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:

,
, 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