Linear Algebra

This book is freely available online. At its home site there are some comments that seem to reflect the notion that “if it’s free it can’t be very good.’ In fact, I chose it one semester for a linear algebra course I taught, and I thought it was excellent — in many ways superior to most commercially published textbooks. Doubters might ponder the major web sites hosted by the free Apache server, FreeBSD and Linux operating systems, or the free R statistical programming language which is now the tool of choice for advanced statistical research.

The book covers the core topics that have been in linear algebra courses for at least 60 years. The author says

The material is standard in that the topics covered are Gaussian reduction, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors.

The book takes an eclectic approach to more recent topics, generally via brief comments or optional “Topics” sections. There is no attempt to capitalize on the latest fashionable topics. There are brief mentions of applications throughout the text and about a dozen applications developed in some detail. I especially enjoyed the applications to balancing chemical equations and to least squares in statistics. All the vector spaces discussed are over fields the students have heard of. While the book is cognizant of computers, it is not a software treatise nor a guide to computational efficiency. Instead, the book’s great strength is an incredibly careful and thoughtful development of concepts.

Another standard is [the] book’s audience: sophomores or juniors, usually with a background of at least one semester of calculus. The help that it gives to students comes from taking a developmental approach — this book’s presentation emphasizes motivation and naturalness, driven home by a wide variety of examples and by extensive and careful exercises. The developmental approach is the feature that most recommends this book so I will say more. Courses in the beginning of most mathematics programs focus less on understanding theory and more on correctly applying formulas and algorithms. Later courses ask for mathematical maturity: the ability to follow different types of arguments, a familiarity with the themes that underlie many mathematical investigations such as elementary set and function facts, and a capacity for some independent reading and thinking. Linear algebra is an ideal spot to work on the transition.

Even at schools where learning to write proofs is mainly carried out in the real analysis course, it is a difficult transition for many, and a second gentle introduction may well help students attain maturity. However, learning to write proofs was not a goal of the course I taught, yet I still liked this book for the very thorough and detailed explanations. Students who think they are too wordy can be referred to Halmos. Even those students may appreciate the extremely detailed solutions manual that runs to over 200 pages. (Of course that means you will need to dream up other problems for tests or collected homework.)

When I used this text I found it helpful to provide students with some means of getting matrix computations done. I offered Octave (also free) on a CD, but many students preferred their graphing calculators for simpler problems. Certainly at this point in time there is no reason to not use technology. Many students also ended up printing out the entire book. Were I to use it again, I might consider having the bookstore print it, as that would be cheaper and faster and the pages would be bound. Still, that would lose the nice feature that the exercises and the answer key are linked so you can bounce back and forth between the two, thereby solving an ancient problem with mathematics textbooks.

I am sure there must be linear algebra courses for which this book would not be a good choice, but I think anyone planning to teach linear algebra should look at this book long enough to see if it fits their course. It deserves to be on the short list for most such courses. I will not say more here because the entire book is but a click away.

After a few years in industry, Robert W. Hayden (bob@statland.org) taught mathematics at colleges and universities for 32 years and statistics for 20 years. In 2005 he retired from full-time classroom work. He now teaches statistics online at statistics.com and does summer workshops for high school teachers of Advanced Placement Statistics. He contributed the chapter on evaluating introductory statistics textbooks to the MAA's Teaching Statistics.