 65</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1456px;top:4102px;word-spacing:8px">Os vetores </span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:4113px"><span class="a" style="left:1826px">i</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1882px;top:4113px">= <span class="w" style="width:14px"></span>(1</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2034px;top:4113px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2067px;top:4113px">0)</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2164px;top:4102px">e </span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:4113px"><span class="a" style="left:2261px">j</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2326px;top:4113px">= <span class="w" style="width:14px"></span>(0</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2478px;top:4113px">;</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2511px;top:4113px">1)</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2608px;top:4102px;word-spacing:9px">geram o espaço vetorial </span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:3410px;top:4117px">R</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:3463px;top:4105px">2</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:3496px;top:4102px">,</span><span class="a" style="left:985px;top:4190px;word-spacing:1px">pois, qualquer </span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4201px"><span class="a" style="left:1451px">(</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1480px;top:4201px">x;<span class="w12"></span>y</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1593px;top:4201px">)</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1642px;top:4201px">2</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:1711px;top:4205px">R</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1764px;top:4193px">2</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1824px;top:4190px;word-spacing:4px">é combinação linear de </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2586px;top:4201px">i</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2638px;top:4190px">e </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2724px;top:4201px">j</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2779px;top:4201px">:</span><span class="g" style="top:4362px"><span class="a" style="left:1311px">(</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:1339px">x;<span class="w12"></span>y</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:1452px">) =</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:1579px">xi</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:1662px">+</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:1736px">yj</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:1829px">=</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:1907px">x</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:1949px">(0</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:2014px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:2047px">1) +</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:2202px">y</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:2240px">(0</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:2306px">;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:2338px;word-spacing:2px;letter-spacing:1px">1) = (</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:2530px">x;</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:2604px">0) + (0</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:4362px"><span class="a" style="left:2825px">;<span class="w11"></span>y</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2896px;top:4362px">) = <span class="w9"></span>(</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3051px;top:4362px">x;<span class="w12"></span>y</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:3164px;top:4362px">)</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:4513px;letter-spacing:-1px">Então:</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1339px;top:4524px">[</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1359px;top:4524px">i;<span class="w11"></span>j</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1452px;top:4524px">] =</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:1570px;top:4528px">R</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1623px;top:4516px">2</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1656px;top:4513px">.</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:4674px;word-spacing:1px;letter-spacing:1px">Exemplo 66</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1456px;top:4675px">Seja </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1623px;top:4685px">V </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1714px;top:4685px">=</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:1802px;top:4689px">R</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1855px;top:4678px">3</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1888px;top:4675px;word-spacing:7px;letter-spacing:1px">. <span class="w" style="width:25px"></span>Deter<span class="l6">mina<span class="l7">r <span class="w6"></span>o <span class="w6"></span>sub<span class="l8">esp<span class="l7">aço gera<span class="l6">do pelo vetor </span></span></span></span></span></span></div>
<div class="ff12" style="font-size:74px">
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<div class="text_layer" style="z-index:2"><div class="ie_fix">
 
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:927px;word-spacing:3px">a) Determine</span></div>
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<span class="a" style="left:1543px;top:938px">[</span></div>
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<span class="a" style="left:1095px;top:1103px;word-spacing:5px">c) Determine uma base para</span></div>
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<span class="a" style="left:1267px;top:1191px;word-spacing:9px">é injetora? E sobrejetora? Justi…que.</span><span class="a" style="left:1095px;top:1464px;letter-spacing:-1px">SOLUÇÃO:</span><span class="a" style="left:1095px;top:1552px">a)</span><span class="a" style="left:985px;top:1817px;word-spacing:5px">Note que</span></div>
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<span class="a" style="left:1714px;top:2147px;word-spacing:21px;letter-spacing:9px">(<span class="l9">1<span class="l10">) <span class="w" style="width:19px"></span>=</span></span></span></div>
