Linear Algebra

Here at Iowa State University, the traditional undergraduate linear algebra course comes in two flavors, one intended primarily for mathematics majors and the other for majors in other disciplines. While both courses cover essentially the same content (linear equations, matrices, determinants, vector spaces, linear transformations, eigenvalues and eigenvectors, and inner product spaces), the method of presentation varies. The majors course is a four-credit, 300-level course that emphasizes proofs; the non-majors course is a three-credit, 200-level course that focuses on routine applications, computations and verifications (e.g., Are these vectors linearly independent? Does this system of linear equations have a solution and, if so, is it unique? Is W a subspace of V?).

The book now under review is intended as a text for the first kind of course. It starts from scratch and does not presuppose any prior knowledge of linear algebra, but it develops the material rigorously and precisely, and expects the student to be able to read and construct proofs. It covers all the content listed above, and then some: there are also discussions of topics not always covered in a first introduction to linear algebra, such as the SVD decomposition and spectral theory of self-adjoint and normal operators.

Topics not covered in the book include the minimal polynomial, the Jordan canonical form, and bilinear and quadratic forms. These omissions seem entirely reasonable in a book that is intended as a text for a first course in the subject.

As will be explained below, there is quite a lot to like about this book, and the few quibbles that I do have are largely a matter of personal taste.

First, the good news. One outstanding feature of the book is the authors’ writing style, which is clear, conversational and humorous, while at the same time rigorous and informative. The level of sophistication of the text rises as the book progresses, but even at the end the book is still quite readable and accessible. I think that this is a book that students will actually read, not just use as a collection of exercises. The authors have clearly given the pedagogical aspects of the subject a lot of thought, and do an excellent job of anticipating problems before they occur and dealing with them. (For example, in discussing the Cayley-Hamilton theorem, the authors give, and explain the error in, the standard incorrect “proof” of just substituting the matrix for the variable in the definition of the characteristic polynomial.)

A useful pedagogical feature is the inclusion of numerous “Quick Exercises” embedded in the text, the vast majority of which have solutions in small, upside-down, orange type at the bottom of the page. These encourage students to pause and reflect while they read, always a good practice for learning how to read mathematics texts. The immediate availability of solutions is ameliorated to some extent by the fact that the reader has to turn the book upside-down (and perhaps use a magnifying glass) to read them.

Another useful feature, especially for students for whom this may be a first “proofs course”, is the inclusion of an appendix on basic logic and proof techniques. This is a short (roughly five pages of text) section, but it is sufficiently clear that it can be assigned as outside reading, without having to spend a class period on the subject.

There are also quite a few examples that illustrate and illuminate the general theory. In addition to examples, every section ends with a selection of exercises of varying degrees of difficulty: some are challenging, but there are certainly quite a few on the easy-to-moderate end of the spectrum. Solutions to most of the odd-numbered ones appear in the back of the book, and the publisher has made available a password-protected solutions manual for all of them. This manual is itself a significant piece of work, weighing in at almost 200 pages of text.

Another nice feature is the successful blending of theory with computation and applications. Several interesting examples (integral kernels, least squares, function approximation) are discussed. There is also a section on linear codes, though this requires consideration of abstract fields (about which, more later). In what I view as a definite plus, the geometrical aspects of the subject are continually stressed. Another definite plus is the discussion of some ideas (like the SVD decomposition) from multiple perspectives.

So, as I said, there is much to like about this book. But then there are the inevitable nits that a reviewer is expected to pick. My biggest quarrel with the book is the very late (last chapter) introduction of determinants. It seems odd to me to have a standard topic like determinants, which is covered in every first course in linear algebra, come after a discussion of the spectral theorem, which is not.

In this regard, the authors may be influenced by Sheldon Axler, whose text Linear Algebra Done Right champions this approach (as did an earlier article of his, appearing in the February 1995 American Mathematical Monthly, called “Down With Determinants!”.) I am not a fan of this approach; I think that determinants are quite useful, both in other branches of mathematics (such as algebraic geometry and algebraic number theory) and also as a tool in an introductory linear algebra course. (More details can be found in the review of Axler’s book that Leslie Hogben and I co-authored in the June 2016 issue of the Monthly.) Suffice it to say that I think that introducing them at the end of the course complicates earlier examples (such as, obviously, calculation of eigenvalues) and deprives the students of a useful tool throughout the semester.

The decision to defer determinants until the very end of the book also results in a very late definition of the characteristic polynomial. So, for example, the fact that an \(n\times n\) complex matrix has \(n\) eigenvalues (counting multiplicity) is not established until about a dozen pages from the end of the book.

While determinants are introduced late, some other topics are introduced early — perhaps a bit too early. Abstract vector spaces are defined in the very first chapter, for example, before the students have had a whole lot of practice dealing with ordinary Euclidean spaces. General linear transformations are introduced in the next chapter, and eigenvalues of transformations and matrices appear very shortly thereafter.

It should also be noted that the authors work in vector spaces over an arbitrary field, not just the real or complex numbers. The notion of an abstract field is also introduced quite early in the book (page 39). My own feeling is that this level of abstraction is unnecessary and quite likely counterproductive in a first course in linear algebra.

Perhaps anticipating this objection, the authors state that the use of arbitrary fields is only strictly necessary in the section on linear codes, and that “readers who are interested only in real or complex scalars can skip to Section 1.5, mentally replacing each instance of \(\mathbb{F}\) with \(\mathbb{R}\) or \(\mathbb{C}\).” I view things differently, however: if abstract fields are really not that necessary, why run the risk of confusing the student with new terminology? I can’t help but wonder if a better solution would be to relegate the material on abstract fields to an appendix, and point out there that most everything done with vector spaces over \(\mathbb{R}\) or \(\mathbb{C}\) applies over arbitrary fields. The section on linear codes can remain in the body of the text, prefaced by a warning that it requires knowledge of the appendix section.

One final complaint: there is no bibliography. In just about every upper-level majors course that I teach, I make a point of encouraging the students to poke around our university library and see what other books (or perhaps even elementary journal articles) are out there on the subject. I view this as a very valuable, if not essential, tool in learning mathematics, and a good bibliography would assist in this enterprise.

So, to summarize and conclude: although I have some concerns with the order of presentation of some of the material, I certainly believe that a student who works through this text will gets an excellent grounding in linear algebra. He or she will actually learn what linear algebra is about and how it can be used, and will be in an excellent position to go on to more advanced courses using this material.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.