Baixe Расчет матриц умножения и их детерминантов e outras Notas de estudo em PDF para Engenharia Química, somente na Docsity! FUNDAMENTOS DE
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Versión 0.1
Andrés Alfonso Caro
Cesar Valero Sepúlveda Linux
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6%7 ! F ' D 24%7 © 6)>6V* > /! ! W .EE ' ! ' W D #X 0 (% ! ! ' Y' .EE 6 *>% 9""000V /" 9""000 ! /! V ! 0 ! ! # * X ! F 9 3 $ $ ! ! D ! ! D ! ' ! ! ! $ /! $ ! % #X 6%7 9 $ ! ! F 4 ! ' (% D ! X F 9 W D % ' /! ' #! ' 0 ! /! ! ! %6)*: 00 % ' ' ! /! F .< 2 , ! $ ! ' 4 F ! ' ! ' ! ! W F 2 ! $ ! 3 ! $ ! 6%7 ! X /! ! ! 0 % ' Y 6%7 < ! ' ()*"%6)*: D /! ! F ;0 #! ! :# ! ! ( % ' %43: ! %8: %6)*: ,. & F ! /! ' 6 6D %! ! 6 %! (! ('Y ! X 6%7 ! ! ! 2! J >Y ! X ! ! ! ; W D ! /! (F 5&&( ! %) $ ! ! ! D /!X # ! < ' # ! ! ! ' +>3&23R*F- /! D ! !! ! $ ! 9 " $ 6%7 ! ()*"%6)*: % ' . F $ ! % ' 5 ;6)&5; $ ! 6%7 ;6)&5; ! ! -.- Z & ! Z ' / /! ! #! .. ,! ..9 ! % ! Z ' 0 ! ' / /! ! #! . /! ! /! 6%7 /!D ,! . 9 ' ! ' ! ! /! ! ! 12& ' 23 -.- ! Z / &! /! ! /! /! ! ! ' ! / D ! /! /! D $ ! 6%7 /! 6%7 /!D % ! ! ! /! ! ' Z / ' ! /! ! /! /! *%$% 6%7 ! ' ' ! 6%7 ;6)&5; Z , ! $ ! 6%7 ;6)&5; F ' !" #$ &! 6%7 ! $ ! 9 D# ()523 [&3 * +,- %4 ' ! * + $ ()523 * + [&3 ' ! Z ' %4 ! #! . ,! .9 $ ! ' $ ! 6%7 ! F ! 9 4 ! $ ! ! D X ! %6)*: ! 9 = A V9 9 V3 9 ! +0 ! /! ! 6%7 "1$%# 2%(3 4 ! D# F ! ' 5+/1 Z ! ! ! ! ! /!W ! /! + %!, 3 ' ! ! ! ' !! ! FX ! F $ 6%7 $ $ Y$ /! ! D 3 6%7 ' /! ' /! ! 1 F /! ! $! ! "" X9 , ' ! ! /! + & - ! /! ! @ % E , 8 ,&8 & 8 3%#"%0%( 067#*/ 6 0%(7#*/ ,&8 & 8 ,&. & . 6%7 ! ! ! /! ! A@ 9 & 8 ! /! /! ! ! BCD # /! /! $ ! ,&89 , ! $ /! ! /! ' % , & . /! ! . *0 ,: & 3- +, , & - " 3*#, ,; & 8 %&%%8( , & .3<- 8. 5 ' ! Y 6%7 ! ! = -. 0 ! ! ! X BE\ BFF\ X9 ,> & 3-8 ,;;- & .< *(#%8( 0(3 9+01/ ,C7&1 2 ,C6&1 2 ,C/& 1 2 ,C& 1 2 ,C$&1 2 ,C&1 2 ,C/!& !1 2 ! .9 ! 6%7 1 (*+0(+"* "*")0"* 6%7 Y X $ +G- ! # 6%7 9G )HIJ XY ! V.