Exercises and Problems in Linear Algebra

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

iii

iv

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

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download