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Exercises and Problems in Linear Algebra
John M. Erdman
Portland State University
Version July 13, 2014
c2010 John M. Erdman
E-mail address:erdman@pdx.edu
Contents
PREFACEvii
Part 1. MATRICES AND LINEAR EQUATIONS1
Chapter 1. SYSTEMS OF LINEAR EQUATIONS
31.1. Background3
1.2. Exercises4
1.3. Problems7
1.4. Answers to Odd-Numbered Exercises
8Chapter 2. ARITHMETIC OF MATRICES
92.1. Background9
2.2. Exercises10
2.3. Problems12
2.4. Answers to Odd-Numbered Exercises
14Chapter 3. ELEMENTARY MATRICES; DETERMINANTS
153.1. Background15
3.2. Exercises17
3.3. Problems22
3.4. Answers to Odd-Numbered Exercises
23Chapter 4. VECTOR GEOMETRY INRn25
4.1. Background25
4.2. Exercises26
4.3. Problems28
4.4. Answers to Odd-Numbered Exercises
29Part 2. VECTOR SPACES31
Chapter 5. VECTOR SPACES
335.1. Background33
5.2. Exercises34
5.3. Problems37
5.4. Answers to Odd-Numbered Exercises
38Chapter 6. SUBSPACES
396.1. Background39
6.2. Exercises40
6.3. Problems44
6.4. Answers to Odd-Numbered Exercises
45Chapter 7. LINEAR INDEPENDENCE
477.1. Background47
7.2. Exercises49
iii iv CONTENTS7.3. Problems51
7.4. Answers to Odd-Numbered Exercises
53Chapter 8. BASIS FOR A VECTOR SPACE
558.1. Background55
8.2. Exercises56
8.3. Problems57
8.4. Answers to Odd-Numbered Exercises
58Part 3. LINEAR MAPS BETWEEN VECTOR SPACES59
Chapter 9. LINEARITY
619.1. Background61
9.2. Exercises63
9.3. Problems67
9.4. Answers to Odd-Numbered Exercises
70Chapter 10. LINEAR MAPS BETWEEN EUCLIDEAN SPACES
7110.1. Background
7110.2. Exercises72
10.3. Problems74
10.4. Answers to Odd-Numbered Exercises
75Chapter 11. PROJECTION OPERATORS
7711.1. Background
7711.2. Exercises78
11.3. Problems79
11.4. Answers to Odd-Numbered Exercises
80Part 4. SPECTRAL THEORY OF VECTOR SPACES81
Chapter 12. EIGENVALUES AND EIGENVECTORS
8312.1. Background
8312.2. Exercises84
12.3. Problems85
12.4. Answers to Odd-Numbered Exercises
86Chapter 13. DIAGONALIZATION OF MATRICES
8713.1. Background
8713.2. Exercises89
13.3. Problems91
13.4. Answers to Odd-Numbered Exercises
92Chapter 14. SPECTRAL THEOREM FOR VECTOR SPACES
9314.1. Background
9314.2. Exercises94
14.3. Answers to Odd-Numbered Exercises
96Chapter 15. SOME APPLICATIONS OF THE SPECTRAL THEOREM 97
15.1. Background
9715.2. Exercises98
15.3. Problems102
15.4. Answers to Odd-Numbered Exercises
103Chapter 16. EVERY OPERATOR IS DIAGONALIZABLE PLUS NILPOTENT 105
CONTENTS v
16.1. Background
10516.2. Exercises106
16.3. Problems110
16.4. Answers to Odd-Numbered Exercises
111Part 5. THE GEOMETRY OF INNER PRODUCT SPACES113
Chapter 17. COMPLEX ARITHMETIC
11517.1. Background
11517.2. Exercises116
17.3. Problems118
17.4. Answers to Odd-Numbered Exercises
119Chapter 18. REAL AND COMPLEX INNER PRODUCT SPACES
12118.1. Background
12118.2. Exercises123
18.3. Problems125
18.4. Answers to Odd-Numbered Exercises
126Chapter 19. ORTHONORMAL SETS OF VECTORS
12719.1. Background
12719.2. Exercises128
19.3. Problems129
19.4. Answers to Odd-Numbered Exercises
131Chapter 20. QUADRATIC FORMS
13320.1. Background
13320.2. Exercises134
20.3. Problems136
20.4. Answers to Odd-Numbered Exercises
137Chapter 21. OPTIMIZATION
13921.1. Background
13921.2. Exercises140
21.3. Problems141
21.4. Answers to Odd-Numbered Exercises
142Part 6. ADJOINT OPERATORS143
Chapter 22. ADJOINTS AND TRANSPOSES
14522.1. Background
14522.2. Exercises146
22.3. Problems147
22.4. Answers to Odd-Numbered Exercises
148Chapter 23. THE FOUR FUNDAMENTAL SUBSPACES
14923.1. Background
14923.2. Exercises151
23.3. Problems155
23.4. Answers to Odd-Numbered Exercises
157Chapter 24. ORTHOGONAL PROJECTIONS
15924.1. Background
15924.2. Exercises160
vi CONTENTS24.3. Problems163
24.4. Answers to Odd-Numbered Exercises
164Chapter 25. LEAST SQUARES APPROXIMATION
16525.1. Background
16525.2. Exercises166
25.3. Problems167
25.4. Answers to Odd-Numbered Exercises
168Part 7. SPECTRAL THEORY OF INNER PRODUCT SPACES169 Chapter 26. SPECTRAL THEOREM FOR REAL INNER PRODUCT SPACES 171
26.1. Background
17126.2. Exercises172
26.3. Problem174
26.4. Answers to the Odd-Numbered Exercise
175Chapter 27. SPECTRAL THEOREM FOR COMPLEX INNER PRODUCT SPACES 177
27.1. Background
17727.2. Exercises178
27.3. Problems181
27.4. Answers to Odd-Numbered Exercises
182Bibliography183
Index185
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. Stu- dents are encouraged to nd books, papers, and web sites whose writing style they nd congenial, whose emphasis matches their interests, and whose price ts their budgets. The short introduc- torybackgroundsection in these exercises, which precede each assignment, are intended only to x notation and provide \ocial" denitions 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 areLinear Algebra[7] by Jim Heeron,quotesdbs_dbs20.pdfusesText_26[PDF] practice 92 parabolas
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