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
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
vi
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
viii
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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