+ √−1- ! $ /! G +π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−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 COSENO EJE X coseno de x ,! 9 (D# ! ' ! ! ! ! ! 9 ,&F3'3'3G9 , 1A 12AI*M* NIAI IAI/(7* (I2 < /! ! D ! ' 1 6%7 ! ' ! # ! ' D ! ! $9 ! '9 nombrefuncion = sen(x) + cos(y) + xy2 −√48x ! 9 AQ (A , 1HF6G& 1A02HAH6&12: 102:;0;;-?1< ;;2H2 , 1<A2 & --4.< L9L LL L(L L L /! ! ' 24
CAPÍTULO 2. PRIMEROS PASOS
) -!& ! X ! ! /! ' ! 6%7 ! ! ! $ ! !Y X ! 3 X ! I % ! $ 6%7 /! D X ! ! /! $ ! 3"+,"* $ X ' ! D ! ' ! ! F ! ! $ /! D ! ' /! ! ! ! ' !Y Y ! & ! : # ! # ! $! ! 9 (x1, x2, x3 · · · xn) = A ,&F3'3'8G9 , &F< < . G9 ,: & = 8 8 . 33 3 = /! ! W ! ,&F< 8 G & = < 8 = ,&F3989.G & = 3 = = 8 = = . = ,: = 3 ! ! ! # ! ! 3 ! ! ! R Z /! ! R9 ka = (ka1, ka2, · · · , kan) 3$9 ,&F< 8 G & = < 8 = ,8 & = - -8 = E @ 3 ! ! F /! ! # ! ! ! ! ! * 6%7 ^_OP A ?Q1 ^_OP 1?1AQ1 ^_` O . B /! ! /! ! ! ! ! ! 9 # ` ! = ! ` # ,; & 3< ,; & = 3 3- = = -< < = = - < < = A 3 ! !Y ! ! ' # ' ! ' 3 6%79 ,&F< 8 4G9 , &F. 3 -G9 ,; & = - 8 3 = < : ! ‖a‖ ! D F 1 ! # ' D 9 a = [a1, a2] ‖a‖ = √ a21 + a 2 2 ! Y ! 9 L H& JL 3$9 ,&F< 8 4G9 , 12 & 33<83 : ! ! ! ! ! ! ! ! ! 8 ' ! 3 6%79 ,&F< 3 G9 ,&1 122 & = . <<<8 34333 8 < < = ! Y & ! 9 , 12 & 3 +,* ",0 ,"* ! ! ! ! ' ! 4 /! ! /! ! << = . 8 = % Y ! F D 9 ,& 01AHH29 ,&F<AA.G9 &F3;;-A8A;;-G9 &F.A8AG9 ,)&F9 9G ) & = -8 . = = = = 3 3 = = = = - = = 8 = = = = -= = 3 3 = = = = . 8 = = = = 3 3 3 = $ Y9 M = 0,25x 1 6 x 7x 1 x2 1 5x 1 3 x2 7 1 5 1 3 1 6! /! ! ! 9 ,&F<AA.G9 &F3A8AG9 &F.A8AG9 ,)13A2& ) & = < = = 3 8 = = . 8 = # . ! < N@N N N < ",0)("* 6*)0*9 A # ! ! # X Y $AM 4N # 4 ! /! F $KAM 4N 4 # ! 3$ 6%79 ,) ) & = < . = = 3 8 = = . 8 = ! Y $ ! ! $K $L ,)H & = < 3 . = = 8 8 = = . = B ! ! ! ! Y ! ' ! /! ! ! ' 3$9 AM 4N SAM 4N 1AMH + J4N 3 6%79 . ,&F<AA.G9 &F3A8AG9 ,: & = 8 33 3 = <= , & = 3 < = . ,&F @ -9 <. 8G9 ,D&F- .<9 .4 G9 ,D&:D D & = 8.