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<div class="text_layer" style="z-index:2"><div class="ie_fix">
 
<div class="ff7" style="font-size:122px">
<span class="a" style="left:985px;top:897px;word-spacing:23px;letter-spacing:1px">3.3 <span class="w" style="width:104px"></span>Co<span class="l6">mpos<span class="l7">içã<span class="l6">o <span class="l11">de <span class="l" style="margin-left:-14px">tra<span class="l7">nsf<span class="l7">orm<span class="l8">açõ<span class="l6">es <span class="l" style="margin-left:-14px">lin<span class="l7">ear<span class="l7">es</span></span></span></span></span></span></span></span></span></span></span></span></span></div>
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<span class="a" style="left:985px;top:1087px;letter-spacing:1px">De…niç<span class="l7">ão <span class="w9"></span>147</span></span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1518px;top:1088px">Se </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1627px;top:1098px">T </span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1670px;top:1128px">1</span></div>
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<div class="ff12" style="font-size:74px">
<span class="a" style="left:1993px;top:1098px">W </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2141px;top:1088px">e </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2242px;top:1098px">T </span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2285px;top:1128px">2</span></div>
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<span class="a" style="left:2521px;top:1098px">!</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2629px;top:1098px">U </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2721px;top:1088px;word-spacing:9px">são duas transformações </span><span class="a" style="left:985px;top:1176px;word-spacing:-1px">linea<span class="l6">res a <span class="w8"></span>com<span class="l6">posta das <span class="w6"></span>duas transformaçõ<span class="l8">es <span class="w6"></span>line<span class="l6">ares é <span class="w7"></span>de…nida do <span class="w8"></span>mesmo modo</span></span></span></span></span><span class="a" style="left:985px;top:1264px;word-spacing:4px;letter-spacing:1px">que a composição de funcões ( lembre-se que um transformação linear é uma </span><span class="a" style="left:968px;top:1352px;word-spacing:5px"> função com a propriedade adicional de ser linear) da seguinte forma </span></div>
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<span class="a" style="left:2006px;top:1716px">2</span></div>
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<span class="a" style="left:2151px;top:1716px">1</span></div>
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<span class="a" style="left:2388px;top:1686px">V </span></div>
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<span class="a" style="left:2562px;top:1686px">U </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1811px;top:1796px">(</span></div>
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<span class="a" style="left:2156px;top:1796px">) <span class="w" style="width:58px"></span>=</span></div>
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<span class="a" style="left:2431px;top:1826px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2464px;top:1796px">(</span></div>
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<span class="a" style="left:985px;top:1946px;word-spacing:10px">Exemplo 148</span></div>
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<span class="a" style="left:1498px;top:1947px">Se </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1598px;top:1957px">T </span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1641px;top:1987px">1</span></div>
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<span class="a" style="left:1788px;top:1950px">2</span></div>
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<span class="a" style="left:1988px;top:1950px">3</span></div>
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<span class="a" style="left:2021px;top:1957px">; <span class="w9"></span>T </span></div>
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<span class="a" style="left:2109px;top:1987px">1</span></div>
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</div>
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<span class="g" style="top:1957px"><span class="a" style="left:2171px">x;<span class="w11"></span>y</span></span>
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<div class="ff17" style="font-size:74px">
<span class="g" style="top:1957px"><span class="a" style="left:2284px">) = <span class="w9"></span>(</span></span>
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<span class="g" style="top:1957px"><span class="a" style="left:2494px"></span></span>
</div>
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<span class="g" style="top:1957px"><span class="a" style="left:2564px">y;<span class="w" style="width:14px"></span>y</span></span>
</div>
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<span class="g" style="top:1957px"><span class="a" style="left:2687px"></span></span>
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<span class="g" style="top:1957px"><span class="a" style="left:2757px">x;<span class="w12"></span>y</span></span>
</div>
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<span class="a" style="left:3049px;top:1947px">e </span></div>
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<span class="a" style="left:3108px;top:1957px">T </span></div>
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<span class="a" style="left:3150px;top:1987px">2</span></div>
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<span class="g" style="top:2045px"><span class="a" style="left:1377px">x</span></span>
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<span class="g" style="top:2045px"><span class="a" style="left:1435px"></span></span>
</div>
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<span class="a" style="left:1508px;top:2045px">y</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1563px;top:2045px"></span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1637px;top:2045px">z</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1700px;top:2035px">então</span></div>
