Advanced Linear Algebra

Contents Preface to the Third Edition, vii Preface to the Second Edition, ix Preface to the First Edition, xi Preliminaries, 1 Part 1: Preliminaries, 1 Part 2: Algebraic Structures, 17 Part I---Basic Linear Algebra, 33 1 Vector Spaces, 35 Vector Spaces, 35 Subspaces, 37 Direct Sums, 40 Spanning Sets and Linear Independence, 44 The Dimension of a Vector Space, 48 Ordered Bases and Coordinate Matrices, 51 The Row and Column Spaces of a Matrix, 52 The Complexification of a Real Vector Space, 53 Exercises, 55 2 Lin ear Transformations, 59 Linear Transformations, 59 The Kernel and Image of a Linear Transformation, 61 Isomorphisms, 62 The Rank Plus Nullity Theorem, 63 Linear Transformations from to , 64 Change of Basis Matrices, 65 The Matrix of a Linear Transformation, 66 Change of Bases for Linear Transformations, 68 Equivalence of Matrices, 68 Similarity of Matrices, 70 Similarity of Operators, 71 Invariant Subspaces and Reducing Pairs, 72 Projection Operators, 73 xiv Contents Topological Vector Spaces, 79 Linear Operators on , 82 Exercises, 83 3 The Isomorphism Theorems, 87 Quotient Spaces, 87 The Universal Property of Quotients and the First Isomorphism Theorem, 90 Quotient Spaces, Complements and Codimension, 92 Additional Isomorphism Theorems, 93 Linear Functionals, 94 Dual Bases, 96 Reflexivity, 100 Annihilators, 101 Operator Adjoints, 104 Exercises, 106 4 Modules I: Basic Properties, 109 Motivation, 109 Modules, 109 Submodules, 111 Spanning Sets, 112 Linear Independence, 114 Torsion Elements, 115 Annihilators, 115 Free Modules, 116 Homomorphisms, 117 Quotient Modules, 117 The Correspondence and Isomorphism Theorems, 118 Direct Sums and Direct Summands, 119 Modules Are Not as Nice as Vector Spaces, 124 Exercises, 125 5 Modules II: Free and Noetherian Modules, 127 The Rank of a Free Module, 127 Free Modules and Epimorphisms, 132 Noetherian Modules, 132 The Hilbert Basis Theorem, 136 Exercises, 137 6 Modules over a Principal Ideal Domain, 139 Annihilators and Orders, 139 Cyclic Modules, 140 Free Modules over a Principal Ideal Domain, 142 Torsion-Free and Free Modules, 145 The Primary Cyclic Decomposition Theorem, 146 The Invariant Factor Decomposition, 156 Characterizing Cyclic Modules, 158 Contents xv Indecomposable Modules, 158 Exercises, 159 7 The Structure of a Linear Operator, 163 The Module Associated with a Linear Operator, 164 The Primary Cyclic Decomposition of , 167 The Characteristic Polynomial, 170 Cyclic and Indecomposable Modules, 171 The Big Picture, 174 The Rational Canonical Form, 176 Exercises, 182 8 Eigenvalues and Eigenvectors, 185 Eigenvalues and Eigenvectors, 185 Geometric and Algebraic Multiplicities, 189 The Jordan Canonical Form, 190 Triangularizability and Schur's Theorem, 192 Diagonalizable Operators, 196 Exercises, 198 9 Real and Complex Inner Product Spaces, 205 Norm and Distance, 208 Isometries, 210 Orthogonality, 211 Orthogonal and Orthonormal Sets, 212 The Projection Theorem and Best Approximations, 219 The Riesz Representation Theorem, 221 Exercises, 223 10 Structure Theory for Normal Operators, 227 The Adjoint of a Linear Operator, 227 Unitary Diagonalizability, 233 Normal Operators, 234 Special Types of Normal Operators, 238 Self-Adjoint Operators, 239 Unitary Operators and Isometries, 