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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

PREFACEvii

Part 1. MATRICES AND LINEAR EQUATIONS1

Chapter 1. SYSTEMS OF LINEAR EQUATIONS

3

1.1. Background3

1.2. Exercises4

1.3. Problems7

1.4. Answers to Odd-Numbered Exercises

8

Chapter 2. ARITHMETIC OF MATRICES

9

2.1. Background9

2.2. Exercises10

2.3. Problems12

2.4. Answers to Odd-Numbered Exercises

14

Chapter 3. ELEMENTARY MATRICES; DETERMINANTS

15

3.1. Background15

3.2. Exercises17

3.3. Problems22

3.4. Answers to Odd-Numbered Exercises

23

Chapter 4. VECTOR GEOMETRY INRn25

4.1. Background25

4.2. Exercises26

4.3. Problems28

4.4. Answers to Odd-Numbered Exercises

29

Part 2. VECTOR SPACES31

Chapter 5. VECTOR SPACES

33

5.1. Background33

5.2. Exercises34

5.3. Problems37

5.4. Answers to Odd-Numbered Exercises

38

Chapter 6. SUBSPACES

39

6.1. Background39

6.2. Exercises40

6.3. Problems44

6.4. Answers to Odd-Numbered Exercises

45

Chapter 7. LINEAR INDEPENDENCE

47

7.1. Background47

7.2. Exercises49

iii iv CONTENTS

7.3. Problems51

7.4. Answers to Odd-Numbered Exercises

53

Chapter 8. BASIS FOR A VECTOR SPACE

55

8.1. Background55

8.2. Exercises56

8.3. Problems57

8.4. Answers to Odd-Numbered Exercises

58

Part 3. LINEAR MAPS BETWEEN VECTOR SPACES59

Chapter 9. LINEARITY

61

9.1. Background61

9.2. Exercises63

9.3. Problems67

9.4. Answers to Odd-Numbered Exercises

70

Chapter 10. LINEAR MAPS BETWEEN EUCLIDEAN SPACES

71

10.1. Background

71

10.2. Exercises72

10.3. Problems74

10.4. Answers to Odd-Numbered Exercises

75

Chapter 11. PROJECTION OPERATORS

77

11.1. Background

77

11.2. Exercises78

11.3. Problems79

11.4. Answers to Odd-Numbered Exercises

80

Part 4. SPECTRAL THEORY OF VECTOR SPACES81

Chapter 12. EIGENVALUES AND EIGENVECTORS

83

12.1. Background

83

12.2. Exercises84

12.3. Problems85

12.4. Answers to Odd-Numbered Exercises

86

Chapter 13. DIAGONALIZATION OF MATRICES

87

13.1. Background

87

13.2. Exercises89

13.3. Problems91

13.4. Answers to Odd-Numbered Exercises

92

Chapter 14. SPECTRAL THEOREM FOR VECTOR SPACES

93

14.1. Background

93

14.2. Exercises94

14.3. Answers to Odd-Numbered Exercises

96
Chapter 15. SOME APPLICATIONS OF THE SPECTRAL THEOREM 97

15.1. Background

97

15.2. Exercises98

15.3. Problems102

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. Exercises106

16.3. Problems110

16.4. Answers to Odd-Numbered Exercises

111

Part 5. THE GEOMETRY OF INNER PRODUCT SPACES113

Chapter 17. COMPLEX ARITHMETIC

115

17.1. Background

115

17.2. Exercises116

17.3. Problems118

17.4. Answers to Odd-Numbered Exercises

119

Chapter 18. REAL AND COMPLEX INNER PRODUCT SPACES

121

18.1. Background

121

18.2. Exercises123

18.3. Problems125

18.4. Answers to Odd-Numbered Exercises

126

Chapter 19. ORTHONORMAL SETS OF VECTORS

127

19.1. Background

127

19.2. Exercises128

19.3. Problems129

19.4. Answers to Odd-Numbered Exercises

131

Chapter 20. QUADRATIC FORMS

133

20.1. Background

133

20.2. Exercises134

20.3. Problems136

20.4. Answers to Odd-Numbered Exercises

137

Chapter 21. OPTIMIZATION

139

21.1. Background

139

21.2. Exercises140

21.3. Problems141

21.4. Answers to Odd-Numbered Exercises

142

Part 6. ADJOINT OPERATORS143

Chapter 22. ADJOINTS AND TRANSPOSES

145

22.1. Background

145

22.2. Exercises146

22.3. Problems147

22.4. Answers to Odd-Numbered Exercises

148

Chapter 23. THE FOUR FUNDAMENTAL SUBSPACES

149

23.1. Background

149

23.2. Exercises151

23.3. Problems155

23.4. Answers to Odd-Numbered Exercises

157

Chapter 24. ORTHOGONAL PROJECTIONS

159

24.1. Background

159

24.2. Exercises160

vi CONTENTS

24.3. Problems163

24.4. Answers to Odd-Numbered Exercises

164

Chapter 25. LEAST SQUARES APPROXIMATION

165

25.1. Background

165

25.2. Exercises166

25.3. Problems167

25.4. Answers to Odd-Numbered Exercises

168
Part 7. SPECTRAL THEORY OF INNER PRODUCT SPACES169 Chapter 26. SPECTRAL THEOREM FOR REAL INNER PRODUCT SPACES 171

26.1. Background

171

26.2. Exercises172

26.3. Problem174

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. Exercises178

27.3. Problems181

27.4. Answers to Odd-Numbered Exercises

182
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