I had to take a linear algebra course in college. The first time through, I flunked. It seemed dull, filled with awful matrix computations, and rather pointless. I had to take it again, but with a different professor who brought the topic to life, explaining not only what we were doing, but some of the history, how it fit into mathematics, and the kinds of problems that could be solved. There was much less emphasis on computation. I loved it and passed with ease (if not an A, it was rather a while back). I’ve been interested in linear algebra since.

This book provides very detailed, very thorough coverage of basic linear algebra. The chapters cover: matrix algebra, square matrices of order 2, matrices and linear equations, vector spaces and subspaces, linear transformations, duality, determinants, polynomial expressions of linear transformations and matrices, diagonalizability, and forms.

The first chapter is a slow introduction to matrices, what they are, and how they work, covering the basic mechanics, including matrix multiplication, inversion, and the transpose. Gaussian elimination is not covered (this will happen in the third chapter), which seems a bit odd.

The second chapter is nicely located. It covers some interesting material with matrices small enough to be easily manipulated by hand. Mention is made of determinants, eigenvalues, and eigenvectors and the Cayley Hamilton theorem over 2x2 matrices over the complex numbers.

The third chapter covers solutions of linear equations and Gaussian reduction. Beginning with chapter four, vector spaces, linear transformations, and other topics are introduced. Typically, each section has some worked problems and a selection of unanswered problems that should give the reader some practice with the material covered. The theorems are stated clearly and the mathematics is precise and well written.

However, there are some things that bother me. First, no mention is made of numerical methods--this is probably appropriate for a pure mathematician, but numerical methods are the workhorses of much applied mathematics. In sections on Gaussian reduction and solutions of linear systems, the mention of numerical methods as well as of the things that can go wrong would be a nice addition.

There is a lot of computation--indeed, several places use the word “brutal” in conjunction with a computation of one sort or another. Some computation is healthy, maybe even a good deal of it, as it helps the reader to hone both computational abilities and instincts about the field. But at the same time, it risks losing readers who don’t care much for symbol pushing.

In several places the matrices are laid out oddly with the columns not lining up nicely (often this occurs where there are negative numbers in the entries). This is a bit visually jarring and tends to make them a bit harder to scan over when looking for structure.

Several times I wondered about a topic or wanted to double check on something, and then went to the back of the book to look it up only to discover there was no index.

This could make a very good junior- or senior-level textbook in a mathematics curriculum where the students had been exposed to formal proof methods and had developed some mathematical maturity. The proofs are nicely done and the problems are appropriate and often interesting.

On the other hand, there is little extraneous material and almost no context given about how the material fits in with the rest of mathematics. There’s an occasional hint in an aside, but generally the tone is no-nonsense mathematics. I’m not sure I would have passed a course using this text.