Advanced Linear Algebra
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Advanced Linear Algebra
Author: Steven Roman
language: en
Publisher: Springer Science & Business Media
Release Date: 2007-09-20
For the third edition, the author has added a new chapter on associative algebras that includes the well known characterizations of the finite-dimensional division algebras over the real field (a theorem of Frobenius) and over a finite field (Wedderburn's theorem); polished and refined some arguments (such as the discussion of reflexivity, the rational canonical form, best approximations and the definitions of tensor products); upgraded some proofs that were originally done only for finite-dimensional/rank cases; added new theorems, including the spectral mapping theorem; corrected all known errors; the reference section has been enlarged considerably, with over a hundred references to books on linear algebra. From the reviews of the second edition: “In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials. ... As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields. ... the exercises are rewritten and expanded. ... Overall, I found the book a very useful one. ... It is a suitable choice as a graduate text or as a reference book.” Ali-Akbar Jafarian, ZentralblattMATH “This is a formidable volume, a compendium of linear algebra theory, classical and modern ... . The development of the subject is elegant ... . The proofs are neat ... . The exercise sets are good, with occasional hints given for the solution of trickier problems. ... It represents linear algebra and does so comprehensively.” Henry Ricardo, MathDL
Advanced Linear Algebra
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.
A Concise Text on Advanced Linear Algebra
This engaging, well-motivated textbook helps advanced undergraduate students to grasp core concepts and reveals applications in mathematics and beyond.