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Linear algebra occupies a central place in modern mathematics. This book provides a rigorous and thorough development of linear algebra at an advanced level, and is directed at graduate students and professional mathematicians. It approaches linear algebra from an algebraic point of view, but its selection of topics is governed not only for their importance in linear algebra itself, but also for their applications throughout mathematics. Students in algebra, analysis, and topology will find much of interest and use to them, and the careful treatment and breadth of subject matter will make this book a valuable reference for mathematicians throughout their professional lives. Topics treated in this book include: vector spaces and linear transformations; dimension counting and applications; representation of linear transformations by matrices; duality; determinants and their uses; rational and especially Jordan canonical form; bilinear forms; inner product spaces; normal linear transformations and the spectral theorem; and an introduction to matrix groups as Lie groups. The book treats vector spaces in full generality, though it concentrates on the finite dimensional case. Also, it treats vector spaces over arbitrary fields, specializing to algebraically closed fields or to the fields of real and complex numbers as necessary.

Tabla de contenidos provista por Syndetics

• Preface
• 1 Vector spaces and linear transformations
• 2 Coordinates
• 3 Determinants
• 4 The structure of a linear transformation I
• 5 The structure of a linear transformation II
• 6 Bilinear, sesquilinear, and quadratic forms
• 7 Real and complex inner product spaces
• 8 Matrix groups as Lie groups
• A Polynomials
• A.1 Basic properties
• A.2 Unique factorization
• A.3 Polynomials as expressions and polynomials as functions
• B Modules over principal ideal domains
• B.1 Definitions and structure theorems
• B.2 Derivation of canonical forms
• Bibliography
• Index

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CHOICE Review

In a crowded market, a book carries the burden to impress, or at least to disclose a unique selling point. This book's title seems wrong. S. Roman's Advanced Linear Algebra (3rd ed., 2011; CH, Feb'93, 30-3313) treats deeper material, and N. Jacobson's broad undergraduate introductory work, Basic Algebra 1 (2nd ed., 1985; 1st ed., 1974), matches the present linear algebra coverage even just as a side business. Here, Weintraub (Lehigh Univ.) focuses on basic structure theorems for linear transformations and forms without module theory. S. Axler's Linear Algebra Done Right (CH, Jul'96, 33-6354) establishes these results and is a much easier read. Axler, an analyst, avoids using determinants because they do not generalize well to infinite dimensions; but Weintraub, an algebraist, just gives them their traditional pride of place without comment. Dreary design and mostly unlabelled theorems make browsing or reference use difficult. Readers must bring mathematical maturity but receive paltry literature pointers, indications of sharper results, current research, or open problems. For example, although the author discusses classifying symmetric bilinear forms, he does not mention the work of C. Riehm and others on classifying general bilinear forms. Lastly, Galois theory never asserts the nonexistence of an algorithm for factoring polynomials. Summing Up: Optional. Only comprehensive mathematics collections serving upper-division undergraduates through researchers/faculty. D. V. Feldman University of New Hampshire