Advanced Linear Algebra: Third Edition

Main subject category: • Linear algebraMathematics Subject Classification (2000): • 15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra This graduate level textbook covers an especially broad range of topics. The book first offers a careful discussion of the basics of linear algebra. It then proceeds to a discussion of modules, emphasizing a comparison with vector spaces, and presents a thorough discussion of inner product spaces, eigenvalues, eigenvectors, and finite dimensional spectral theory, culminating in the finite dimensional spectral theorem for normal operators.The new edition has been revised and contains a chapter on the QR decomposition, singular values and pseudoinverses, and a chapter on convexity, separation and positive solutions to linear systems. Cover......Page 1 Title Page......Page 5 Preface to the Third Edition......Page 9 Preface to the Second Edition......Page 11 Preface to the First Edition......Page 13 Contents......Page 15 Part 1: Preliminaries......Page 21 Part 2: Algebraic Structures......Page 37 Part I—Basic Linear Algebra......Page 53 Vector Spaces......Page 55 Subspaces......Page 57 Direct Sums......Page 60 Spanning Sets and Linear Independence......Page 64 The Dimension of a Vector Space......Page 68 Ordered Bases and Coordinate Matrices......Page 71 The Row and Column Spaces of a Matrix......Page 72 The Complexification of a Real Vector Space......Page 73 Exercises......Page 75 Linear Transformations......Page 79 The Kernel and Image of a Linear Transformation......Page 81 Isomorphisms......Page 82 The Rank Plus Nullity Theorem......Page 83 Linear Transformations from F^n to F^m......Page 84 Change of Basis Matrices......Page 85 The Matrix of a Linear Transformation......Page 86 Equivalence of Matrices......Page 88 Similarity of Matrices......Page 90 Similarity of Operators......Page 91 Invariant Subspaces and Reducing Pairs......Page 92 Projection Operators......Page 93 Topological Vector Spaces......Page 99 Linear Operators on V^C......Page 102 Exercises......Page 103 Quotient Spaces......Page 107 The Universal Property of Quotients and the First Isomorphism Theorem......Page 110 Quotient Spaces, Complements and Codimension......Page 112 Additional Isomorphism Theorems......Page 113 Linear Functionals......Page 114 Dual Bases......Page 116 Reflexivity......Page 120 Annihilators......Page 121 Operator Adjoints......Page 124 Exercises......Page 126 Modules......Page 129 Submodules......Page 131 Spanning Sets......Page 132 Linear Independence......Page 134 Annihilators......Page 135 Free Modules......Page 136 Quotient Modules......Page 137 The Correspondence and Isomorphism Theorems......Page 138 Direct Sums and Direct Summands......Page 139 Modules Are Not as Nice as Vector Spaces......Page 144 Exercises......Page 145 The Rank of a Free Module......Page 147 Noetherian Modules......Page 152 The Hilbert Basis Theorem......Page 156 Exercises......Page 157 Annihilators and Orders......Page 159 Cyclic Modules......Page 160 Free Modules over a Principal Ideal Domain......Page 162 Torsion-Free and Free Modules......Page 165 The Primary Cyclic Decomposition Theorem......Page 166 The Invariant Factor Decomposition......Page 176 Indecomposable Modules......Page 178 Exercises......Page 179 7 The Structure of a Linear Operator......Page 183 The Module Associated with a Linear Operator......Page 184 The Primary Cyclic Decomposition of V_τ......Page 187 The Characteristic Polynomial......Page 190 Cyclic and Indecomposable Modules......Page 191 The Big Picture......Page 194 The Rational Canonical Form......Page 196 Exercises......Page 202 Eigenvalues and Eigenvectors......Page 205 Geometric and Algebraic Multiplicities......Page 209 The Jordan Canonical Form......Page 210 Triangularizability and Schur's Lemma......Page 212 Diagonalizable Operators......Page 216 Exercises......Page 218 9 Real and Complex Inner Product Spaces......Page 225 Norm and Distance......Page 228 Isometries......Page 230 Orthogonality......Page 231 Orthogonal and Orthonormal Sets......Page 232 The Projection Theorem and Best Approximations......Page 239 The Riesz Representation