*Introduction to Linear Algebra, 5th Edition*

by Gilbert Strang

Wellesley - Cambridge Press, 2016, ISBN 978-0-9802327-7-6, x+574 pages.

Reviewed by Douglas Farenick, University of Regina, douglas.farenick@uregina.ca

Undergraduate mathematics textbooks are not what they used to be, and Gilbert Strang’s superb new edition of *Introduction to Linear Algebra* is an example of everything that a modern textbook could possibly be, and more.

First, let us consider the book itself. As with his classic *Linear Algebra and its Applications* (Academic Press) from forty years ago, Strang’s new edition of *Introduction to Linear Algebra* keeps one eye on the theory, the other on applications, and has the stated goal of “opening linear algebra to the world” (Preface, page x). Aimed at the serious undergraduate student – though not just those undergraduates who fill the lecture halls of MIT, Strang’s home institution – the writing is engaging and personal, and the presentation is exceptionally clear and informative (even seasoned instructors may benefit from Strang’s insights). The first six chapters offer a traditional first course that covers vector algebra and geometry, systems of linear equations, vector spaces and subspaces, orthogonality, determinants, and eigenvalues and eigenvectors. The next three chapters are devoted to the singular value decomposition, linear transformations, and complex numbers and complex matrices, followed by chapters that address a wide range of contemporary applications and computational issues. The book concludes with a brief but cogent treatment of linear statistical analysis.

I would like to stress that there is a richness to the material that goes beyond most texts at this level. Included are guides to websites and to OpenCourseWare, which I shall comment upon later in this review. The final page lists “Six Great Theorems of Linear Algebra.” Chapter 7 begins with an informative account of image compression, and would be wonderful material for an undergraduate student to present in a seminar to other students.

Strang’s experience at writing and teaching linear algebra is apparent in the layout of the typeset. For example, on page 5, after developing material on linear combinations of vectors, we find the heading “The Important Questions.” On page 149, after studying the null space, there is a subsection with the heading “Elimination: The Big Picture.” Each section contains the headings “Review of the Key Ideas,” “Worked Examples,” “Problems,” and “Challenge Problems.” These sections are essential reading for the instructor, not just the student. The Worked Examples include material such as the Gershgorin Circle Theorem, while the Problems and Challenge Problems offer the student a chance to master basic ideas and to think much more mathematically about the concepts under study. For example, Problem 29 of Chapter 6 asks for the computation of the eigenvalues of three matrices (not just generic matrices, but matrices with structure and, thus, a chance to learn something about how the features of the matrix influence the eigenvalues), while Problem 39 of the same chapter asks for the possible values of the determinants, traces, and eigenvalues of the six 3 X 3 permutation matrices. There is nothing here that can be said to be dry, uninteresting, or irrelevant; rarely does an undergraduate mathematics text feel so alive as this one.

This review appeared in the Bulletin of the International Linear Algebra Society, IMAGE Vol.58 (2017) 18-19