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.

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 C0mplexification of a Real Vector Space, 53

 Exercises, 55

 Linear 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 Fn to Fm, 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

 Topological Vector Spaces, 79

 Linear Operators on Vc, 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

 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

 lndecomposable Modules, 158

  Exercises, 159

 The Structure of a Linear Operator, 163

 The Module Associated with a Linear Operator, 164

 The Primary Cyclic Decomposition of Vr, 167

 The Characteristic Polynomial, 170

 Cyclic and Indecomposable Modules, 171

 The Big Picture, 174

 The Rational Canonical Form, 176

 Exercises, 182

 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

 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

 Orthogonal Projections, 231

 Unitary Diagonalizability, 233

 Normal Operators, 234

 Special Types of Normal Operators, 238

 Self-Adjoint Operators, 239

 Unitary Operators and Isometrics, 240

 The Structure of Normal Operators, 245

 Functional Calculus, 247

 Positive Operators, 250

 The Polar Decomposition of an Operator, 252

 Exercises, 254

Part ll--Topics, 257

11 Metric Vector Spaces: The Theory of Bilinear Forms, 259

 Symmetric, Skew-Symmetric and Alternate Forms, 259

 The Matrix ofa Bilinear Form, 261

 Quadratic Forms, 264

 Orthogonality, 265

 Linear Functionals, 268

 Orthogonal Complements and Orthogonal Direct Sums, 269

 Isometries, 271

 Hyperbolic Spaces, 272

 Nonsingular Completions ofa 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, 30 l

 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

 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

     Tensor Algebras, 392

 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, 45 l

 Division Algebras, 462

 Exercises, 469

19  The Umbral Calculus, 471

 Formal Power Series, 471

 The Umbral Algebra, 473

  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, 49

 The Transfer Formulas, 504

 A Final Remark, 505

 Exercises, 506

Referenees, 507

Index of Symbols, 513

Index, 515