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<span class="a" style="left:1901px;top:2045px">T </span></div>
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<span class="a" style="left:2089px;top:2076px">1</span></div>
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<span class="a" style="left:2142px;top:2045px">:</span></div>
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<span class="a" style="left:1482px;top:2207px">(</span></div>
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<span class="a" style="left:1511px;top:2207px">T </span></div>
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<span class="a" style="left:1656px;top:2207px">T </span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1699px;top:2237px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2207px"><span class="a" style="left:1732px">)<span class="w12"></span>(</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1801px;top:2207px">x;<span class="w12"></span>y</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1915px;top:2207px">) <span class="w" style="width:58px"></span>=</span></div>
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<span class="a" style="left:2147px;top:2207px">T </span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2190px;top:2237px">2</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2223px;top:2207px">(</span></div>
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<span class="a" style="left:2469px;top:2207px">))</span><span class="a" style="left:2017px;top:2317px">=</span></div>
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<span class="a" style="left:2190px;top:2347px">2</span></div>
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<span class="g" style="top:2317px"><span class="a" style="left:2223px">(</span></span>
</div>
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<span class="g" style="top:2317px"><span class="a" style="left:2251px">x</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="g" style="top:2317px"><span class="a" style="left:2310px"></span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2317px"><span class="a" style="left:2383px">y;<span class="w" style="width:14px"></span>y</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="g" style="top:2317px"><span class="a" style="left:2509px"></span></span>
</div>
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<span class="g" style="top:2317px"><span class="a" style="left:2583px">x;<span class="w11"></span>y</span></span>
</div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2712px;top:2317px"></span></div>
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<span class="a" style="left:2786px;top:2317px">x</span></div>
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<span class="a" style="left:2828px;top:2317px">)</span><span class="g" style="top:2427px"><span class="a" style="left:2017px">= <span class="w" style="width:57px"></span>(</span></span>
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<span class="g" style="top:2427px"><span class="a" style="left:2176px">x</span></span>
</div>
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<span class="g" style="top:2427px"><span class="a" style="left:2234px"></span></span>
</div>
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<span class="g" style="top:2427px"><span class="a" style="left:2307px">y</span></span>
</div>
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<div class="ff16" style="font-size:85px">
<span class="a" style="left:2617px;top:4098px;word-spacing:6px">a uma transformação linear </span></div>
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<span class="a" style="left:1448px;top:4186px;word-spacing:11px;letter-spacing:1px">que é <span class="w6"></span>inje<span class="l6">tor<span class="l7">a e <span class="w8"></span>sobr<span class="l8">ejet<span class="l6">or<span class="l6">a ao <span class="w7"></span>mesm<span class="l6">o temp<span class="l6">o. <span class="w" style="width:27px"></span>Quan<span class="l7">do há <span class="w6"></span>um </span></span></span></span></span></span></span></span></span><span class="a" style="left:985px;top:4274px;word-spacing:6px">isomor…smo entre dois espaços vetoriais dizemos que estes são</span></div>
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<span class="a" style="left:3484px;top:4523px">é </span><span class="a" style="left:985px;top:4611px;word-spacing:5px;letter-spacing:1px">a identidade em </span></div>
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<div class="ff16" style="font-size:85px">
<span class="a" style="left:2160px;top:4611px;word-spacing:3px;letter-spacing:2px">é a inversa a direita de </span></div>
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<span class="a" style="left:3065px;top:4611px;word-spacing:6px;letter-spacing:1px">Se existe uma </span><span class="a" style="left:985px;top:4700px;letter-spacing:-1px">transformação</span></div>
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<div class="ff16" style="font-size:85px">
<span class="a" style="left:3077px;top:4700px;word-spacing:1px">é a identidade </span><span class="a" style="left:985px;top:4788px">em </span></div>
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<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:926px;letter-spacing:1px">De…niç<span class="l7">ão <span class="w9"></span>153</span></span></div>
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<span class="a" style="left:1518px;top:927px">Seja </span></div>
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<span class="a" style="left:1405px;top:1018px">1</span></div>
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<span class="a" style="left:2692px;top:1018px">1</span></div>
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<span class="a" style="left:3064px;top:1015px;word-spacing:6px">então dizemos </span><span class="a" style="left:985px;top:1103px">que </span></div>
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<span class="a" style="left:985px;top:1264px;word-spacing:1px;letter-spacing:1px">Proposição 154</span></div>
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<span class="a" style="left:1984px;top:1275px">!</span></div>
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<span class="a" style="left:2091px;top:1275px">W </span></div>
