Book Reviews: Three Books on Linear Algebra

Linear Algebra is one of the foundational topics for all applied mathematics. But compared to Analysis, it initially often feels stranger and less familiar. Although technically not hard, the level of abstraction is higher, making it hard to see what all the formalism is supposed to achieve.

The following three books do an outstanding job at presenting this material to the reader.

Introduction to Linear Algebra (5th ed)

This is a well-known beginner’s book on Linear Algebra: a first introduction to the topic for early undergraduate students. It’s a bit of a standard, now in its 5th edition.

It is particularly strong on the mechanics of Linear Algebra: working with vectors and matrices, manipulating rows and columns, operating with expressions and equations. It is unrivaled as a hands-on course to become familiar with vector and matrix expression. I only came across this book late, after using Linear Algebra almost daily for 25 years, but nevertheless learned a new trick or technique in nearly every chapter. That’s pretty impressive.

The emphasis is clearly on the solution of linear systems of equations, which the author considers “the central problem of linear algebra” (Chapter 2, first sentence). This view may not be universally shared: as physicist, at least, I cannot recall a single instance in physics where the solution hinges on the solution of a linear system of equations. In contrast, matrix diagonalization (and its relatives and descendants in Hilbert spaces) are foundational to much of theoretical physics: all of quantum mechanics depends on it, and so do many problems involving oscillation or vibrations. To be sure, the author covers diagonalization and eigentheory very well, too, but compared to solving systems of equations, they do take a backseat.

Nevertheless, I strongly recommend this book to learn about working with vectors and matrices from the ground up.

Linear Algebra Done Right (3rd ed)

Sheldon Axler’s book is in many ways the exact complement and follow-on to the book by Strang. It is intended for advanced undergraduates, and it intentionally avoids most of the “mechanics” of vectors, matrices, and (particularly) determinants. Instead, it adopts an abstract, “algebraic” approach, where one manipulates symbols and expressions, regardless of what they represent. Matrices are considered as operators, representing transformations on vector spaces, and eigentheory occupies a much more prominent role than in Strang’s book.

Since learning this abstract, algebraic way of thinking is one of the stated purposes of any introductory Linear Algebra class, it is important and necessary to get an introduction to this perspective on the field. This book helps, by not cluttering the presentation with calculational details.

I used to be a big fan of this book, but have found it less convincing of late. Some of the formalism now strikes me as unnecessarily fussy and obscuring the main points. The author’s decision to treat diagonalization separately for real and complex vector spaces introduces an additional distinction that matters little to most application-minded students. But it is still the most accessible introduction to the algebraic way of doing Linear Algebra that I am aware of.

Matrix Analysis and Applied Linear Algebra

Meyer’s book is a comprehensive, serious text book for graduate students and and practitioners. The treatment is rigorous, but the book’s emphasis and selection of topics is not geared towards pure mathematicians, but instead towards an audience consisting of scientists and engineers, who are going to use Linear Algebra as part of their day-to-day toolbox.

Of course, all the standard topics are covered, but so are several exciting non-standard topics, often related to applications. (Gershgorin circles. Resolvent of an operator. Matrix functions.)

The presentation is uncommonly careful and precise. (This is the only book I found that gives a complete algorithm for calculating the Jordan Normal Form of a matrix, correct in full generality and all edge cases — if you should really need it!)

If I was allowed only one book on Linear Algebra, it would doubtlessly be this one: for the range of topics, depth of the treatment, and the focus on applications. But I am not certain the book would be suitable as a first introduction to the topic for a beginner (even a diligent one).

(The book is published by a professional organization, the Society of Industrial and Applied Mathematics, and can be difficult to obtain from Amazon. It might be easier to buy it directly from the publisher.)