# Linear Algebra Problem Book, 16

MAA, 1995 - 336 y[W
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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 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 쌠