Features

• Discusses the structure theory of an operator, various topics on inner product spaces, and the trace and determinant functions of a linear operator
• Covers bilinear forms with a full treatment of symplectic spaces and orthogonal spaces
• Explains the construction of tensor, symmetric, and exterior algebras

Pedagogical Features

• Takes a gentle approach that gradually builds from simple concepts to complex ideas and results
• Begins each section with an outline of previously introduced concepts and results necessary for mastering the new material
• Includes a wide variety of exercises and problems, with selected solutions in the appendices

Solutions manual available for qualifying instructors

Summary

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.