Exercises and Problems in Linear Algebra

[Pages:196]Exercises and Problems in Linear Algebra John M. Erdman

Portland State University Version July 13, 2014 c 2010 John M. Erdman

E-mail address: erdman@pdx.edu

Contents

PREFACE

vii

Part 1. MATRICES AND LINEAR EQUATIONS

1

Chapter 1. SYSTEMS OF LINEAR EQUATIONS

3

1.1. Background

3

1.2. Exercises

4

1.3. Problems

7

1.4. Answers to Odd-Numbered Exercises

8

Chapter 2. ARITHMETIC OF MATRICES

9

2.1. Background

9

2.2. Exercises

10

2.3. Problems

12

2.4. Answers to Odd-Numbered Exercises

14

Chapter 3. ELEMENTARY MATRICES; DETERMINANTS

15

3.1. Background

15

3.2. Exercises

17

3.3. Problems

22

3.4. Answers to Odd-Numbered Exercises

23

Chapter 4. VECTOR GEOMETRY IN Rn

25

4.1. Background

25

4.2. Exercises

26

4.3. Problems

28

4.4. Answers to Odd-Numbered Exercises

29

Part 2. VECTOR SPACES

31

Chapter 5. VECTOR SPACES

33

5.1. Background

33

5.2. Exercises

34

5.3. Problems

37

5.4. Answers to Odd-Numbered Exercises

38

Chapter 6. SUBSPACES

39

6.1. Background

39

6.2. Exercises

40

6.3. Problems

44

6.4. Answers to Odd-Numbered Exercises

45

Chapter 7. LINEAR INDEPENDENCE

47

7.1. Background

47

7.2. Exercises

49

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CONTENTS

7.3. Problems

51

7.4. Answers to Odd-Numbered Exercises

53

Chapter 8. BASIS FOR A VECTOR SPACE

55

8.1. Background

55

8.2. Exercises

56

8.3. Problems

57

8.4. Answers to Odd-Numbered Exercises

58

Part 3. LINEAR MAPS BETWEEN VECTOR SPACES

59

Chapter 9. LINEARITY

61

9.1. Background

61

9.2. Exercises

63

9.3. Problems

67

9.4. Answers to Odd-Numbered Exercises

70

Chapter 10. LINEAR MAPS BETWEEN EUCLIDEAN SPACES

71

10.1. Background

71

10.2. Exercises

72

10.3. Problems

74

10.4. Answers to Odd-Numbered Exercises

75

Chapter 11. PROJECTION OPERATORS

77

11.1. Background

77

11.2. Exercises

78

11.3. Problems

79

11.4. Answers to Odd-Numbered Exercises

80

Part 4. SPECTRAL THEORY OF VECTOR SPACES

81

Chapter 12. EIGENVALUES AND EIGENVECTORS

83

12.1. Background

83

12.2. Exercises

84

12.3. Problems

85

12.4. Answers to Odd-Numbered Exercises

86

Chapter 13. DIAGONALIZATION OF MATRICES

87

13.1. Background

87

13.2. Exercises

89

13.3. Problems

91

13.4. Answers to Odd-Numbered Exercises

92

Chapter 14. SPECTRAL THEOREM FOR VECTOR SPACES

93

14.1. Background

93

14.2. Exercises

94

14.3. Answers to Odd-Numbered Exercises

96

Chapter 15. SOME APPLICATIONS OF THE SPECTRAL THEOREM

97

15.1. Background

97

15.2. Exercises

98

15.3. Problems

102

15.4. Answers to Odd-Numbered Exercises

103

Chapter 16. EVERY OPERATOR IS DIAGONALIZABLE PLUS NILPOTENT

105

CONTENTS

v

16.1. Background

105

16.2. Exercises

106

16.3. Problems

110

16.4. Answers to Odd-Numbered Exercises

111

Part 5. THE GEOMETRY OF INNER PRODUCT SPACES

113

Chapter 17. COMPLEX ARITHMETIC

115

17.1. Background

115

17.2. Exercises

116

17.3. Problems

118

17.4. Answers to Odd-Numbered Exercises

119

Chapter 18. REAL AND COMPLEX INNER PRODUCT SPACES

121

18.1. Background

121

18.2. Exercises

123

18.3. Problems

125

18.4. Answers to Odd-Numbered Exercises

126

Chapter 19. ORTHONORMAL SETS OF VECTORS

127

19.1. Background

127

19.2. Exercises

128

19.3. Problems

129

19.4. Answers to Odd-Numbered Exercises

131

Chapter 20. QUADRATIC FORMS

133

20.1. Background

133

20.2. Exercises

134

20.3. Problems

136

20.4. Answers to Odd-Numbered Exercises

137

Chapter 21. OPTIMIZATION

139

21.1. Background

139

21.2. Exercises

140

21.3. Problems

141

21.4. Answers to Odd-Numbered Exercises

142

Part 6. ADJOINT OPERATORS

143

Chapter 22. ADJOINTS AND TRANSPOSES

145

22.1. Background

145

22.2. Exercises

146

22.3. Problems

147

22.4. Answers to Odd-Numbered Exercises

148

Chapter 23. THE FOUR FUNDAMENTAL SUBSPACES

149

23.1. Background

149

23.2. Exercises

151

23.3. Problems

155

23.4. Answers to Odd-Numbered Exercises

157

Chapter 24. ORTHOGONAL PROJECTIONS

159

24.1. Background

159

24.2. Exercises