240 The Structure of Normal Operators, 245 Functional Calculus, 247 Positive Operators, 250 The Polar Decomposition of an Operator, 252 Exercises, 254 Part II---Topics, 257 11 Metric Vector Spaces: The Theory of Bilinear Forms, 259 Symmetric, Skew-Symmetric and Alternate Forms, 259 The Matrix of a Bilinear Form, 261 Orthogonal Projections, 231 xvi Contents Quadratic Forms, 264 Orthogonality, 265 Linear Functionals, 268 Orthogonal Complements and Orthogonal Direct Sums, 269 Isometries, 271 Hyperbolic Spaces, 272 Nonsingular Completions of a Subspace, 273 The Witt Theorems: A Preview, 275 The Classification Problem for Metric Vector Spaces, 276 Symplectic Geometry, 277 The Structure of Orthogonal Geometries: Orthogonal Bases, 282 The Classification of Orthogonal Geometries: Canonical Forms, 285 The Orthogonal Group, 291 The Witt Theorems for Orthogonal Geometries, 294 Maximal Hyperbolic Subspaces of an Orthogonal Geometry, 295 Exercises, 297 12 Metric Spaces, 301 The Definition, 301 Open and Closed Sets, 304 Convergence in a Metric Space, 305 The Closure of a Set, 306 Dense Subsets, 308 Continuity, 310 Completeness, 311 Isometries, 315 The Completion of a Metric Space, 316 Exercises, 321 13 Hilbert Spaces, 325 A Brief Review, 325 Hilbert Spaces, 326 Infinite Series, 330 An Approximation Problem, 331 Hilbert Bases, 335 Fourier Expansions, 336 A Characterization of Hilbert Bases, 346 Hilbert Dimension, 346 A Characterization of Hilbert Spaces, 347 The Riesz Representation Theorem, 349 Exercises, 352 14 Tensor Products, 355 Universality, 355 Bilinear Maps, 359 Tensor Products, 361 Contents xvii When Is a Tensor Product Zero?, 367 Coordinate Matrices and Rank, 368 Characterizing Vectors in a Tensor Product, 371 Defining Linear Transformations on a Tensor Product, 374 The Tensor Product of Linear Transformations, 375 Change of Base Field, 379 Multilinear Maps and Iterated Tensor Products, 382 Tensor Spaces, 385 Special Multilinear Maps, 390 Graded Algebras, 392 The Symmetric and Antisymmetric The Determinant, 403 Exercises, 406 15 Positive Solutions to Linear Systems: Convexity and Separation, 411 Convex, Closed and Compact Sets, 413 Convex Hulls, 414 Linear and Affine Hyperplanes, 416 Separation, 418 Exercises, 423 16 Affine Geometry, 427 Affine Geometry, 427 Affine Combinations, 428 Affine Hulls, 430 The Lattice of Flats, 431 Affine Independence, 433 Affine Transformations, 435 Projective Geometry, 437 Exercises, 440 17 Singular Values and the Moore--Penrose Inverse, 443 Singular Values, 443 The Moore--Penrose Generalized Inverse, 446 Least Squares Approximation, 448 Exercises, 449 18 An Introduction to Algebras, 451 Motivation, 451 Associative Algebras, 451 Division Algebras, 462 Exercises, 469 19 The Umbral Calculus, 471 Formal Power Series, 471 The Umbral Algebra, 473 Tensor Algebras, 392 xviii Contents Formal Power Series as Linear Operators, 477 Sheffer Sequences, 480 Examples of Sheffer Sequences, 488 Umbral Operators and Umbral Shifts, 490 Continuous Operators on the Umbral Algebra, 492 Operator Adjoints, 493 Umbral Operators and Automorphisms of the Umbral Algebra, 494 Umbral Shifts and Derivations of the Umbral Algebra, 499 The Transfer Formulas, 504 A Final Remark, 505 Exercises, 506 References, 507 Index of Symbols, 513 Index, 515