<384-. 3 = = 3- 38 = % ! ' ! Y $ ! > ! ! Y , ! 9 Fnxm = kMnxm, donde fij = kmij 3$ 6%79 ,)&F< 89 < . 9 3 .G9 ,O&9 ,E&O;) E & = 3- -< 38 = = 3- -3 4 = = -3 = ! 7 /! /! ! Y NN ! # ! Y N7N 3 !V ' ! Y ! ! # ! ! ! 7 ! D ! ! F' <A -- & = < 4 = = = , -&F< 89 G - & = < 8 = = = , 3&F8 49 < G 3 & = 8 4 = = < = , -&F< 49 . G - & = < 4 = = . = , &F< 89 . <G & = < 8 = = . < = 4 ! Cij (Cij)T * 6%7 Cij ,33& 1 332;132;;- 33 & ,3-& 1 3-2;132;; 3- & ,3& 1 32;132;;< 3 & 3 ,-3& 1 -32;132;; -3 & <E 4 ,--& 1 --2;132;;< -- & ,-& 1 -2;132;;8 - & 3 ,3& 1 32;132;;< 3 & -3 ,-& 1 -2;132;;8 - & 83 ,& 1 2;132;; & 34 /J&F33 3- 39 -3 -- -9 3 - G /J & = 3 = = 4 3 = = -3 83 34 = * 6%7 $ (Cij)T ,/J$&/JH /J$ & = 4 -3 = = 83 = = 3 3 34 = * 6%7 (Cij ) T det A MainvA = CijT de 9 ,)&/J$ ) & = .<.<3 < 3< 38 -84-84 = = .. .<.<. -4-4 = = 3-<8 34.84 -<8.4 = 3 6%7 Y ! ! ! Y ! ' L&L $ 9 L&H ' JL 3$9 ,&F< 8 49. < 9 G9 ,D&12 D & = .<.<3 < 3< 38 -84-84 = = .. .<.<. -4-4 = = 3-<8 34.84 -<8.4 = ,/&1;D2 & = 3 = = 3 = = 3 = , &1D;2 & = 3 = = 3 = = 3 = $ Y A−1 = B ! V Y S ! ! Y & < 4 ) ' ' ' ' ' < T ' & ( A % Y 6 /! V Y ! ! ! /! D In 3$9 In = 1 0 00 1 0 0 0 1 < 3 6%7 ! Y ! Y ! ' L RHJL 3$ 6%79 Y V ,M&F3 - 39- 9 3 8G M & = 3 - 3 = = - = = 3 8 = ,"1M2 & - A Y A = [ a11 a12 a21 a22 ] H- -J Y 9 det A = |A| = a11a22 − a21a12 |A| 6%7 ! Y ! ' L'H JL $ 3 6%79 ,&F< 4 .9 3 8 49 . 8 -G9 , 12 & 344 * $ 9 ,D&F< 4 9 8 -G9 , 1D2 & . A /! ! Y ! ! Y ! 3 6%7 ! Y ! ! ' L 'H JL ! ! ! ! Y ! ! 3$ 6%7 .- , 7!A 5 , &F3 -9- 89 . 3- <G9 , 1 2 & 3--*:3 , 1 2 & 44*38 - , !A 5 ,E&F3 -9 3- - 9- 3 - 893 3 G9 , 1E2 & 38 333 , 1E2 & 3 : 6 : Y ! H 4 J F ! & 1 /!9 &A& 3 & = 6%7 ! Y ! LH JL 3$ 6%79 , -- J J! ,&F 893 3G & = 8 = = 3 3 = , 12 = & = 3 : = = 3 = O +.a-+ .V- , ,&F3 3 <9 - 39 - 3 3G & = 3 3 < = = - 3 = = - 3 3 = , 12 & = = = 3 = = - = *')5( " "'0)("* )("0"* /%( /&+! ! ' & 6%7 ! ' ! ! ! ! ' 9 M< RNA &H< M<NJ Y na # ma ! ! na # ! ' R Y F R ! 