Joint Review of:

Linear Algebra is a subject taught in the service of several masters. A craving for the abstractness of vector spaces and a desire to showcase the art of proof tugs against a desire for computational tools. In classical terms, this dichotomy is often framed as offering a course for students of mathematics versus a course for students of engineering. These two texts display the broad range of approaches one might take to a linear algebra course.

Andrilli and Hecker’s *Elementary Linear Algebra* looks intimidating and at 750 pages it is certainly hefty. This stems from their focus on providing numerous exercises and plenty of examples. The authors also state that easing the transition from computation to abstract thinking is one of their goals. It is not clear to me that they succeed in this, although they do provide plenty of proofs in the text, and a few opportunities for writing proofs in the exercises. Instead, they succeed at building an algorithmic feel for the subject, with many summaries of how to do things like diagonalizing a matrix in six easy steps or testing a set of vectors for linear independence in three steps.

Each section in Andrilli and Hecker’s text ends with a list of new vocabulary and highlights. Of course this is a sound idea, but it is a bit overwhelming to find 10 pages of text followed by as many highlighted statements, and twice as many new vocabulary words. Most sections have similar proportions. Andrilli and Hecker apply this same diligence to the exercises, and there the effect is lovely. It allows the reader to choose more practice when desired, without it being necessary to complete every exercise.

Happily, the text includes a chapter devoted to applications and another on numerical methods. Coding Theory, Ohm’s Law, Computer Graphics as well as several applications to other mathematical topics are presented and mention is given throughout the book when the necessary background for an application has been completed. The last chapter, on numerical methods, collects the topics of LDU, QR, SVD decompositions and also deals with partial pivoting and finding dominant eigenvalues, all relevant topics for today’s average student.

At the other end of the spectrum is the book *A First Course in Linear Algebra* by Eie and Chang. Where Andrilli and Hecker tip the scales towards computation and leaving nothing unwritten, Eie and Chang take a more general approach. Chapter 1 defines a vector space over a field, and it defines the term field too. This tendency towards abstraction continues throughout the text. There are not enough exercises. For the sophisticated reader, the text provides some concreteness along with the general theory. For example, the first chapter not only defines abstract vector space; it also describes solving systems of linear equations and introduces echelon form. And the examples provided really are for those just learning the subject.

Eie and Chang don’t trouble themselves to think about applications of linear algebra, but they do discuss the Jordan Canonical Form and the Spectral Theorem. The text itself is well-written. There are only a few places where the phrasing seems a bit awkward, and this is no way diminishes the readability. Each chapter begins with a brief synopsis which nicely captures the heart of the chapter, and there are enough boldface headings and vocabulary to keep the reader oriented to the task at hand.

Each of these texts will appeal to a subset of students and faculty. While the subject matter is the same, the approaches differ tremendously. Which one will you choose?

Michele Intermont is Associate Professor of Mathematics at Kalamazoo College.