Linear Algebra Problem Book

Dolciani Mathematical Expositions

by Paul R. Halmos

Can one learn linear algebra solely by solving problems? Paul Halmos thinks so, and you will too once you read this book. The Linear Algebra Problem Book is an ideal text for a course in linear algebra. It takes the student step by step from the basic axioms of a field through the notion of vector spaces, on to advanced concepts such as inner product spaces and normality. All of this occurs by way of a series of 164 problems, each with hints and, at the back of the book, full solutions. This book is a marvelous example of how to teach and learn mathematics by 'doing' mathematics. It will work well for classes taught in small groups and can also be used for self-study. After working their way through the book, students will understand not only the theorems of linear algebra, but also some of the questions which were asked which enabled the theorems to be discovered in the first place. They will gain confidence in their problem solving abilities and be better prepared to understand more advanced courses. As the author explains, 'I don't think I understand a subject until I know the questions … I wrote this book to organize those questions, problems, in my own mind.' Try this book with your students and they too will be able to organize and understand the questions of linear algebra.

Genres: Mathematics, Algebra, Textbooks

349 pages, Paperback

First published January 1, 1995

Reviews

[annalcoholic · Rating 5 out of 5]

ma pire passion

[Ben Bullock · Rating 5 out of 5]

The Linear Algebra Problem Book by Halmos is one of a series of "problem books" which were either created or edited by Halmos in the 1980s and 1990s, such as Halmos's "Hilbert Space Problem Book" or the "Polynomials Problem Book" by Barbeau. This one is published by the Dolciani Mathematical Expositions, a rather miscellaneous collection of texts.

The structure of the book is a set of questions and answers divided into several chapters. Each chapter has an introductory section and then the problems. In the middle of the book there are short hints to turn to if you get stuck, and then in the latter half of the book there are full solutions. The solutions are actually where the main content of the book is. Halmos gives not only the solution, but also some discussion about it.

I worked through most of the problems in this book except for the final chapter. Some of the problems were easy and some difficult, but one thing which is very noticeable is that the explanation given by Halmos in the introduction to the chapters was not enough to attempt the problems. The suggested answers, like the problems, tend to assume that you know a lot of material which he did not cover at all in the introductory material, so I think realistically this book can only be used as a supplementary text.

I didn't find any major errors in the book beyond a few typos, but in a couple of cases I found some much more concise solutions than the ones that Halmos gives.

[Viktor Velev · Rating 5 out of 5]

Honestly, the best book if you really want to understand Linear Algebra. Halmos makes you learn the concepts by doing the math yourself, which is the only way to learn math! Supplementing this read with Halmos' Finite-dimensional vector spaces is highly recommended!

[Adam Lantos · Rating 5 out of 5]

Wow, I was really surprised from the quality of the exercises! There are so many stuff that other books don't deal with that this book has problems on that it got me very excited!
Looking for something important that is not dealt or dealt in a poor manner in other books? Then check in this book, you will probably find it here!

[Chris Herdt]

Professor John Johnstone said this book, while containing just linear algebra problems and their solutions, is useful, as the solutions are well-explained.