When Life is Linear: From Computer Graphics to Bracketology

One of the nice things about linear algebra, I’ve always thought, is that there is something in the subject for just about everybody. There’s a lot of beautiful theory, but at the same time those people who like to roll up their sleeves and get their hands dirty with computations, particularly in aid of interesting applications, will find much here to interest them as well.

At Iowa State University, we offer two different introductory undergraduate courses in linear algebra — one is a proof-based course intended for mathematics majors, the other is a more computational course with applications for non-majors. (There is also a more sophisticated joint undergraduate/graduate course in applied linear algebra.) I’ve taught the non-major course a couple of times, and enjoyed it, but have noted that most introductory texts are usually so busy developing the ideas behind linear algebra that they don’t really have time or space in which to really discuss the applications in any depth. Typically an application will just be developed rather briefly, which may result in it appearing somewhat contrived and artificial. The book under review does an excellent job of addressing these concerns, and would make a very useful supplement to a first course in linear algebra.

What the author has done in this slim (about 130 pages of text) book is to assemble a potpourri of interesting applications of linear algebra and discuss each one, if not exhaustively, in at least sufficient detail to give a reader a sense of how linear algebra enters the picture. The primary focus is on computer graphics and data mining, but the reader will also see discussions of cryptography, least squares approximation, compressed sensing, Markov processes and the Google pagerank algorithm, fractals, and sports ranking “bracketology”.

In addition to discussing applications, Chartier also develops the linear algebra that is necessary to understand them, though not in the kind of depth that one sees in a standard (four times as large) introductory text. This book is, therefore, not intended as a substitute for such texts; instead, linear algebra topics are introduced only to the extent that they are necessary to help explain some application. So, for example, matrix multiplication is defined, but determinants are not (at least not for general square matrices; the 2×2 case is mentioned, however). The author’s development of this background linear algebra is quite concise and not accompanied by the myriad examples that we see in the usual textbooks on the subject. There are also no exercises.

I think that it is the attempt to actually teach some linear algebra in the book that is its weakest feature. Presumably the author did so to make the book as broadly accessible as possible, but because the introduction of linear algebra concepts in this text is so concise, I have some doubts as to whether most people who are seeing this for the very first time will benefit greatly from the treatment here. So, while the author suggests in his Preface that he sees high school students as a possible intended audience, I suspect that, for most such students, this is a bit optimistic.

Moreover, I’ve always thought that applications are best introduced after the student has had some time to actually mull over and assimilate the main ideas that are being applied. In addition, had the author not spent time and pages discussing these basic facts about linear algebra, he could have put in more applications, which seem to me to be the real lifeblood of this book.

I therefore see this book as being most useful as an adjunct to an introductory course in linear algebra — either as an assigned supplementary text or as a desk reference for instructors wanting to spice up their own lectures with interesting examples that likely go beyond the coverage of the assigned text. Used in this way, I think it could be very valuable. The writing is clear, the applications are interesting, and a student with some knowledge of linear algebra would benefit greatly from knowing the material in this book.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.