Theorem......Page 241 Exercises......Page 243 The Adjoint of a Linear Operator......Page 247 Orthogonal Projections......Page 251 Unitary Diagonalizability......Page 253 Normal Operators......Page 254 Special Types of Normal Operators......Page 258 Self-Adjoint Operators......Page 259 Unitary Operators and Isometries......Page 260 The Structure of Normal Operators......Page 265 Functional Calculus......Page 267 Positive Operators......Page 270 The Polar Decomposition of an Operator......Page 272 Exercises......Page 274 Part II—Topics......Page 277 Symmetric, Skew-Symmetric and Alternate Forms......Page 279 The Matrix of a Bilinear Form......Page 281 Quadratic Forms......Page 284 Orthogonality......Page 285 Linear Functionals......Page 288 Orthogonal Complements and Orthogonal Direct Sums......Page 289 Isometries......Page 291 Hyperbolic Spaces......Page 292 Nonsingular Completions of a Subspace......Page 293 The Witt Theorems: A Preview......Page 295 The Classification Problem for Metric Vector Spaces......Page 296 Symplectic Geometry......Page 297 The Structure of Orthogonal Geometries: Orthogonal Bases......Page 302 The Classification of Orthogonal Geometries: Canonical Forms......Page 305 The Orthogonal Group......Page 311 The Witt Theorems for Orthogonal Geometries......Page 314 Maximal Hyperbolic Subspaces of an Orthogonal Geometry......Page 315 Exercises......Page 317 The Definition......Page 321 Open and Closed Sets......Page 324 Convergence in a Metric Space......Page 325 The Closure of a Set......Page 326 Dense Subsets......Page 328 Continuity......Page 330 Completeness......Page 331 Isometries......Page 335 The Completion of a Metric Space......Page 336 Exercises......Page 341 A Brief Review......Page 345 Hilbert Spaces......Page 346 Infinite Series......Page 350 An Approximation Problem......Page 351 Hilbert Bases......Page 355 Fourier Expansions......Page 356 Hilbert Dimension......Page 366 A Characterization of Hilbert Spaces......Page 367 The Riesz Representation Theorem......Page 369 Exercises......Page 372 Universality......Page 375 Bilinear Maps......Page 379 Tensor Products......Page 381 When Is a Tensor Product Zero?......Page 387 Coordinate Matrices and Rank......Page 388 Characterizing Vectors in a Tensor Product......Page 391 Defining Linear Transformations on a Tensor Product......Page 394 The Tensor Product of Linear Transformations......Page 395 Change of Base Field......Page 399 Multilinear Maps and Iterated Tensor Products......Page 402 Tensor Spaces......Page 405 Special Multilinear Maps......Page 410 The Symmetric and Antisymmetric Tensor Algebras......Page 412 The Determinant......Page 423 Exercises......Page 426 15 Positive Solutions to Linear Systems: Convexity and Separation......Page 431 Convex, Closed and Compact Sets......Page 433 Convex Hulls......Page 434 Linear and Affine Hyperplanes......Page 436 Separation......Page 438 Exercises......Page 443 Affine Geometry......Page 447 Affine Combinations......Page 448 Affine Hulls......Page 450 The Lattice of Flats......Page 451 Affine Independence......Page 453 Affine Transformations......Page 455 Projective Geometry......Page 457 Exercises......Page 460 Singular Values......Page 463 The Moore–Penrose Generalized Inverse......Page 466 Least Squares Approximation......Page 468 Exercises......Page 469 Associative Algebras......Page 471 Division Algebras......Page 482 Exercises......Page 489 Formal Power Series......Page 491 The Umbral Algebra......Page 493 Formal Power Series as Linear Operators......Page 497 Sheffer Sequences......Page 500 Examples of Sheffer Sequences......Page 508 Umbral Operators and Umbral Shifts......Page 510 Continuous Operators on the Umbral Algebra......Page 512 Operator Adjoints......Page 513 Umbral Operators and Automorphisms of the Umbral Algebra......Page 514 Umbral Shifts and Derivations of the Umbral Algebra......Page 519 The Transfer Formulas......Page 524 A Final Remark......Page 525 Exercises......Page 526 References......Page 527 Index of Symbols......Page 533 Index......Page 535 This is a graduate textbook covering an especially broad range of topics. The first part of the book contains