Solutions to the exercises from T.M.Apostol - Calculus, vol.1

(Parte 1 de 3)

Solutions to the exercises from T.M.Apostol,

Calculus, vol. 1 assigned to doctoral students in years 2002-2003 andrea battinelli dipartimento di scienze matematiche e informatiche “R.Magari” dell’università di Siena via del Capitano 15 - 53100 Siena tel: +39-0577-233769/02 fax: /01/30 e-mail: battinelli @unisi.it web: http//w.batman vai li

Contents

12.1 Historical introduction	25
12.2 The vector space of n-tuples of real numbers	25
12.3 Geometric interpretation for n ≤ 3	25
12.4 Exercises	25
12.4.1 n. 1 (p. 450)	25
12.4.2 n. 2 (p. 450)	25
12.4.3 n. 3 (p. 450)	26
12.4.4 n. 4 (p. 450)	27
12.4.5 n. 5 (p. 450)	28

12.4.7 n. 7 (p. 451)	29
12.4.8 n. 8 (p. 451)	29
12.4.9 n. 9 (p. 451)	30
12.4.10 n. 10 (p. 451)	30
12.4.1 n. 1 (p. 451)	30
12.4.12 n. 12 (p. 451)	32
12.5 The dot product	3
12.6 Length or norm of a vector	3
12.7 Orthogonality of vectors	3
12.8 Exercises	3
12.8.1 n. 1 (p. 456)	3
12.8.2 n. 2 (p. 456)	3
12.8.3 n. 3 (p. 456)	34
12.8.4 n. 5 (p. 456)	34
12.8.5 n. 6 (p. 456)	34
12.8.6 n. 7 (p. 456)	35
12.8.7 n. 10 (p. 456)	36
12.8.8 n. 13 (p. 456)	36
12.8.9 n. 14 (p. 456)	37
12.8.10 n. 15 (p. 456)	39
12.8.1 n. 16 (p. 456)	39
12.8.12 n. 17 (p. 456)	39
12.8.13 n. 19 (p. 456)	40
12.8.14 n. 20 (p. 456)	40
12.8.15 n. 21 (p. 457)	40
12.8.16 n. 2 (p. 457)	41
12.8.17 n. 24 (p. 457)	42
12.8.18 n. 25 (p. 457)	42
12.9 Projections. Angle between vectors in n-space	43
12.10 The unit coordinate vectors	43
12.1 Exercises	43
12.1.1 n. 1 (p. 460)	43
12.1.2 n. 2 (p. 460)	43
12.1.3 n. 3 (p. 460)	43
12.1.4 n. 5 (p. 460)	4
12.1.5 n. 6 (p. 460)	45
12.1.6 n. 8 (p. 460)	46
12.1.7 n. 10 (p. 461)	46
12.1.8 n. 1 (p. 461)	47
12.1.9 n. 13 (p. 461)	48
12.1.10 n. 17 (p. 461)	48

4 CONTENTS 12.12 The linear span of a finite set of vectors .......... ...... 50

12.13 Linear independence	50
12.14 Bases	50
12.15 Exercises	50
12.15.1 n. 1 (p. 467)	50
12.15.2 n. 3 (p. 467)	50
12.15.3 n. 5 (p. 467)	51
12.15.4 n. 6 (p. 467)	51
12.15.5 n. 7 (p. 467)	51
12.15.6 n. 8 (p. 467)	51
12.15.7 n. 10 (p. 467)	52
12.15.8 n. 12 (p. 467)	53
12.15.9 n. 13 (p. 467)	5
12.15.10 n. 14 (p. 468)	56
12.15.1 n. 15 (p. 468)	56
12.15.12 n. 17 (p. 468)	56
12.15.13 n. 18 (p. 468)	57
12.15.14 n. 19 (p. 468)	57
12.15.15 n. 20 (p. 468)	58
12.16 The vector space Vn (C) of n-tuples of complex numbers	59
12.17 Exercises	59

CONTENTS 5

13.1 Introduction	61
13.2 Lines in n-space	61
13.3 Some simple properties of straight lines	61
13.4 Lines and vector-valued functions	61
13.5 Exercises	61
13.5.1 n. 1 (p. 477)	61
13.5.2 n. 2 (p. 477)	61
13.5.3 n. 3 (p. 477)	62
13.5.4 n. 4 (p. 477)	62
13.5.5 n. 5 (p. 477)	62
13.5.6 n. 6 (p. 477)	63
13.5.7 n. 7 (p. 477)	64
13.5.8 n. 8 (p. 477)	64
13.5.9 n. 9 (p. 477)	65
13.5.10 n. 10 (p. 477)	65
13.5.1 n. 1 (p. 477)	6
13.5.12 n. 12 (p. 477)	67
13.6 Planes in euclidean n-spaces	67
13.7 Planes and vector-valued functions	67

