## Advanced Linear AlgebraThis 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. The second edition contains two new chapters: a chapter on convexity, separation and positive solutions to linear systems and a chapter on the QR decomposition, singular values and pseudoinverses. The treatments of tensor products and the umbral calculus have been greatly expanded and there is now a discussion of determinants (in the chapter on tensor products), the complexification of a real vector space, Schur's lemma and Gersgorin disks. |

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### Contents

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 |

51 | |

55 | |

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 |

473 | |

475 | |

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### Common terms and phrases

According to Theorem adjoint affine associated sequence base field bijective bilinear form called cardinality Cauchy sequence Chapter char coefficients complement complete complex inner product complexification contains converges convex coordinate countable cyclic decomposition cyclic submodule Definition Let denoted diagonal diagonalizable dim dim dim dimension direct sum eigenvalues eigenvectors elementary divisors elements Example exists finite free module Hence Hilbert basis Hilbert space holds hyperbolic implies injective inner product space invariant factors invertible irreducible isometry isotropic ker ker lemma linear combination linear functional linear map linear operator linear transformation linearly independent map defined matrix metric space metric vector space minimal polynomial multilinear multiset nonempty nonsingular nonzero ordered basis orthogonal geometry pair positive principal ideal domain Proof Prove rational canonical form real inner product rk rk scalar multiplication Sheffer sequence Show spanning set submodule subset subspace suppose surjective symmetric symplectic geometries tensor product totally degenerate unitary universal property upper triangular

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