a careful but rapid discussion of the basics of linear algebra, including vector spaces, linear transformations, quotient spaces, and isomorphism theorems. The author then proceeds to modules, emphasizing a comparison with vector spaces. A thorough discussion of inner product spaces, eigenvalues, eigenvectors, and finite dimensional spectral theory follows, culminating in the finite dimensional spectral theorem for normal operators. The second part of the book is a collection of topics, including metric vector spaces, metric spaces, Hilbert spaces, tensor products, and affine geometry. The last chapter discusses the umbral calculus, an area of modern algebra with important applications. For the third edition, the author has added a new chapter on associative algebras that includes the well known characterizations of the finite-dimensional division algebras over the real field (a theorem of Frobenius) and over a finite field (Wedderburn's theorem); polished and refined some arguments (such as the discussion of reflexivity, the rational canonical form, best approximations and the definitions of tensor products); upgraded some proofs that were originally done only for finite-dimensional/rank cases; added new theorems, including the spectral mapping theorem; and considerably expanded the reference section with over a hundred references to books on linear algebra. From the reviews of the second edition: "In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials. ... As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields. ... the exercises are rewritten and expanded. ... Overall, I found the book a very useful one. ... It is a suitable choice as a graduate text or as a reference book." - Ali-Akbar Jafarian, ZentralblattMATH "This is a formidable volume, a compendium of linear algebra theory, classical and modern ... . The development of the subject is elegant ... . The proofs are neat ... . The exercise sets are good, with occasional hints given for the solution of trickier problems. ... It represents linear algebra and does so comprehensively." -Henry Ricardo, MathDL For the third edition, the author has added a new chapter on associative algebras that includes the well known characterizations of the finite-dimensional division algebras over the real field (a theorem of Frobenius) and over a finite field (Wedderburn's theorem); polished and refined some arguments (such as the discussion of reflexivity, the rational canonical form, best approximations and the definitions of tensor products); upgraded some proofs that were originally done only for finite-dimensional/rank cases; added new theorems, including the spectral mapping theorem; corrected all known errors; the reference section has been enlarged considerably, with over a hundred references to books on linear algebra. From the reviews of the second edition: “In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials. ... As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields. ... the exercises are rewritten and expanded. ... Overall, I found the book a very useful one. ... It is a suitable choice as a graduate text or as a reference book.” Ali-Akbar Jafarian, ZentralblattMATH “This is a formidable volume, a compendium of linear algebra theory, classical and modern .... The development of the subject is elegant .... The proofs are neat .... The exercise sets are good, with occasional hints given for the solution of trickier problems. ... It represents linear algebra and does so comprehensively.” Henry Ricardo, MathDL This graduate level textbook covers an especially broad range of topics. The book first offers a careful discussion of the basics of linear algebra. It then proceeds to a discussion of modules, emphasizing a comparison with vector spaces, and presents a thorough discussion of inner product spaces, eigenvalues, eigenvectors, and finite dimensional spectral theory, culminating in the finite dimensional spectral theorem for normal operators. The new edition has been revised and contains a chapter on the QR decomposition, singular values and pseudoinverses, and a chapter on convexity, separation and positive solutions to linear systems.