This little book is a pleasure to read, but it certainly takes a special kind of beginning student to enjoy it. As the authors state in their preface, the book has a very limited set of mathematical prerequisites (the bare minimum can be reduced to basic high school algebra). However the mathematical maturity expected is most likely going to imply that a student reading this book has already been introduced to the basic tools and ideas of linear algebra before. If the "background" material introduced in the latter appendices (on groups, group actions, rings and algebras) is not in itself a good enough hint, the first section of the book, on groups and fields, will probably provide the right perspective. In fact, the group-theoretic terminology and examples can be avoided or skipped in a first reading, but the book clearly assumes a rather mature audience, presumably an advanced undergraduate.
The book is terse, in the sense that there is no detailed explication of the ideas and notions introduced, one after another, in rapid fire formation. However, there are several examples and exercises, and the details in the proofs are not scarce. There are many places where the reader is invited to fill in some details or supply the proof of a statement, but the exposition is clear and consistently elegant.
I would certainly not recommend this as a textbook for an introductory linear algebra course, unless the students are exceptionally motivated and sufficiently interested in abstraction and formal reasoning. On the other hand, I believe that this little book could be a perfect guide for that special wunderkind in your vector calculus class, who insists on majoring in engineering but cannot help being interested in mathematics proper. Students who are planning to go into graduate school in mathematics and need to brush up on their linear algebra or fill in gaps in their basics will definitely find this book a welcome treat. I can also see how the book may be used to jumpstart students in basic research in fields like operator theory and representation theory. In short, there are many who can get a substantial amount out of this little book, and I am definitely glad to have read it; I would just keep my ordinary freshmen away from it unless they are up for a challenge.
Gizem Karaali is assistant professor of Mathematics at Pomona College.