This is more than a problem book: it is a complete inquiry-based course in linear algebra. It operates by looking at specific examples (usually matrices, sometimes vector spaces) to derive some conjectures and generalizations and then asking the reader to prove or disprove these. The problems are quite hard! Even if you know the subject well you may have to think a bit about some of them.
The book is much more chatty than the usual problem book. It’s full of statements like “Try this…” or “That doesn’t work because…”. The structure is the same as Halmos’s A Hilbert Space Problem Book, with a long problem section, a short hint section, and a long solution section. The narrative weaves back and forth between the problem section and the solution section, so you should read the solutions even if you solve the problems by yourself.
The book is to a large extent a re-working of Halmos’s earlier book Finite-Dimensional Vector Spaces, covering the same topics, but rearranged as a linked series of problems. There are no numerical exercises, but usually numerical examples are used to help discover the theorems. I find it much more accessible than the earlier book, both because of the extra narrative and because it is driven by examples.
One weakness of this book is that it is very insular: it never tells us where linear algebra came from or where it can go. The various properties defined and explored here were historically driven by problems in other areas of mathematics, in particular Hilbert spaces and operator theory, but we are not shown that. Rather than tying the subject to outside problems, the book uses a “characterization” approach: it defines a series of properties, and for each one attempts to characterize the objects that have that property and relate the properties to other properties that have been defined.
A Very Bad Feature of the book is that it has no index. If, for example, you read in the hint to problem 148 that you should use the spectral theorem but don’t know what that is, you may have a hard time finding out.
Is this a complete textbook, or just a handy supplement? That depends on your purpose. For most courses today it would be inadequate by itself because it doesn’t have any applications or any numerical techniques. If your course is really Matrix Algebra and not Linear Algebra, as many are, this is not the book for you! But it does cover everything that is strictly within the theory of linear algebra, in an interesting and fun way, and if you have decided you are going to focus on proofs and problem-solving, then this book has everything you need.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.