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Linear Algebra Concepts and Applications

Przemyslaw Bogacki
Publication Date: 
Number of Pages: 
AMS/MAA Textbooks
[Reviewed by
Benjamin Linowitz
, on

This textbook is, in many ways, a standard one. It is meant for a one-semester introduction to linear algebra at the sophomore level. It is written in the usual "definition/theorem/proof" style, yet with a conversational tone. Like many books at this level, both matrices and vector spaces are discussed, though the emphasis is on matrices. Numerous applications are mentioned, though few are covered in depth. Many exercises are given at the end of each chapter, with the answers to the odd numbered ones appearing at the end of the book. You probably get the point. This is a perfectly good, well-written introduction to linear algebra that most students would benefit from. But it is not the type of book that is going to inspire you to drop your favorite text when you next teach linear algebra.

Unlike most textbooks meant for a one-semester course in linear algebra, this one actually contains a semester's worth of material (not two or three semester's worth) and could probably be covered in it's entirety without having to go at a frantic pace. All of the usual suspects are covered -- it begins with chapters on the geometry of vectors in \(\mathbb R^2\) and \(\mathbb R^3\) and systems of linear equations, and ends with a chapter on eigenstuff and the SVD --  but not much else. Examples of omitted topics include inner product spaces, Markov processes and real quadratic forms.

The author rightly emphasizes the connections between linear algebra and geometry throughout the text. He moreover gives several genuinely interesting applications of the material to multivariable calculus. My favorites involve applications of the SVD to surface integrals and to Jacobian matrices and the geometry of change of variables for double integrals. In fact, I think that the author's 30 page treatment of the SVD and its applications is probably the high point of the text.

One disappointing aspect of the book is its physical appearance. The book is entirely in black and white, which is especially surprising given the text's emphasis on geometry. Moreover, statements of theorems and corollaries tend not to be shaded, causing them to blend in with the surrounding exposition. A final nit to pick regards the typesetting. The text is full of expressions like \[f(\begin{bmatrix}x \\ y \\ z\end{bmatrix})\] rather than \[f\left(\begin{bmatrix}x \\ y \\ z\end{bmatrix}\right),\] and \[\{\begin{bmatrix}x \\ y\end{bmatrix} : x,y\in\mathbb R\}\] rather than \[\left\{\begin{bmatrix}x \\ y\end{bmatrix} : x,y\in\mathbb R\right\}.\] Perhaps I'm a curmudgeon, but expressions like these look awful.

So will I use this book the next time I teach linear algebra? Probably not. It's a perfectly well-written, conventional text, but at the end of the day there simply isn't very much about the book that stands out from the many other linear algebra books on the market.

Benjamin Linowitz ( is an Assistant Professor of Mathematics at Oberlin College. His research concerns the theory of arithmetic groups, a fascinating area lying at the intersection of algebraic number theory and differential geometry. He is also interested in the history of mathematics, though as a spectator rather than an active participant. His website can be found at