# Linear Algebra

Springer Science & Business Media, Jan 26, 1987 - Mathematics - 285 pages
"Linear Algebra" is intended for a one-term course at the junior or senior level. It begins with an exposition of the basic theory of vector spaces and proceeds to explain the fundamental structure theorem for linear maps, including eigenvectors and eigenvalues, quadratic and hermitian forms, diagnolization of symmetric, hermitian, and unitary linear maps and matrices, triangulation, and Jordan canonical form. The book also includes a useful chapter on convex sets and the finite-dimensional Krein-Milman theorem. The presentation is aimed at the student who has already had some exposure to the elementary theory of matrices, determinants and linear maps. However the book is logically self-contained. In this new edition, many parts of the book have been rewritten and reorganized, and new exercises have been added.

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### Contents

 Vector Spaces 1 1 DEFINITIONS 2 2 BASES 10 3 DIMENSION OF A VECTOR SPACE 15 4 SUMS AND DIRECT SUMS 19 Matrices 23 2 LINEAR EQUATIONS 29 3 MULTIPLICATION OF MATRICES 31
 8 INVERSE OF A MATRIX 174 9 THE RANK OF A MATRIX AND SUBDETERMINANTS 177 Symmetric Hermitian and Unitary Operators 180 2 HERMITIAN OPERATORS 184 3 UNITARY OPERATORS 188 Eigenvectors and Eigenvalues 194 2 THE CHARACTERISTIC POLYNOMIAL 200 3 EIGENVALUES AND EIGENVECTORS OF SYMMETRIC MATRICES 213

 Linear Mappings 43 2 LINEAR MAPPINGS 51 3 THE KERNEL AND IMAGE OF A LINEAR MAP 59 4 COMPOSITION AND INVERSE OF LINEAR MAPPINGS 66 5 GEOMETRIC APPLICATIONS 72 Linear Maps and Matrices 81 2 THE MATRIX ASSOCIATED WITH A LINEAR MAP 82 3 BASES MATRICES AND LINEAR MAPS 87 Scalar Products and Orthogonality 95 2 ORTHOGONAL BASES POSITIVE DEFINITE CASE 103 3 APPLICATION TO LINEAR EQUATIONS THE RANK 113 4 BILINEAR MAPS AND MATRICES 118 5 GENERAL ORTHOGONAL BASES 123 6 THE DUAL SPACE AND SCALAR PRODUCTS 125 7 QUADRATIC FORMS 132 8 SYLVESTERS THEOREM 135 Determinants 140 2 EXISTENCE OF DETERMINANTS 143 3 ADDITIONAL PROPERTIES OF DETERMINANTS 150 4 CRAMERS RULE 157 5 TRIANGULATION OF A MATRIX BY COLUMN OPERATIONS 161 6 PERMUTATIONS 163 7 EXPANSION FORMULA AND UNIQUENESS OF DETERMINANTS 168
 4 DIAGONALIZATION OF A SYMMETRIC LINEAR MAP 218 5 THE HERMITIAN CASE 225 6 UNITARY OPERATORS 227 Polynomials and Matrices 231 2 POLYNOMIALS OF MATRICES AND LINEAR MAPS 233 Triangulation of Matrices and Linear Maps 237 2 THEOREM OF HAMILTONCAYLEY 241 3 DIAGONALIZATION OF UNITARY MAPS 243 Polynomials and Primary Decomposition 245 2 GREATEST COMMON DIVISOR 248 3 UNIQUE FACTORIZATION 251 4 APPLICATION TO THE DECOMPOSITION OF A VECTOR SPACE 255 5 SCHURS LEMMA 260 6 THE JORDAN NORMAL FORM 262 Convex Sets 268 2 SEPARATING HYPERPLANES 270 3 EXTREME POINTS AND SUPPORTING HYPERPLANES 272 4 THE KREINMILMAN THEOREM 274 Complex Numbers 277 Iwasawa Decomposition and Others 283 Index 293 Copyright