Linear Algebra Gems: Assets for Undergraduate Mathematics

Around 1990, the teaching of linear algebra came under examination, following the trend of calculus reform. The Linear Algebra Curriculum Study Group has been a dominant voice in leading the discussion since that time. The group, under the capable editorship of David Carlson, Charles Johnson, David Lay and Duane Porter, has once again provided a valuable resource for revitalizing one of the most basic courses in the math curriculum with this publication.

Having just finished teaching linear algebra for the first time last spring, and preparing to embark on my second opportunity, I was delighted to find Linear Algebra Gems: Assets for Undergraduate Mathematics, just before the start of classes. As with most who teach this course, I imagine, I found that only some of my students were mathematics majors. Many were computer science majors, some were economics majors, and several were undecided about which of these fields to pursue. When I picked up the book, I expected to find some ideas not only about the topics in the course, but also some reflections on the relevance of linear algebra outside of mathematics. I'm not sure where I could have gotten the idea that the book would provide some motivation to give my students; that's clearly not something the editors were attempting to do in this volume. (In fact, as mentioned below, there are such articles in a different MAA volume). But I was definitely rewarded in the mathematics I found.

Let me here explain the format of the book. It is most definitely a resource book, and not a text. The book is divided into ten parts, each organized around a central topic. Each part consists of many short, self-contained articles by various authors. About a third of the articles have not appeared in print before; the others are almost exclusively from the American Mathematical Monthly or the College Mathematics Journal. They are all short — most are only a page or two. This format makes it easy to pick up the book and expand (or recall!) one's knowledge or find a different viewpoint from standard textbook treatment without having to devote a large chunk of time. This was the intent of the editors, and they have succeeded. The sections of the book are:

• Partitioned Matrix Multiplication
• Determinants
• Eigenanalysis
• Geometry
• Matrix Forms
• Polynomials and Matrices
• Linear Systems, Inverses and Rank
• Applications
• Other Topics
• Problems

Most of these sections speak for themselves, but it is worth noting that "Geometry" refers to projections and least squares etc, that "Applications" refers to applications of linear algebra to other branches of mathematics rather than to other disciplines; that "Problems" contains over one hundred problems which could be used as homework and exam problems, although the editors warn that the "problems vary greatly in difficulty and subtlety, and we have not attempted to grade this difficulty." (p 293)

As I read, I recorded my comments about particular gems. To be expected, some of the articles left me unenthusiastic, wondering just why I was supposed to consider this a "gem". But the majority of them engendered very positive reactions.

My comments included:

"True! I didn't know this."
"This would make a great exam question or basis for a project."
"Cool! Use this in class."
"I wish I had read this before class started."
"Sweet! A simple argument for something basic presented as a fact in our text."

One of the gems that caught my attention pointed out that although matrix multiplication is not commutative, given matrices A and B, the products AB and BA do share eigenvalues. The proof supplied relies on partitioning matrices into blocks and on the fact that the eigenvalues of similar matrices are the same. This is intriguing and yet simple enough to use in a class. Other notable gems described constructing matrices with integral eigenvalues and constructing matrices with nice eigenspaces. For those readers like myself who are just beginning to think about this material again — and this time as an expositor rather than a student — these tips are enlightening and time saving. These are just two examples of the gems this book contains.

Linear Algebra Gems complements an earlier MAA publication, Resources for Teaching Linear Algebra (edited by Ann Watkins and William Watkins in addition to Carlson, Johnson, Lay and Porter, and previously reviewed in Read This!). Resources for Teaching Linear Algebra is a book about the pedagogy of linear algebra. It contains some wonderful essays by people who use linear algebra outside of academe (eg, Margaret Wright at AT&T Bell Labs, David Young at Boeing), who reflect on which topics in basic linear algebra they consider most useful in their careers now. Resources also contains essays regarding the struggle of teaching certain concepts, reflections on how to approach linear algebra in innovative ways, descriptions of activities which can be used in class, and thoughts on the exposition of linear algebra. Linear Algebra Gems, on the other hand, is all mathematics. I exaggerate only in that I have neglected to point out that each of the ten parts begins with a paragraph or two of introduction regarding the topic, sometimes pointing out particular difficulties students experience, or which articles are of a more advanced nature. Indeed, it is worth noting that while the focus of the book is on a first course in linear algebra, there are some articles of interest for a more advanced course.

In sum, both Linear Algebra Gems and its companion, Resources for Teaching Linear Algebra, belong on your bookshelf or within easy reach of all the faculty in your department. The latter will help foster the discussion of how we teach linear algebra and Linear Algebra Gems will have us talking over coffee in the lounge about neat mathematical tidbits.

 Michele Intermont (intermon@kzoo.edu) is Associate Professor of Mathematics at Kalamazoo College. Her area of specialty is algebraic topology.