Is it fun to solve problems, and is solving problems about something a good way to learn something? The answers seem to be yes, provided the problems are neither too hard nor too easy.

The book is addressed to students (and teachers) of undergraduate linear algebra—it might supplement but not (I hope) replace my old Finite- Dimensional Vector Spaces. It largely follows that old book in organization and level and order—but only “largely”—the principle is often violated. This is not a step-by-step textbook—the problems vary back and forth between subjects, they vary back and forth from easy to hard and back again. The location of a problem is not always a hint to what methods might be appropriate to solve it or how hard it is.

Words like “hard” and “easy” are subjective of course. I tried to make some of the problems accessible to any interested grade school student, and at the same time to insert some that might stump even a professional expert (at least for a minute or two). Correspondingly, the statements of the problems, and the introductions that precede and the solutions that follow them sometimes laboriously explain elementary concepts, and, at other times assume that you are at home with the language and attitude of mathematics at the research level. Example: sometimes I assume that you know nothing, and carefully explain the associative law, but at other times I assume that the word “topology”, while it may not refer to something that you are an expert in, refers to something that you have heard about.