13 Applications of vector algebra to analytic geometry 61 13.8 Exercises ....... .......... ........... ...... 67

13.8.1 n. 2 (p. 482)	67
13.8.2 n. 3 (p. 482)	68
13.8.3 n. 4 (p. 482)	69
13.8.4 n. 5 (p. 482)	69
13.8.5 n. 6 (p. 482)	70
13.8.6 n. 7 (p. 482)	71
13.8.7 n. 8 (p. 482)	72
13.8.8 n. 9 (p. 482)	72
13.8.9 n. 10 (p. 483)	73
13.8.10 n. 1 (p. 483)	74
13.8.1 n. 12 (p. 483)	75
13.8.12 n. 13 (p. 483)	75
13.8.13 n. 14 (p. 483)	75
13.9 The cross product	76
13.10 The cross product expressed as a determinant	76
13.1 Exercises	76
13.1.1 n. 1 (p. 487)	76
13.1.2 n. 2 (p. 487)	76
13.1.3 n. 3 (p. 487)	76
13.1.4 n. 4 (p. 487)	7
13.1.5 n. 5 (p. 487)	7
13.1.6 n. 6 (p. 487)	7
13.1.7 n. 7 (p. 488)	78
13.1.8 n. 8 (p. 488)	79
13.1.9 n. 9 (p. 488)	79
13.1.10 n. 10 (p. 488)	80
13.1.1 n. 1 (p. 488)	80
13.1.12 n. 12 (p. 488)	82
13.1.13 n. 13 (p. 488)	82
13.1.14 n. 14 (p. 488)	83
13.1.15 n. 15 (p. 488)	84
13.12 The scalar triple product	85
13.13 Cramer’s rule for solving systems of three linear equations	85
13.14 Exercises	85
13.15 Normal vectors to planes	85
13.16 Linear cartesian equations for planes	85
13.17 Exercises	86
13.17.1 n. 1 (p. 496)	86
13.17.2 n. 2 (p. 496)	86
13.17.3 n. 3 (p. 496)	87
13.17.4 n. 4 (p. 496)	87

13.17.6 n. 6 (p. 496)	8
13.17.7 n. 8 (p. 496)	8
13.17.8 n. 9 (p. 496)	8
13.17.9 n. 10 (p. 496)	89
13.17.10 n. 1 (p. 496)	90
13.17.1 n. 13 (p. 496)	90
13.17.12 n. 14 (p. 496)	90
13.17.13 n. 15 (p. 496)	90
13.17.14 n. 17 (p. 497)	91
13.17.15 n. 20 (p. 497)	91
13.18 The conic sections	91
13.19 Eccentricity of conic sections	91
13.20 Polar equations for conic sections	91
13.21 Exercises	91
13.2 Conic sections symmetric about the origin	92
13.23 Cartesian equations for the conic sections	92
13.24 Exercises	92
13.25 Miscellaneous exercises on conic sections	92

CONTENTS 7 14 Calculus of vector-valued functions 93

15.1 Introduction	95
15.2 The definition of a linear space	95
15.3 Examples of linear spaces	95
15.4 Elementary consequences of the axioms	95
15.5 Exercises	95
15.5.1 n. 1 (p. 5)	95
15.5.2 n. 2 (p. 5)	96
15.5.3 n. 3 (p. 5)	96
15.5.4 n. 4 (p. 5)	96
15.5.5 n. 5 (p. 5)	97
15.5.6 n. 6 (p. 5)	97
15.5.7 n. 7 (p. 5)	98
15.5.8 n. 1 (p. 5)	98
15.5.9 n. 13 (p. 5)	98
15.5.10 n. 14 (p. 5)	9
15.5.1 n. 16 (p. 5)	9
15.5.12 n. 17 (p. 5)	101
15.5.13 n. 18 (p. 5)	101
15.5.14 n. 19 (p. 5)	102

15.5.16 n. 23 (p. 5)	102
15.5.17 n. 24 (p. 5)	102
15.5.18 n. 25 (p. 5)	102
15.5.19 n. 26 (p. 5)	102
15.5.20 n. 27 (p. 5)	103
15.5.21 n. 28 (p. 5)	103
15.6 Subspaces of a linear space	103
15.7 Dependent and independent sets in a linear space	103
15.8 Bases and dimension	103
15.9 Exercises	103
15.9.1 n. 1 (p. 560)	103
15.9.2 n. 2 (p. 560)	104
15.9.3 n. 3 (p. 560)	104
15.9.4 n. 4 (p. 560)	104
15.9.5 n. 5 (p. 560)	104
15.9.6 n. 6 (p. 560)	104
15.9.7 n. 7 (p. 560)	104
15.9.8 n. 8 (p. 560)	105
15.9.9 n. 9 (p. 560)	105
15.9.10 n. 10 (p. 560)	105
15.9.1 n. 1 (p. 560)	105
15.9.12 n. 12 (p. 560)	106
15.9.13 n. 13 (p. 560)	106
15.9.14 n. 14 (p. 560)	106
15.9.15 n. 15 (p. 560)	107
15.9.16 n. 16 (p. 560)	107
15.9.17 n. 2 (p. 560)	108
15.9.18 n. 23 (p. 560)	108
15.10 Inner products. Euclidean spaces. Norms	1
15.1 Orthogonality in a euclidean space	11
15.12 Exercises	112
15.12.1 n. 9 (p. 567)	112
15.12.2 n. 1 (p. 567)	12
15.13 Construction of orthogonal sets. The Gram-Schmidt process	115
15.14 Orthogonal complements. projections	115

finite-dimensional subspace	115
15.16 Exercises	115
15.16.1 n. 1 (p. 576)	115
15.16.2 n. 2 (p. 576)	116
15.16.3 n. 3 (p. 576)	117

