Springer Science & Business Media, Dec 31, 2007 - Mathematics - 486 pages
Let me begin by thanking the readers of the first edition for their many helpful comments and suggestions. The second edition represents a major change from the first edition. Indeed, one might say that it is a totally new book, with the exception of the general range of topics covered. The text has been completely rewritten. I hope that an additional 12 years and roughly 20 books worth of experience has enabled me to improve the quality of my exposition. Also, the exercise sets have been completely rewritten. The second edition contains two new chapters: a chapter on convexity, separation and positive solutions to linear systems (Chapter 15) and a chapter on the QR decomposition, singular values and pseudoinverses (Chapter 17). The treatments of tensor products and the umbral calculus have been greatly expanded and I have included discussions of determinants (in the chapter on tensor products), the complexification of a real vector space, Schur's lemma and Geršgorin disks. Steven Roman Irvine, California February 2005 Preface to the First Edition This book is a thorough introduction to linear algebra, for the graduate or advanced undergraduate student. Prerequisites are limited to a knowledge of the basic properties of matrices and determinants. However, since we cover the basics of vector spaces and linear transformations rather rapidly, a prior course in linear algebra (even at the sophomore level), along with a certain measure of “mathematical maturity,” is highly desirable.

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### Contents

 Vector Spaces 33 Subspaces 35 Direct Sums 38 Spanning Sets and Linear Independence 41 The Dimension of a Vector Space 44 Ordered Bases and Coordinate Matrices 47 The Row and Column Spaces of a Matrix 48 The Complexification of a Real Vector Space 49
 Spectral Resolutions and Functional Calculus 228 Positive Operators 230 The Polar Decomposition of an Operator 232 Exercises 234 Part IITopics 236 Metric Vector Spaces 237 The Matrix of a Bilinear Form 242 Quadratic Forms 244

 Exercises 51 Linear Transformations 55 The Kernel and Image of a Linear Transformation 57 The Rank Plus Nullity Theorem 59 Change of Basis Matrices 60 The Matrix of a Linear Transformation 61 Change of Bases for Linear Transformations 63 Equivalence of Matrices 64 Similarity of Matrices 65 Similarity of Operators 66 Invariant Subspaces and Reducing Pairs 68 Linear Operators on 71 Exercises 72 The Isomorphism Theorems 75 The Universal Property of Quotients and the First Isomorphism Theorem 77 Quotient Spaces Complements and Codimension 79 Additional Isomorphism Theorems 80 Linear Functionals 82 Dual Bases 83 Reflexivity 84 Annihilators 86 Operator Adjoints 88 Exercises 90 Modules I Basic Properties 92 Submodules 95 Spanning Sets 96 Linear Independence 98 Torsion Elements 99 Homomorphisms 100 Quotient Modules 101 The Correspondence and Isomorphism Theorems 102 Modules Are Not As Nice As Vector Spaces 106 Exercises 107 Modules II Free and Noetherian Modules 109 Free Modules and Epimorphisms 114 Noetherian Modules 115 The Hilbert Basis Theorem 118 Exercises 119 Modules over a Principal Ideal Domain 121 Cyclic Modules 122 Free Modules over a Principal Ideal Domain 123 TorsionFree and Free Modules 125 Cyclic Modules 126 The First Decomposition 127 The Primary Decomposition 128 The Cyclic Decomposition of a Primary Module 130 The Primary Cyclic Decomposition Theorem 134 The Invariant Factor Decomposition 135 Exercises 138 The Structure of a Linear Operator 141 The Module Associated with a Linear Operator 142 Orders and the Minimal Polynomial 144 Cyclic Submodules and Cyclic Subspaces 145 Summary 147 The Rational Canonical Form 148 Exercises 151 Eigenvalues and Eigenvectors 153 Eigenvalues and Eigenvectors 155 Geometric and Algebraic Multiplicities 157 The Jordan Canonical Form 158 Triangularizability and Schurs Lemma 160 Diagonalizable Operators 165 Projections 166 The Algebra of Projections 167 Resolutions of the Identity 170 Spectral Resolutions 172 Projections and Invariance 173 Exercises 174 Real and Complex Inner Product Spaces 181 Norm and Distance 183 Isometries 186 Orthogonality 187 Orthogonal and Orthonormal Sets 188 The Projection Theorem and Best Approximations 192 Orthogonal Direct Sums 194 The Riesz Representation Theorem 195 Exercises 196 Structure Theory for Normal Operators 200 Unitary Diagonalizability 204 Normal Operators 205 Special Types of Normal Operators 207 SelfAdjoint Operators 208 Unitary Operators and Isometries 210 The Structure of Normal Operators 215 Matrix Versions 222 Orthogonal Projections 223 Orthogonal Resolutions of the Identity 226 The Spectral Theorem 227
 Orthogonality 245 Linear Functionals 248 Orthogonal Complements and Orthogonal Direct Sums 249 Isometries 252 Hyperbolic Spaces 253 Nonsingular Completions of a Subspace 254 A Preview 256 The Classification Problem for Metric Vector Spaces 257 Symplectic Geometry 258 Orthogonal Bases 264 Canonical Forms 266 The Orthogonal Group 272 The Witts Theorems for Orthogonal Geometries 275 Maximal Hyperbolic Subspaces of an Orthogonal Geometry 277 Exercises 279 Metric Spaces 283 Open and Closed Sets 286 Convergence in a Metric Space 287 The Closure of a Set 288 Dense Subsets 290 Continuity 292 Completeness 293 Isometries 297 The Completion of a Metric Space 298 Exercises 303 Hilbert Spaces 307 Hilbert Spaces 308 Infinite Series 312 An Approximation Problem 313 Hilbert Bases 317 Fourier Expansions 318 A Characterization of Hilbert Bases 328 A Characterization of Hilbert Spaces 329 The Riesz Representation Theorem 331 Exercises 334 Tensor Products 337 Bilinear Maps 341 Tensor Products 343 When Is a Tensor Product Zero? 348 Coordinate Matrices and Rank 350 Characterizing Vectors in a Tensor Product 354 Defining Linear Transformations on a Tensor Product 355 The Tensor Product of Linear Transformations 357 Change of Base Field 359 Multilinear Maps and Iterated Tensor Products 363 Tensor Spaces 366 Special Multilinear Maps 371 Graded Algebras 372 The Symmetric Tensor Algebra 374 The Exterior Product Space 380 The Determinant 387 Exercises 391 Positive Solutions to Linear Systems Convexity and Separation 394 Convex Closed and Compact Sets 398 Convex Hulls 399 Linear and Affine Hyperplanes 400 Separation 402 Exercises 407 Affine Geometry 409 Affine Combinations 411 Affine Hulls 412 The Lattice of Flats 413 Affine Independence 416 Affine Transformations 417 Projective Geometry 419 Exercises 423 Operator Factorizations QR and Singular Value 425 Singular Values 428 The MoorePenrose Generalized Inverse 430 Least Squares Approximation 433 Exercises QRFactorization 434 The Umbral Calculus 437 The Umbral Algebra 439 Formal Power Series as Linear Operators 443 Sheffer Sequences 446 Examples of Sheffer Sequences 454 Umbral Operators and Umbral Shifts 456 Continuous Operators on the Umbral Algebra 458 Operator Adjoints 459 Umbral Operators and Automorphisms of the Umbral Algebra 460 Umbral Shifts and Derivations of the Umbral Algebra 465 The Transfer Formulas 470 A Final Remark 471 Exercises 472 References 473 Index 475 Copyright

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