Advanced linear algebra / Steven Roman.

Format:

Book

Author/Creator:

Roman, Steven

Series:

Graduate texts in mathematics ; 135.

Graduate texts in mathematics ; 135

Language:

English

Subjects (All):

Algebras, Linear.

Physical Description:

xvi, 482 pages : illustrations ; 24 cm.

Edition:

Second edition.

Place of Publication:

New York : Springer, 2005.

Summary:

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 many important applications.

The new edition has been thoroughly rewritten, both in the text and exercise sets, and contains new chapters on convexity and separation, positive solutions to linear systems, singular values and QR decomposition. Treatments of tensor products and the umbral calculus have been greatly expanded and discussions of determinants, complexification of a real vector space, Schur's lemma and Gersgorin disks have been added.

The author is Emeritus Professor of Mathematics, having taught at a number of universities, including MIT, UC Santa Barabara, the University of South Florida, the California State University at Fullerton and UC Irvine. He has written 27 books in mathematics at various levels and 9 books on computing. His interests lie mostly in the areas of algebra, set theory and logic, probability and finance.

Contents:

Part 2 Algebraic Structures 16

Part I Basic Linear Algebra 31

1 Vector Spaces 33

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

2 Linear Transformations 55

Linear Transformations 55

Isomorphisms 57

The Kernel and Image of a Linear Transformation 57

Linear Transformations from F[superscript n] to F[superscript m] 59

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 Paris 68

Topological Vector Spaces 68

Linear Operators on V[superscript C] 71

3 The Isomorphism Theorems 75

Quotient Spaces 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

4 Modules I: Basic Properties 93

Modules 93

Motivation 93

Submodules 95

Spanning Sets 96

Linear Independence 98

Torsion Elements 99

Annihilators 99

Free Modules 99

Homomorphisms 100

Quotient Modules 101

The Correspondence and Isomorphism Theorems 102

Direct Sums and Direct Summands 102

Modules Are Not As Nice As Vector Spaces 106

5 Modules II: Free and Noetherian Modules 109

The Rank of a Free Module 109

Free Modules and Epimorphisms 114

Noetherian Modules 115

The Hilbert Basis Theorem 118

6 Modules over a Principal Ideal Domain 121

Annihilators and Orders 121

Cyclic Modules 122

Free Modules over a Principal Ideal Domain 123

Torsion-Free and Free Modules 125

Prelude to Decomposition: Cyclic Modules 126

The First Decomposition 127

A Look Ahead 127

The Primary Decomposition 128

The Cyclic Decomposition of a Primary Module 130

The Primary Cyclic Decomposition Theorem 134

The Invariant Factor Decomposition 135

7 The Structure of a Linear Operator 141

A Brief Review 141

The Module Associated with a Linear Operator 142

Orders and the Minimal Polynomial 144

Cyclic Submodules and Cyclic Subspaces 145

The Decomposition of V[subscript tau] 147

The Rational Canonical Form 148

8 Eigenvalues and Eigenvectors 153

The Characteristic Polynomial of an Operator 153

Eigenvalues and Eigenvectors 155

Geometric and Algebraic Multiplicities 157

The Jordan Canonical Form 158

Triangularizability and Schur's Lemma 160

Diagonalizable Operators 165

Projections 166

The Algebra of Projections 167

Resolutions of the Identity 170

Spectral Resolutions 172

Projections and Invariance 173

9 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

10 Structure Theory for Normal Operators 201

The Adjoint of a Linear Operator 201

Unitary Diagonalizability 204

Normal Operators 205

Special Types of Normal Operators 207

Self-Adjoint 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

Spectral Resolutions and Functional Calculus 228

Positive Operators 230

The Polar Decomposition of an Operator 232

11 Metric Vector Spaces: The Theory of Bilinear Forms 239

Symmetric, Skew-Symmetric and Alternate Forms 239

The Matrix of a Bilinear Form 242

Quadratic Forms 244

Orthogonality 245

Linear Functionals 248

Orthogonal Complements and Orthogonal Direct Sums 249

Isometries 252

Hyperbolic Spaces 253

Nonsingular Completions of a Subspace 254

The Witt Theorems: A Preview 256

The Classification Problem for Metric Vector Spaces 257

Symplectic Geometry 258

The Structure of Orthogonal Geometries: Orthogonal Bases 264

The Classification of Orthogonal Geometries: Canonical Forms 266

The Orthogonal Group 272

The Witt's Theorems for Orthogonal Geometries 275

Maximal Hyperbolic Subspaces of an Orthogonal Geometry 277

12 Metric Spaces 283

The Definition 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

13 Hilbert Spaces 307

A Brief Review 307

Hilbert Spaces 308

Infinite Series 312

An Approximation Problem 313

Hilbert Bases 317

Fourier Expansions 318

A Characterization of Hilbert Bases 328

Hilbert Dimension 328

A Characterization of Hilbert Spaces 329

The Riesz Representation Theorem 331

14 Tensor Products 337

Universality 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 Antisymmetric Tensor Algebra: The Exterior Product Space 380

The Determinant 387

15 Positive Solutions to Linear Systems: Convexity and Separation 395

Convex, Closed and Compact Sets 398

Convex Hulls 399

Linear and Affine Hyperplanes 400

Separation 402

16 Affine Geometry 409

Affine Geometry 409

Affine Combinations 41

Affine Hulls 412

The Lattice of Flats 413

Affine Independence 416

Affine Transformations 417

Projective Geometry 419

17 Operator Factorizations: QR and Singular Value 425

The QR Decomposition 425

Singular Values 428

The Moore-Penrose Generalized Inverse 430

Least Squares Approximation 433

18 The Umbral Calculus 437

Formal Power Series 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.

Notes:

Includes bibliographical references (pages [473]-474) and index.

ISBN:

0387247661

OCLC:

57452536