CONTENTS 9

16.1 Linear transformations	121
16.2 Null space and range	121
16.3 Nullity and rank	121
16.4 Exercises	121
16.4.1 n. 1 (p. 582)	121
16.4.2 n. 2 (p. 582)	122
16.4.3 n. 3 (p. 582)	122
16.4.4 n. 4 (p. 582)	122
16.4.5 n. 5 (p. 582)	123
16.4.6 n. 6 (p. 582)	123
16.4.7 n. 7 (p. 582)	123
16.4.8 n. 8 (p. 582)	123
16.4.9 n. 9 (p. 582)	124
16.4.10 n. 10 (p. 582)	124
16.4.1 n. 16 (p. 582)	125
16.4.12 n. 17 (p. 582)	125
16.4.13 n. 23 (p. 582)	126
16.4.14 n. 25 (p. 582)	126
16.4.15 n. 27 (p. 582)	127
16.5 Algebraic operations on linear transformations	128
16.6 Inverses	128
16.7 One-to-one linear transformations	128
16.8 Exercises	128
16.8.1 n. 15 (p. 589)	128
16.8.2 n. 16 (p. 589)	128
16.8.3 n. 17 (p. 589)	128
16.8.4 n. 27 (p. 590)	129
16.9 Linear transformations with prescribed values	129
16.10 Matrix representations of linear transformations	129
16.1 Construction of a matrix representation in diagonal form	129
16.12 Exercises	129
16.12.1 n. 3 (p. 596)	129
16.12.2 n. 4 (p. 596)	131
16.12.3 n. 5 (p. 596)	131
16.12.4 n. 7 (p. 597)	132
16.12.5 n. 8 (p. 597)	133

10 CONTENTS 10 CONTENTS

Part I Volume 1

Chapter 1 CHAPTER 1

4 Chapter 1 4 Chapter 1

Chapter 2 CHAPTER 2

6 Chapter 2 6 Chapter 2

Chapter 3 CHAPTER 3

8 Chapter 3 8 Chapter 3

Chapter 4 CHAPTER 4

10 Chapter 4 10 Chapter 4

Chapter 5 CHAPTER 5

12 Chapter 5 12 Chapter 5

Chapter 6 CHAPTER 6

14 Chapter 6 14 Chapter 6

Chapter 7 CHAPTER 7

16 Chapter 7 16 Chapter 7

Chapter 8 CHAPTER 8

18 Chapter 8 18 Chapter 8

Chapter 9 CHAPTER 9

20 Chapter 9 20 Chapter 9

Chapter 10 CHAPTER 10

Chapter 1 CHAPTER 1

Chapter 12 VECTORALGEBRA

12.1 Historical introduction

12.2 The vector space of n-tuples of real numbers

12.3 Geometric interpretation for n ≤ 3

The seven points to be drawn are the following:µ 7

The purpose of the exercise is achieved by drawing, as required, a single picture, containing all the points (included the starting points A and B,I would say).

26 Vector algebra

It can be intuitively seen that, by letting t vary in all R, the straight line through point A with direction given by the vector b ≡ −→ OB is obtained.

The seven points this time are the following:µ 5

It can be intuitively seen that, by letting t vary in all R, the straight line through B with direction given by the vector a ≡ −→ OA is obtained.

Exercises 27

12.4.4 n. 4 (p. 450) (a) The seven points to be drawn are the following:

The whole purpose of this part of the exercise is achieved by drawing a single picture, containing all the points (included the starting points A and B,I would say). This is made clear, it seems to me, by the question immediately following.

(b) It is hard not to notice that all points belong to the same straight line; indeed, as it is going to be clear after the second lecture, all the combinations are affine.

(c)I f the value of x is fixed at 0 and y varies in [0,1],t he segment OB is obtained; the same construction with the value of x fixed at 1 yields the segment AD,w here−→ OD = −→

OA+−→ OB, and hence D is the vertex of the parallelogram with three vertices at O, A,a nd B. Similarly, when x = 12 the segment obtained joins the midpoints of the two sides OA and BD; and it is enough to repeat the construction a few more times to convince oneself that the set

is matched by the set of all points of the parallelogram OADB. The picture below

28 Vector algebra

(d) All the segments in the above construction are substituted by straight lines, and the resulting set is the (infinite) stripe bounded by the lines containing the sides OB and AD,o fe quation 3x − y =0 and 3x − y =5 respectively. (e) The whole plane.

Exercises 29

(Parte 1 de 3)