Advanced Linear Algebra

Designed to meet the requirements of a second course in linear algebra or a beginning graduate course, this book starts with familiar topics and builds to deeper results. Each section begins with definitions and results and offers extensive exercises with hints and selected solutions. The material adds motivation by referring to other fields of mathematics, such as functional analysis, representation of groups and algebras, field theory, and Lie algebras. Additionally, the text defines and proves the existence of tensor products and delves into the symmetric and exterior algebras.

Symbol Description	p. xv
Vector Spaces	p. 1
Fields	p. 2
The Space Fn	p. 7
Vector Spaces over an Arbitrary Field	p. 11
Subspaces of Vector Spaces	p. 15
Span and Independence	p. 25
Bases and Finite-Dimensional Vector Spaces	p. 31
Bases and Infinite-Dimensional Vector Spaces	p. 38
Coordinate Vectors	p. 42
Linear Transformations	p. 47
Introduction to Linear Transformations	p. 48
The Range and Kernel of a Linear Transformation	p. 56
The Correspondence and Isomorphism Theorems	p. 63
Matrix of a Linear Transformation	p. 67
The Algebra of L(V, W) and M_mn(F)	p. 74
Invertible Transformations and Matrices	p. 80
Polynomials	p. 87
The Algebra of Polynomials	p. 88
Roots of Polynomials	p. 99
Theory of a Single Linear Operator	p. 105
Invariant Subspaces of an Operator	p. 106
Cyclic Operator	p. 114
Maximal Vectors	p. 119
Indecomposable Linear Operators	p. 123
Invariant Factors and Elementary Divisors	p. 129
Canonical Forms	p. 138
Operators on Real and Complex Vector Spaces	p. 145
Inner Product Spaces	p. 151
Inner Products	p. 152
Geometry in Inner Product Spaces	p. 156
Orthonormal Sets and the Gram-Schmidt Process	p. 164
Orthogonal Complements and Projections	p. 172
Dual Spaces	p. 178
Adjoints	p. 182
Linear Operators on Inner Product Spaces	p. 189
Self-Adjoint and Normal Operators	p. 190
Spectral Theorems	p. 194
Normal Operators on Real Inner Product Spaces	p. 198
Unitary and Orthogonal Operators	p. 204
Polar Decomposition and Singular Value Decomposition	p. 210
Trace and Determinant of a Linear Operator	p. 217
Trace of a Linear Operator	p. 218
Determinant of a Linear Operator and Matrix	p. 224
Uniqueness of the Determinant of a Linear Operator	p. 241
Bilinear Maps and Forms	p. 249
Basic Properties of Bilinear Maps	p. 250
Symplectic Spaces	p. 261
Quadratic Forms and Orthogonal Space	p. 270
Real Quadratic Forms	p. 283
Tensor Products	p. 289
Introduction to Tensor Products	p. 290
Properties of Tensor Products	p. 300
The Tensor Algebra	p. 309
The Symmetric and Exterior Algebras	p. 316
Answers to Selected Exercises	p. 329
Hints to Selected Problems	p. 335
Index	p. 343
Table of Contents provided by Ingram. All Rights Reserved.

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Advanced Linear Algebra

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