A Guide to Advanced Linear Algebra

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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.

A thorough development of a topic at the core of mathematics, ideal for graduate students and professional mathematicians.

Steven H. Weintraub is Professor of Mathematics at Lehigh University. He is the author of over fifty research papers and this is his ninth book.

Linear algebra has for decades been the course in the undergraduate
math major where the transition was made from the formulaic (often
called ""plug and chug"") solution of problems to where proofs are
the norm. To many, it is really two courses, the first linear
algebra without proofs and the second with proofs. Recently, a
third course has been added, computational linear algebra where the
emphasis is on computer graphics and the generation of images.
Nothing impresses potential math majors more than when they are
told how many mathematicians work in the film industry. This book
fits into the high end of the second course; it is almost
exclusively proofs, although the author cannot resist putting in a
few pages of computation. It is not suitable for the traditional
linear algebra course, not even for the last segment where proofs
take over. It is a book for a special topics course in linear
algebra for students at the junior/senior level, students preparing
for a qualifying exam or for working mathematicians that need an
overview of the main results of linear algebra and how they are
usually proved. In that niche, this book is excellent, Weintraub
keeps the math flowing, appropriately directional and justified,
and it is a rare occasion when he passes on including the detailed
proof. There are some times when a part of the proof is not
included, but it is rare and generally inconsequential. If you have
any current or potential need for understanding the theory of
linear algebra, this is a book that you need to have on your easy
access shelf."" - Charles Ashbacher, *Journal of Recreational
Mathematics*

""Linear algebra occupies a central place in pure mathematics. It
plays an essential role in such widely differing fields as Galois
theory, function spaces and homological algebra. In this context
linear algebra is about vector spaces and linear transformations,
not about matrices. The natural and perhaps most enlightening
approach to the canonical forms for linear transformations on
finite-dimensional vector spaces, one of the main goals of this
book, is via the basic structure theorems for modules over a
principal ideal domain, but the author has not gone that far. The
infinite-dimensional case is, however, treated. Thus, the book is
for advanced students of pure mathematics and not for those
requiring a textbook on numerically applicable linear algebra. It
is, moreover, written in a style to which a student of pure
mathematics is fully accustomed. Although the book is for advanced
students, it begins with the basics, but it does not deal with
matrix operations or the solution of systems of linear equations.
The chapter headings are: (1) Vector spaces and linear
transformations, (2) Coordinates, (3) Determinants, (4) and (5) The
structure of a linear transformation I and II, (6) Bilinear,
sesquilinear and quadratic forms, (7) Real and complex inner
product spaces, (8) Matrix groups as Lie groups, Appendix A:
Polynomials. Appendix B: Modules over principal ideal domains. This
book can be warmly recommended to any student of pure mathematics
requiring a precise and concise treatment of all the important and
""well-known"" results of linear algebra. The student will be
grateful for the direct route that it follows and for the
occasional explanations of the ""right"" way to understand the
material."" - Rabe von Randow, *Zentrallblatt*

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