Book of Smut, Vol 1

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. I. Vector Spaces -- Ii. Matrices -- Iii. Linear Mappings -- Iv. Linear Maps And Matrices -- V. Scalar Products And Orthogonality -- Vi. Determinants -- Vii. Symmetric, Hermitian, And Unitary Operators -- Viii. Eigenvectors And Eigenvalues -- Ix. Polynomials And Matrices -- X. Triangulation Of Matrices And Linear Maps -- Xi. Polynomials And Primary Decomposition -- Xii. Convex Sets -- Appendix: Complex Numbers. Serge Lang. Includes Index. Includes Bibliographical References And Index. Foreword 6 Contents 8 I Vector Spaces 12 §1. DEFINITIONS 13 §2. BASES 21 §3. DIMENSION OF A VECTOR SPACE 26 §4. SUMS AND DIRECT SUMS 30 II Matrices 34 §1. THE SPACE OF MATRICES 34 §2. LINEAR EQUATIONS 40 §3. MULTIPLICATION OF MATRICES 42 III Linear Mappings 54 §1. MAPPINGS 54 §2. LINEAR MAPPINGS 62 §3. THE KERNEL AND IMAGE OF A LINEAR MAP 70 §4. COMPOSITION AND INVERSE OF LINEAR MAPPINGS 77 §5. GEOMETRIC APPLICATIONS 83 IV Linear Maps and Matrices 92 §1. THE LINEAR MAP ASSOCIATED WITH A MATRIX 92 §2. THE MATRIX ASSOCIATED WITH A LINEAR MAP 93 §3. BASES, MATRICES, AND LINEAR MAPS 98 V Scalar Products and Orthogonality 106 §1. SCALAR PRODUCTS 106 §2. ORTHOGONAL BASES, POSITIVE DEFINITE CASE 114 §3. APPLICATION TO LINEAR EQUATIONS; THE RANK 124 §4. BILINEAR MAPS AND MATRICES 129 §5. GENERAL ORTHOGONAL BASES 134 §6. THE DUAL SPACE AND SCALAR PRODUCTS 136 §7. QUADRATIC FORMS 143 §8. SYLVESTER'S THEOREM 146 VI Determinants 151 §1. DETERMINANTS OF ORDER 2 151 §2. EXISTENCE OF DETERMINANTS 154 §3. ADDITIONAL PROPERTIES OF DETERMINANTS 161 §4. CRAMER'S RULE 168 §5. TRIANGULATION OF A MATRIX BY COLUMN OPERATIONS 172 §6. PERMUTATIONS 174 §7. EXPANSION FORMULA AND UNIQUENESS OF DETERMINANTS 179 §8. INVERSE OF A MATRIX 185 §9. THE RANK OF A MATRIX AND SUBDETERMINANTS 188 VII Symmetric, Hermitian, and Unitary Operators 191 §1. SYMMETRIC OPERATORS 191 §2. HERMITIAN OPERATORS 195 §3. UNITARY OPERATORS 199 VIII Eigenvectors and Eigenvalues 205 §1. EIGENVECTORS AND EIGENVALUES 205 §2. THE CHARACTERISTIC POLYNOMIAL 211 §3. EIGENVALUES AND EIGENVECTORS OF SYMMETRIC MATRICES 224 §4. DIAGONALIZATION OF A SYMMETRIC LINEAR MAP 229 §5. THE HERMITIAN CASE 236 §6. UNITARY OPERATORS 238 IX Polynomials and Matrices 242 §1. POLYNOMIALS 242 §2. POLYNOMIALS OF MATRICES AND LINEAR MAPS 244 X Triangulation of Matrices and Linear Maps 248 §1. EXISTENCE OF TRIANGULATION 248 §2. THEOREM OF HAMILTON-CAYLEY 252 §3. DIAGONALIZATION OF UNITARY MAPS 254 XI Polynomials and Primary Decomposition 256 §1. THE EUCLIDEAN ALGORITHM 256 §2. GREATEST COMMON DIVISOR 259 §3. UNIQUE FACTORIZATION 262 §4. APPLICATION TO THE DECOMPOSITION OF A VECTOR SPACE 266 §5. SCHUR'S LEMMA 271 §6. THE JORDAN NORMAL FORM 273 XII Convex Sets 279 §1. DEFINITIONS 279 §2. SEPARATING HYPERPLANES 281 §3. EXTREME POINTS AND SUPPORTING HYPERPLANES 283 §4. THE KREIN-MILMAN THEOREM 285 APPENDIX I Complex Numbers 288 APPENDIX II Iwasawa Decomposition and Others 294 Index 304 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 theorems for linear maps, including eigenvectors and eigenvalues, quadric and hermitian forms, diagonalization 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. 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 theorems 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 finie-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. Intended as an introductory textbook for a one-term course in linear and multilinear algebra, this begins with an exposition of the basic theory of vector spaces and proceeds to explain the fundamental structure theorems for linear maps, including eigenvectors and eigenvalues. As usual, a collection of objects will be called a set.