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Linear Algebra: Challenging Problems for Students

Fuzhen Zhang
Johns Hopkins University Press
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Fabio Mainardi
, on

When I was a student, I suffered from the lack of sufficiently interesting exercises on linear algebra: the problems in my textbooks were sometimes boring, with a lot of calculation but little need of imagination. This book offers a remedy for this situation and is therefore very welcome.

Zhang gathers about 400 linear algebra problems of varying difficulty, mainly at undergraduate level. It is aimed at students familiar with the elementary techniques and notions (e.g. linear combinations, matrix computations). The problems are often of the “show that…” kind, that is, the reader is required to provide a proof. Here is a typical example:

Problem 4.111. Let A, B be square complex matrices. Show that, if AB + BA = 0 and B is not nilpotent, then the matrix equation AX + XA = B has no solution.

Such a problem might be, I guess, hard to solve for many students, although it can be solved by completely elementary methods.

Some of the problems are actually theorems or propositions that can be found in advanced linear algebra books: these are the “challenging problems” referred to in the title. For this reason, this book is especially suited for mathematics students.

Each chapter starts with a reminder for the necessary background; hints and solutions are available at the end of the book. Of course, students are expected to refer to solutions as little as possible.

How to use this textbook: a) if you are an instructor, you may simply pick some problems for regular assignments; b) if you are a student, you can use it for reading, browsing freely through the problems, selecting some of them at random to work on. The problems are independent from each other within a chapter, so you don’t need to respect the order. It is a useful and stimulating book, ideal to prepare for an exam or a contest.

Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are mainly Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. At present, he works in a "classe préparatoire" in Geneva. He may be reached at

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