June 23, 2010
by CRC Press
Textbook - 364 Pages - 2 B/W Illustrations
ISBN 9781439829660 - CAT# K11457
Series: Textbooks in Mathematics
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Solutions manual available for qualifying instructors
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.
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
Theory of a Single Linear Operator
Inner Product Spaces
Linear Operators on Inner Product Spaces
Trace and Determinant of a Linear Operator
Bilinear Maps and Forms
Appendix A: Answers to Selected Exercises
Appendix B: Hints to Selected Problems
… The book is well written, and the examples are appropriate. … Each section contains relevant problems at the end. The ‘What You Need to Know’ feature at the beginning of each section outlining the knowledge required to grasp the material is useful. Summing Up: Recommended.
—CHOICE, January 2011
Pedagogically, a structural and general approach is taken and, topically, the material has been chosen in order to cover the material a beginning graduate student would be expected to know when taking a first course in group or field theory or functional analysis. …
—SciTech Book News, February 2011
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