Elementary Linear Algebra enters its tenth edition as one of the most popular linear algebra texts in use today. All of the standard material you would expect in a sophomore-level linear algebra course can be found here, plus some nice additional topics. The authors recognize several points of view in the narrative — geometrical, algebraic, abstract, computational, and applied — without focusing on any one approach.
The text is appropriate for a beginning course. In particular, the expectations for proof-reading are modest and the student need not have seen vectors or other topics before their introduction. Any exercises requiring calculus are clearly marked and are not essential.
The development is pretty well illustrated by the table of contents. The last chapter, “Applications of Linear Algebra,” is available only in the Applications Edition (discussed below), but most chapters in the main text also have a section or two dedicated specifically to timely example applications.
There are also two appendices: “How to Read Theorems” (in just two pages!) and “Complex Numbers,” which provides background for the section on complex vector spaces.
The writing is clear, concise, and appropriate for the audience. A typical section will begin with an overview and a few motivating examples and then present relevant theorems with proofs, discussion, and more examples. New to this edition, each section closes with a brief concept review and skills summary.
There are lots of exercises for each chapter. In addition to the standard practice problems, there is a sufficient number of more interesting computations and proofs. The author saves some of the proofs from the chapter text for the exercises so that students — many of whom will be new proof-writers — have a model, and there will also be some more challenging problems and proofs. Most exercise sets close with some true/false questions (new to this edition). Solutions to selected exercises are in the back.
There is not much reference in the text to the use of computer algebra systems or other software, but there is a collection of special technology exercises on the publisher’s website available for free without a login. These exercises are conveniently presented in pdf format with data files available in Maple, MATLAB, and Mathematica. A full solution guide and other instructor resources are also available at the publisher’s website.
Changes in the Tenth Edition. Beyond the changes already discussed, the new edition features some much-needed reorganization of the material. The chapters on vectors and Euclidean spaces have been consolidated into one, addressing a specific frustration this reviewer had with the previous edition. Eigenvectors have been moved up in the book and complex vector spaces are now introduced right after eigenvalues, which makes sense. Applications have also been expanded considerably and have been more deliberately and clearly integrated into the main text. Overall, the changes meaningfully improve the previous edition.
As mentioned earlier, one significant change is that the technology exercises have been removed from the print edition altogether and moved online.
Applications Edition. The text is sold in two versions, with the Applications Edition being identical to the regular edition except for the inclusion of an additional chapter entitled “Applications of Linear Algebra.” These are (mostly) stand-alone sections on a variety of applications from biology (e.g., genetics, population growth), computer science (cryptography, computer graphics), and other areas (economic models, forest management, fractals and chaos). Prerequisites for each section are clearly indicated and the pages in the extra sections are tinted for easy access. They make good projects
The sections are the same as the previous edition except that the unit on electric networks (this reviewer’s favorite) has been integrated into the main text as an example of applications of linear systems.
Closing thoughts. Linear algebra is a topic that can be approached in many different ways, and anyone who has taught the course a few times will gladly tell you which approach is correct. Anton and Rorres is the sort of book that has enough in it that even if you don’t subscribe to their exact development of the material, you can probably find your own path through it and find good exercises. Moreover, this new edition improves on its predecessor enough to compensate for the annoyance of updating.
Bill Wood is a freelance mathematician living in Conway, Arkansas.