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Linear Algebra in Action

Harry Dym
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics
[Reviewed by
Ludovick Bouthat
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The present book is intended for students with limited exposure to linear algebra, although mathematicians at any level can genuinely appreciate its content. The prerequisites encompass a foundation in real and complex vector spaces theory, coupled with some matrix theory and a modest familiarity with linear transformations. Indeed, the initial twelve chapters largely constitute the material typically covered in a traditional introductory linear algebra course.
This excellent manuscript primarily draws its inspiration from graduate courses that the author conducted over a span of 40 years. In contrast to more exhaustive classical linear algebra texts like Linear Algebra and Its Applications by Strang, this book's aim is not to be comprehensive but to provide a well-rounded selection of tools and applications within the realm of linear algebra. The author articulates this intention succinctly, expressing that the book contains results he "wishes he had encountered during his graduate studies."
One standout feature of this book is its exceptional writing. It boasts a high degree of readability, particularly for those unaccustomed to such academic texts. The author's writing style is eloquently encapsulated by a recurring quote in the book: "I have tried to present the material in the way that most of the mathematicians that I know work rather than the way they write." This manifests itself in the book's approach, where the validity of theorems or lemmas is often “demonstrated” not through conventional proofs but via well-chosen examples. This approach fosters a robust intuition for problems in linear algebra, arguably one of the book's greatest strengths.
Throughout the book, readers will encounter numerous "keep in mind" and "warning" environments, used to call attention to helpful remarks and to conventions that have been introduced over time, respectively. These helpful entries are indexed for easy reference. Furthermore, the book features a plethora of exercises spanning various levels of difficulty, although solutions are nonexistent.
If there is one critique to offer, it would be the sparse inclusion of references, which are mainly confined to the supplementary notes sections at the conclusion of each chapter. Nonetheless, this minor shortcoming does not overshadow the multitude of exceptional qualities this book has to offer.
About the reviewer: Ludovick Bouthat ( is a PhD student at Université Laval.