Advanced Linear Algebra focuses on vector spaces and the maps between them that preserve their structure (linear transformations). It starts with familiar concepts and then slowly builds to deeper results. Along with including many exercises and examples, each section reviews what students need to know before studying the material. The book first introduces vector spaces over fields as well as the fundamental concepts of linear combinations, span of vectors, linear independence, basis, and dimension. After covering linear transformations, it discusses the algebra of polynomials with coefficients in a field, concentrating on results that are consequences of the division algorithm. The author then develops the whole structure theory of a linear operator on a finite dimensional vector space from a collection of some simple results. He also explores the entire range of topics associated with inner product spaces, from the Gram–Schmidt process to the spectral theorems for normal and self-adjoint operators on an inner product space. The text goes on to rigorously describe the trace and determinant of linear operators and square matrices. The final two chapters focus on bilinear forms and tensor products and related material. Designed for advanced undergraduate and beginning graduate students, this textbook shows students the beauty of linear algebra. It also prepares them for further study in mathematics.

Table of Contents:
Vector Spaces Fields The Space Fn Vector Spaces over an Arbitrary Field Subspaces of Vector Spaces Span and Independence Bases and Finite Dimensional Vector Spaces Bases and Infinite Dimensional Vector Spaces Coordinate Vectors Linear Transformations Introduction to Linear Transformations The Range and Kernel of a Linear Transformation The Correspondence and Isomorphism Theorems Matrix of a Linear Transformation The Algebra of L(V, W) and Mmn(F) Invertible Transformations and Matrices Polynomials The Algebra of Polynomials Roots of Polynomials Theory of a Single Linear Operator Invariant Subspaces of an Operator Cyclic Operators Maximal Vectors Indecomposable Linear Operators Invariant Factors and Elementary Divisors Canonical Forms Operators on Real and Complex Vector Spaces Inner Product Spaces Inner Products Geometry in Inner Product Spaces Orthonormal Sets and the Gram–Schmidt Process Orthogonal Complements and Projections Dual Spaces Adjoints Linear Operators on Inner Product Spaces Self-Adjoint and Normal Operators Spectral Theorems Normal Operators on Real Inner Product Spaces Unitary and Orthogonal Operators Polar Decomposition and Singular Value Decomposition Trace and Determinant of a Linear Operator Trace of a Linear Operator Determinant of a Linear Operator and Matrix Uniqueness of the Determinant of a Linear Operator Bilinear Maps and Forms Basic Properties of Bilinear Maps Symplectic Spaces Quadratic Forms and Orthogonal Space Real Quadratic Forms Tensor Products Introduction to Tensor Products Properties of Tensor Products The Tensor Algebra The Symmetric and Exterior Algebras Appendix A: Answers to Selected Exercises Appendix B: Hints to Selected Problems Index