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

Scalars
1
Half double addition
2
Exponentiation
3
Complex numbers
4
Affine transformations
5
Matrix multiplication 7 Modular multiplication 8 Small operations 9 Identity elements
6
Complex inverses
7
Affine inverses 12 Matrix inverses 13 Abelian groups
8
Projections
82
Sums of projections
83
not quite idempotence
84
Duality
85
Linear functional s 76 Dual spaces
86
Solution of equations
87
Reflexivity
88
Annihilators 80 Double annihilators
92

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
Spans
25
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
Bases
39
Examples of independence 36 Independence over K 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
Domain and range
55
Kernel
56
Composition
57
Range inclusion and factorization 59 Transformations as vectors
65
Invertibility
67
Invertibility examples
69
n x
72
Zeroone matrices
75
Invertible matrix bases
76
Finitedimensional invertibility 67 Matrices
77
Diagonal matrices
79
Universal commutativity
80
Invariance 71 Invariant complements
81
Adjoints
93
Adjoints of projections
95
Matrices of adjoints
96
Similarity
97
coordinates 86 Similarity transformations
98
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
Canonical Forms
107
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
Inner Product Spaces
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
Vanishing quadratic forms 128 Hermitian transformations
140
Skew transformations
141
Real Hermitian forms 131 Positive transformations
142
positive inverses 133 Perpendicular projections
144
Projections on C x C
145
Orthogonal projections
146
Normality
149
Hints
169
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