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<span class="a" style="left:985px;top:1514px;word-spacing:1px;letter-spacing:1px">Proposição 155</span></div>
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<div class="ff12" style="font-size:74px">
<span class="a" style="left:1981px;top:1525px">W </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2088px;top:1514px;word-spacing:7px">é um isomomor…smo, então</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3014px;top:1525px">T </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:3094px;top:1514px;word-spacing:2px;letter-spacing:1px">é inversível e </span><span class="a" style="left:985px;top:1602px;word-spacing:7px">além disso</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1345px;top:1613px">T </span></div>
<div class="ff14" style="font-size:78px">
<span class="a" style="left:1399px;top:1613px"></span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1444px;top:1605px">1</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1504px;top:1602px;word-spacing:4px">também é um isomor…smo.</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:1763px;word-spacing:1px;letter-spacing:1px">Proposição 156</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1580px;top:1764px">Se </span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:1775px"><span class="a" style="left:1681px">T </span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:1775px"><span class="a" style="left:1755px">:</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1796px;top:1775px">V </span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1875px;top:1775px">!</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1969px;top:1775px">W </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2075px;top:1764px;word-spacing:3px">uma transformação linear invertível ( </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3280px;top:1775px">T </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:3359px;top:1764px">é <span class="w9"></span>um </span><span class="a" style="left:985px;top:1852px;word-spacing:5px">isomor…smo) e </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1489px;top:1863px"></span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1562px;top:1852px">e </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1622px;top:1863px"> </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1694px;top:1852px;word-spacing:6px">são bases de </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2111px;top:1863px">V </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2196px;top:1852px">e </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2256px;top:1863px">W;</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2370px;top:1852px;letter-spacing:-1px">então:</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:1934px;top:2123px"></span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1965px;top:2131px">T </span></div>
<div class="ff14" style="font-size:78px">
<span class="a" style="left:2018px;top:2127px"></span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2064px;top:2120px">1</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:2097px;top:2123px"></span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2127px;top:2104px"></span><span class="a" style="left:2127px;top:2179px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2190px;top:2131px">=</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:2267px;top:2101px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2311px;top:2131px">[</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2331px;top:2131px">T </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2384px;top:2131px">]</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2405px;top:2112px"></span><span class="a" style="left:2405px;top:2171px"></span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:2447px;top:2101px"></span></div>
<div class="ff14" style="font-size:78px">
<span class="a" style="left:2490px;top:2091px"></span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2536px;top:2084px">1</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1095px;top:2270px;word-spacing:-2px">Observ<span class="l6">ação: <span class="w12"></span>Quando estamos trabalhando com o espaço</span></span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:2877px;top:2285px">R</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2930px;top:2273px">n</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2987px;top:2270px;word-spacing:-2px">e a base canônica</span><span class="a" style="left:985px;top:2358px">de</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:1078px;top:2373px">R</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:1131px;top:2361px">n</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:1190px;top:2358px">por simplicidade omitimos as bases e a matriz de</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2746px;top:2369px">T </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2819px;top:2369px">:</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:2860px;top:2373px">R</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2913px;top:2361px">n</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2973px;top:2369px">!</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:3067px;top:2373px">R</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:3120px;top:2361px">n</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3160px;top:2369px">;</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:3181px;top:2358px;word-spacing:-2px">em relação</span><span class="a" style="left:985px;top:2446px;word-spacing:10px;letter-spacing:1px">a base canôni<span class="l7">ca, <span class="w" style="width:33px"></span>é denota<span class="l6">da simpl<span class="l6">esme<span class="l6">nte por</span></span></span></span></span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2457px"><span class="a" style="left:2575px">[</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2595px;top:2457px">T </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2648px;top:2457px">]</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2681px;top:2457px">:</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2733px;top:2446px;word-spacing:5px;letter-spacing:1px">Neste caso a proposição</span><span class="a" style="left:985px;top:2535px;word-spacing:5px">acima é escrita na forma mais conveniente: "Se</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2529px;top:2545px">T </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2602px;top:2545px">:</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:2643px;top:2549px">R</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2696px;top:2537px">n</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:2756px;top:2545px">!</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:2850px;top:2549px">R</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:2903px;top:2537px">n</span></div>