160

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CONTENTS

24.3. Problems

163

24.4. Answers to Odd-Numbered Exercises

164

Chapter 25. LEAST SQUARES APPROXIMATION

165

25.1. Background

165

25.2. Exercises

166

25.3. Problems

167

25.4. Answers to Odd-Numbered Exercises

168

Part 7. SPECTRAL THEORY OF INNER PRODUCT SPACES

169

Chapter 26. SPECTRAL THEOREM FOR REAL INNER PRODUCT SPACES

171

26.1. Background

171

26.2. Exercises

172

26.3. Problem

174

26.4. Answers to the Odd-Numbered Exercise

175

Chapter 27. SPECTRAL THEOREM FOR COMPLEX INNER PRODUCT SPACES

177

27.1. Background

177

27.2. Exercises

178

27.3. Problems

181

27.4. Answers to Odd-Numbered Exercises

182

Bibliography

183

Index

185

PREFACE

This collection of exercises is designed to provide a framework for discussion in a junior level linear algebra class such as the one I have conducted fairly regularly at Portland State University. There is no assigned text. Students are free to choose their own sources of information. Students are encouraged to find books, papers, and web sites whose writing style they find congenial, whose emphasis matches their interests, and whose price fits their budgets. The short introductory background section in these exercises, which precede each assignment, are intended only to fix notation and provide "official" definitions and statements of important theorems for the exercises and problems which follow.

There are a number of excellent online texts which are available free of charge. Among the best are Linear Algebra [7] by Jim Hefferon,

and A First Course in Linear Algebra [2] by Robert A. Beezer,

Another very useful online resource is Przemyslaw Bogacki's Linear Algebra Toolkit [3].

And, of course, many topics in linear algebra are discussed with varying degrees of thoroughness in the Wikipedia [12]

and Eric Weisstein's Mathworld [11].

Among the dozens and dozens of linear algebra books that have appeared, two that were written before "dumbing down" of textbooks became fashionable are especially notable, in my opinion, for the clarity of their authors' mathematical vision: Paul Halmos's Finite-Dimensional Vector Spaces [6] and Hoffman and Kunze's Linear Algebra [8]. Some students, especially mathematically inclined ones, love these books, but others find them hard to read. If you are trying seriously to learn the subject, give them a look when you have the chance. Another excellent traditional text is Linear Algebra: An Introductory Approach [5] by Charles W. Curits. And for those more interested in applications both Elementary Linear Algebra: Applications Version [1] by Howard Anton and Chris Rorres and Linear Algebra and its Applications [10] by Gilbert Strang are loaded with applications. If you are a student and find the level at which many of the current beginning linear algebra texts are written depressingly pedestrian and the endless routine computations irritating, you might examine some of the more advanced texts. Two excellent ones are Steven Roman's Advanced Linear Algebra [9] and William C. Brown's A Second Course in Linear Algebra [4]. Concerning the material in these notes, I make no claims of originality. While I have dreamed up many of the items included here, there are many others which are standard linear algebra exercises that can be traced back, in one form or another, through generations of linear algebra texts, making any serious attempt at proper attribution quite futile. If anyone feels slighted, please contact me. There will surely be errors. I will be delighted to receive corrections, suggestions, or criticism at

vii

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PREFACE

erdman@pdx.edu

I have placed the the LATEX source files on my web page so that those who wish to use these exercises for homework assignments, examinations, or any other noncommercial purpose can download the material and, without having to retype everything, edit it and supplement it as they wish.

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