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
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