<div class="ff4" style="font-size:85px">
<span class="a" style="left:2967px;top:2535px">é inversível então</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:985px;top:2625px"></span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1016px;top:2633px">T </span></div>
<div class="ff14" style="font-size:78px">
<span class="a" style="left:1069px;top:2633px"></span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1115px;top:2626px">1</span></div>
<div class="ff1" style="font-size:221px">
<span class="a" style="left:1148px;top:2625px"></span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1199px;top:2633px">= [</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1297px;top:2633px">T </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1350px;top:2633px">]</span></div>
<div class="ff14" style="font-size:78px">
<span class="a" style="left:1370px;top:2623px"></span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1416px;top:2616px">1</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1461px;top:2633px">"</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:2784px;word-spacing:1px;letter-spacing:1px">Proposição 157</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1580px;top:2784px">Seja </span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:2795px"><span class="a" style="left:1747px">T </span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:2795px"><span class="a" style="left:1830px">:</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1880px;top:2795px">V </span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1970px;top:2795px">!</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2073px;top:2795px">W </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2184px;top:2784px;word-spacing:7px">uma transformação linear, com </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:3237px;top:2795px">dim</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:3372px;top:2795px">V </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:3461px;top:2795px">=</span><span class="a" style="left:985px;top:2883px">dim</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1120px;top:2883px">W;</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1240px;top:2872px">e </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1307px;top:2883px"></span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1387px;top:2872px">e </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1454px;top:2883px"> </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1532px;top:2872px;word-spacing:10px">bases de </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1830px;top:2883px">V </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1922px;top:2872px">e </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1989px;top:2883px">W </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2101px;top:2872px;letter-spacing:-1px">respectivamente </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2601px;top:2883px">:</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2654px;top:2872px">Então</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2877px;top:2883px">T </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2963px;top:2872px;word-spacing:3px;letter-spacing:2px">é inversível se, e </span><span class="a" style="left:985px;top:2960px;word-spacing:1px;letter-spacing:1px">somente se det </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1484px;top:2971px">[</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1505px;top:2971px">T </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1558px;top:2971px">]</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:1578px;top:2953px"></span><span class="a" style="left:1578px;top:3012px"></span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1640px;top:2971px">6</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1640px;top:2971px">= 0</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1755px;top:2971px">:</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:3121px;word-spacing:6px">Observação 158</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1595px;top:3122px;word-spacing:6px;letter-spacing:1px">Se na proposição acima tivermos </span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3133px"><span class="a" style="left:2724px">V </span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:2820px;top:3133px">=</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2914px;top:3133px">W </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:3030px;top:3133px">=</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:3124px;top:3137px">R</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:3177px;top:3124px">n</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:3252px;top:3122px;letter-spacing:-2px">podemos </span><span class="a" style="left:985px;top:3210px;word-spacing:12px">escrever: Seja </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1462px;top:3221px">T </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1537px;top:3221px">:</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:1580px;top:3225px">R</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:1633px;top:3212px">n</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1694px;top:3221px">!</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:1790px;top:3225px">R</span></div>
<div class="ff15" style="font-size:48px">
