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.

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 Source: Journal of Recreational Mathematics