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# The Art of Knowing When and How to Apply Linear Algebra in the Real World

When Life Is Linear: From Computer Graphics to Bracketology. By Tim Chartier, Mathematical Association of America, Washington, DC, 140 pages, 2015, \$50.00.

When students ask me what the most useful mathematics course is, I always respond, “Linear algebra.” The students are sometimes surprised, as their experiences in linear algebra appeared to them anything but useful. The students’ reactions stress that linear algebra courses vary in scope and purpose. Some are computationally focused, while others are theoretical and abstract and aim to develop students’ proof-writing abilities.

When our graduates visit after a few semesters of graduate school, however, they often echo comments about the usefulness and importance of this course. These are often the students who go into applied mathematics or related graduate programs – which usually present all of the applications of linear algebra and its role as the mathematical underpinnings for many theoretical and algorithmic results.

Tim Chartier’s book When Life Is Linear highlights the main concepts of a first course in linear algebra while developing their power through applications. The applications are engaging because Chartier uses examples that are accessible to most students, even high school students. Most inquisitive students have asked questions like: What goes into producing 3D animation? How does Google produce a ranking of web pages following a search inquiry? It is Chartier’s hope that by seeing applications of math and learning how it relates to the world, students will be enlightened, inspired, and motivated to find the usefulness of linear algebra.

The book is also a useful resource for the instructor, in that it shows how different techniques affect an application. For instance, one of the applications presented is in the area of classification and similarity measure. Chartier shows how the techniques of distance and angle calculation, least-squares analysis, eigenvectors, and principal component analysis all play roles in answering the question of how closely related items are, whether those items are movies, colleges, or purchases on Amazon. In addition, even simple matrix operations—such as addition, subtraction, elimination, and multiplication—are motivated and applied to the areas of graphics and image manipulation. The book shows how the use of each operation affects an image differently, providing motivation and application for operations whose coverage in a standard textbook is usually just a definition and some computational examples. The book also breaks images down into component parts, using the singular value decomposition and PCA to give the reader ideas about ways in which images can be compressed for storage.

What is remarkable about the book is that Chartier not only explains how these techniques are applied in the real world, but also inspires readers to be creative both in applying the techniques themselves and in choosing the types of problems to which they can be applied. As he says, “The difficulty in applying math isn’t in the complexity of the mathematical method but more in recognizing that a method, which can be quite simple, can be applied.” A great example occurs in his discussion of modeling college basketball game results as a simple system of linear equations and using Gaussian elimination to produce ratings for college basketball teams. He then shows how these ratings can be used to predict future (tournament) games. Chartier has gained celebrity in the media for his and his students’ research in predicting NCAA Basketball Tournament games. In the book, Chartier outlines some of the simpler ranking methods, such as the Colley and the Massey methods, and tells how some of his students were sufficiently inspired by the application to be successful in predicting previous tournament games.

The closest rank 1 (far left) and rank 10 (second from left) approximations to the images of the Dürer print shown in the two right-hand images, one of which is the closest rank 50 approximation and the other the original print. Can you tell which is which? Image credit: Figure 9.2 in When Life Is Linear.

One of the strengths of the book lies in its appropriateness as a companion text for either a computationally based or a proof-based course in linear algebra. Chartier wrote it as a companion text to his MOOC at Davidson College, and it would be perfect for a course that requires students to find and analyze real-world projects. It would serve equally well as outside reading to give students in more theoretical classes an idea of the applicability of the methods.

The book is not intended, though, as a stand-alone text. Chartier does not explain the derivation or calculation of certain methods in detail, and he often assumes the existence of calculator buttons that will perform certain operations, such as the SVD and least-squares analysis. Such material would need to be introduced in a course through lecture or supplied via a more traditional textbook or journal article. Moreover, some applications not presented in the book might serve as background material for other courses, notably techniques related to systems of differential equations and linear programming. However, I don’t think the purpose of the book is to give a comprehensive listing of all applications of linear algebra, and the missing applications can be supplied by more traditional texts if desired.

I find the book highly readable and think it would be enjoyable especially for students who have been exposed to some elementary mathematical topics, such as cosine functions, vectors and matrices, and distance functions. It is thus certainly appropriate for mathematically oriented high school students as well as more advanced students. Chartier succeeds in making the book simultaneously entertaining for students and informative for instructors. A lot of the applications are novel and not included in standard linear algebra texts, and I think many instructors of linear algebra will find the topics presented here new and interesting. Fans of Chartier’s earlier Math Bytes1 will see several similar topics, including the prediction of March Madness games and image manipulation. These topics are covered in more detail in When Life Is Linear and more of the linear algebra underpinnings are explained. But the main strength of the text is its ability to inspire creative thinking about ways to apply mathematics in real-world settings. For students beginning their journey of mathematical discovery, this book will serve as a valuable and inspirational resource.

1 T. Chartier, Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing, Princeton University Press, Princeton, New Jersey, 2014.

Kevin Hutson is an associate professor of mathematics at Furman University.