<span class="a" style="left:1843px;top:3212px">n</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1910px;top:3210px;word-spacing:3px">uma transformação linear, então</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2991px;top:3221px">T </span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:3071px;top:3210px;word-spacing:1px;letter-spacing:1px">é invertível se </span></div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3309px"><span class="a" style="left:985px;letter-spacing:4px">det[</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3309px"><span class="a" style="left:1120px">T </span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1173px;top:3309px">]</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1214px;top:3309px">6</span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1214px;top:3309px">= 0</span></div>
<div class="ff7" style="font-size:85px">
<span class="a" style="left:985px;top:3459px;word-spacing:10px">Exemplo 159</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:1498px;top:3460px">Seja </span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:1660px;top:3470px">T </span></div>
<div class="ff17" style="font-size:74px">
<span class="a" style="left:1735px;top:3470px">:</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:1777px;top:3474px">R</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:1830px;top:3463px">3</span></div>
<div class="ff9" style="font-size:126px">
<span class="a" style="left:1884px;top:3470px">!</span></div>
<div class="ff5" style="font-size:66px">
<span class="a" style="left:1979px;top:3474px">R</span></div>
<div class="ff0" style="font-size:43px">
<span class="a" style="left:2032px;top:3463px">3</span></div>
<div class="ff12" style="font-size:74px">
<span class="a" style="left:2065px;top:3470px">;</span></div>
<div class="ff16" style="font-size:85px">
<span class="a" style="left:2113px;top:3460px;word-spacing:3px">dada por </span></div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3470px"><span class="a" style="left:2446px">T </span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3470px"><span class="a" style="left:2499px">(</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3470px"><span class="a" style="left:2527px;letter-spacing:7px">x;y;z</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3470px"><span class="a" style="left:2710px">) <span class="w7"></span>= (</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3470px"><span class="a" style="left:2868px">x</span></span>
</div>
<div class="ff17" style="font-size:74px">
<span class="g" style="top:3470px"><span class="a" style="left:2926px">+ 2</span></span>
</div>
<div class="ff12" style="font-size:74px">
<span class="g" style="top:3470px"><span class="a" style="left:3037px">y</span></span>
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<span class="a" style="left:1501px;top:3548px;word-spacing:4px">determine a transformação inversa </span></div>
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<span class="a" style="left:1095px;top:3709px;word-spacing:5px">Solução: Facilmente podemos ver que</span></div>
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<span class="a" style="left:1496px;top:3870px">2</span><span class="a" style="left:1496px;top:4002px">4</span></div>
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<span class="a" style="left:2736px;top:4180px;word-spacing:2px">Como exercício veri…que</span><span class="a" style="left:985px;top:4268px;word-spacing:2px">que vale</span></div>
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<span class="a" style="left:1816px;top:4279px">) = <span class="w10"></span>(</span></div>
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<span class="a" style="left:2155px;top:4279px">)</span></div>
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<span class="a" style="left:1095px;top:4356px;word-spacing:-4px">Podemos também neste caso <span class="w6"></span>calcular a inversa usando diretamente a di…nição</span><span class="a" style="left:985px;top:4444px;word-spacing:3px">de transformação inversa da seguinte forma</span><span class="a" style="left:1095px;top:4533px;word-spacing:2px">Sabemos que</span></div>
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LINEARES</li><li>3.1 Propriedades das Transformações Lineares</li><li>3.2 Transformações Lineares e Matrizes</li><li>3.2.1 Transformação linear associada a uma matriz</li><li>3.2.2 Matriz de uma transformação linear</li><li><a href="http://pt.scribd.com/doc/36872047/37/Composicao-de-transformacoes-lineares?sh=e86b212a1a565291" data-first-page="80" data-section="37">3.3 Composição de transformações lineares</a></li><li><a href="http://pt.scribd.com/doc/36872047/38/A-Inversa-de-uma-transformacao-linear?sh=b55dba19955b05fc" data-first-page="80" data-section="38">3.4 A Inversa de uma transformação linear</a></li><li>3.5 Nona lista de exercícios</li><li>4.1 Transformações especiais no plano e no es-
paço</li><li>4.2 Propriedades dos operadores inversíveis</li><li>4.2.1 Matrizes Semelhantes</li><li>4.3 Operadores autoadjuntos e ortogonais</li><li>4.4 Décima lista de exercícios</li><li>4.5 Autovalores e Autovetores</li><li>4.5.1 Autovalores e autovetores de uma matriz</li><li>4.5.2 Polinômio Característico</li><li>4.6 Décima primeira lista de exercícios</li><li>APLICAÇÕES</li><li>5.1 Aplicações da Álgebra Linear na Engenharia
Cartográ…ca</li><li>5.2 Aplicações de espaços vetoriais na computação
grá…ca</li><li>5.3 Aplicações de autovalores e autovetores na
engenharia civil</li></ul></div></div><div class="autogen_class_views_read_document_info" id="autogen_id_447531259"><div class="document_info"><h2>Informação e Avaliação</h2><table><tbody><tr><td class="row_title">Categoria:</td><td><em>Sem categoria</em></td></tr><tr><td class="row_title">Classificação:</td><td><div class="value" id="rating"><span id="document_starbox" class="starbox"><div class="positioner"><div class="stars"><div class="ghost" style="display: none; width: 34px;"></div><div class="colorbar" style="width: 34px;"></div><div class="star_holder"><div class="star star-0"></div><div class="star star-1"></div><div class="star star-2"></div><div class="star star-3"></div><div class="star star-4"></div></div></div></div></span><span class="count" itemprop="aggregateRating">(<span class="notranslate">1</span> Rating)</span></div></td></tr><tr><td class="row_title">Data de envio:</td><td itemprop="dateCreated">09/04/2010</td></tr><tr><td class="row_title">Direitos Autorais:</td><td>Attribution Non-commercial</td></tr><tr><td class="row_title">Tags:</td><td><div class="autogen_class_views_read_tags" id="autogen_id_447530803"><em>Este documento não tem etiquetas.</em><script type="text/javascript">
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