Linear Algebra Problem Book, 第 16 巻

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MAA, 1995 - 336 ページ
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Can one learn linear algebra solely by solving problems? Paul Halmos thinks so, and you will too once you read this book. The Linear Algebra Problem Book is an ideal text for a course in linear algebra. It takes the student step by step from the basic axioms of a field through the notion of vector spaces, on to advanced concepts such as inner product spaces and normality. All of this occurs by way of a series of 164 problems, each with hints and, at the back of the book, full solutions. This book is a marvelous example of how to teach and learn mathematics by 'doing' mathematics. It will work well for classes taught in small groups and can also be used for self-study. After working their way through the book, students will understand not only the theorems of linear algebra, but also some of the questions which were asked which enabled the theorems to be discovered in the first place. They will gain confidence in their problem solving abilities and be better prepared to understand more advanced courses. As the author explains, 'I don't think I understand a subject until I know the questions ... I wrote this book to organize those questions, problems, in my own mind.' Try this book with your students and they too will be able to organize and understand the questions of linear algebra.
 

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目次

Preface Chapter 1 Scalars 1 Double addition
1
Half double addition
2
Exponentiation
3
Complex numbers
4
Affine transformations 6 Matrix multiplication
6
Modular multiplication
7
Small operations
8
Identity elements
9
matrices
99
real and complex
100
Rank and nullity 91 Similarity and rank 92 Similarity of transposes
101
Ranks of sums 94 Ranks of products 95 Nullities of sums and products
102
Some similarities 97 Equivalence
103
Rank and equivalence
104
Chapter 7 Canonical Forms
107
Eigenvalues
109

Complex inverses
10
Affine inverses
11
Matrix inverses
12
Abelian groups 14 Groups
14
Independent group axioms
15
Fields 17 Addition and multiplication in fields 18 Distributive failure 19 Finite fields
16
Vectors
17
Vector spaces
20
Examples 22 Linear combinations
21
Subspaces
23
Unions of subspaces
26
Equalities of spans
28
Some special spans
30
Sums of subspaces
31
Distributive subspaces 30 Total sets
34
Dependence
35
Independence
37
Chapter 3 Bases 33 Exchanging bases 34 Simultaneous complements
39
Examples of independence 36 Independence over R and Q 37 Independence in
40
Vectors common to different bases 39 Bases in C3 40 Maximal independent sets 41 Complex as real
41
Subspaces of full dimension 43 Extended bases
42
Finitedimensional subspaces 45 Minimal total sets 46 Existence of minimal total sets 47 Infinitely total sets
43
Relatively independent sets 49 Number of bases in a finite vector space
44
Direct sums
45
Quotient spaces
46
Dimension of a quotient space 53 Additivity of dimension
49
Transformations
51
Linear transformations 55 Domain and range
55
Kernel
58
Composition
61
Range inclusion and factorization 59 Transformations as vectors
65
Invertibility
67
Invertibility examples
69
n xn 64 Zeroone matrices
75
Invertible matrix bases
76
Finitedimensional invertibility 67 Matrices
77
Diagonal matrices
79
Universal commutativity 70 Invariance
80
Invariant complements
81
Projections
82
Sums of projections
83
not quite idempotence
84
Chapter 5 Duality 75 Linear functionals
85
Dual spaces
86
Solution of equations
87
Reflexivity
88
Annihilators 80 Double annihilators
92
Adjoints
93
Adjoints of projections
95
Matrices of adjoints
96
vectors
97
coordinates 86 Similarity transformations
98
Sums and products of eigenvalues 101 Eigenvalues of products 102 Polynomials in eigenvalues 103 Diagonalizing permutations
110
Polynomials in eigenvalues converse
112
Multiplicities 106 Distinct eigenvalues 107 Comparison of multiplicities
115
Triangularization 109 Complexification
118
Unipotent transformation 111 Nipotence 112 Nilpotent products
120
Nilpotent direct sums
121
Jordan form
122
Minimal polynomials
124
Noncommutative Lagrange interpolation
126
Chapter 8 Inner Product Spaces 117 Inner products
129
Polarization
130
The Pythagorean theorem 120 The parallelogram law 121 Complete orthonormal sets
131
Schwarz inequality
132
Orthogonal complements
133
More linear functionals 125 Adjoints on inner product spaces
134
Quadratic forms
137
Distinct eigenvalues
138
Normality 139 Unitary transformations
139
Vanishing quadratic forms 128 Hermitian transformations
140
Skew transformations
141
Real Hermitian forms 131 Positive transformations
142
Hermitian diagonalization
143
positive inverses 133 Perpendicular projections
144
Projections on CC
145
Orthogonal projections
146
Normal diagonalizability
147
Normal commutativity
148
Adjoint commutativity
149
Adjoint intertwining
150
Normal products
151
Functions of transformations
152
Gramians
153
Monotone functions
154
Reducing ranges and kernels
155
Truncated shifts
156
Nonpositive square roots
157
Similar normal transformations
158
Unitary equivalence of transposes
159
Unitary and orthogonal equivalence
160
Null convergent powers
161
Power boundedness
162
Reduction and index 2
166
Chapter 1
169
Chapter 2
173
Chapter 3
183
Chapter 4
189
Chapter 5
199
Chapter 6
219
Chapter 7
228
2 x 2
242
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