4 ! & 9 F + A & ' ! ! ' ! D ! 3$9 9 2x + y − 2z = 10 3x + 2y + 2z = 1 5x + 4y + 3z = 4 ! LL 2x + y − 2z − 10 = 0 3x + 2y + 2z − 1 = 0 5x + 4y + 3z − 4 = 0 A # : ; B ! ! #! R5 E3 R1 R3 R2 R4 I3 I1 I2E1 E2 − + −+ − + ,! <.9 ! < * %[ +% $ [ b- Y $ −E1aE2 a+>.a> -i1 V> i2 V>.i3O −E2 V+> -i1 a+> a><a> -i2 V+><-i3O E3 V+>.-i1 V+><-i2 a+>.a><a>=-i3O 5 +>.a> -i1 V> i2 V>.i3 O E1 − E2 V+> -i1 a+> a><a> -i2 V+><-i3 O E2 V+>.-i1 V+><-i2 a+>.a><a>=-i3 O −E3 ! ! 9 R1 = 4Ω R2 = 8Ω R3 = 7Ω R4 = 9Ω E R5 = 2Ω E1 = 9V E2 = 6V E3 = 8V 5Y AS 12 −8 −4−8 24 7 4 −7 13 i1i2 i3 O 36 −8 3 6%79 ,6&F3- <9 -< .9< . 3GA C&F99 G 6 & = 3- < = = -< . = = < . 3 = C & = = = = = = ,5&162;C 5 & = -<< 3 = = -<< = = <- 48<< = ! '9 i1O ?<A. i2O ? < < i3O V AE= )) . % (*+,')5( " )(.)* 3 6%7 ! ! # ! ' L (L ! 9 = a = [1 2 3 7 8] ! ! ! /! ! ! # ! ! 1 + 2x + 3x2 + 7x3 + 8x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cN , ! ,< ; @ & < ,!3&-; @ !3 & - ,!-& @; @ !- & 3<384-. ( ! $ ! ! ! V z = x + yi ! $ ( (< F ! 9 =E Re(z) = x Im(z) = y % ' Y /!X 9 Re(z) Im(z) z=7+3i ,! < 9 >+Y- 6+Y- % $ A 9HJ $ ( A0HJ 3 #! < $ /! ! z = 7 + 3i Re(z) = 7 Im(z) = 3 4 ! ! $ ! ! 6%7 FV ! ! X9 , ! J ,3&3-8:; @ 3 & ? 3-8 : , 132 & 3-8 ,!132 & , J ,-& @:-; @; @ - & 3<384-. : - 3 8 , 1-2 & 3<384-. ,!1-2 & - 3 8 Y ! ! # D! /! $ ?< = - : - = = = , ,H & = - = = - = Y $! Y ! ! R +- , J! 0 , 3& J12 3 & = - - = = = , 3H & = - = = : - = , J! Q , Q12 & = - = = : - = * = : , J ,&3: @ & 3 : ,-&:; @ - & : ,&:- & < : < ? , J , &- & - : - = # : , J ,-&<:; @9 &:<; @9 , &-; & 3- : < , J , 3&- 3 & 4-.. : <38 : 3 X ' $ ! V ! ' ! ' F $ ejθ ejθ = cosθ + jsenθ , ! '! 3! ! 3! 6%7 ! ! ! /! D ! D! ! 5o 85o ?= Re(z) Im(z) r=2 r=5 r=8 θ=14 θ=45ο ο θ=74ο ,! < 9 ! /! D ! /! < !9 - @ D# ! < ! < ! D! ! ! ! /! D ! /! ! ! D ! ! X /! . # A ! ! .U ! , ! -A 8 0 ,&F-; 13A-.2A 8; 13A-.2A ; 13A-2G9 , 12 & = 3 = ! /! 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