If you’re a mathematician, reading a book that is written by, and primarily intended for, physicists can sometimes be a frustrating experience. Notation can be unusual and awkward (especially if the Einstein summation convention is used), definitions can be imprecise, and theorems may not always be supported by proof. Michael Spivak, in volume 5 of his five-volume magnum opus on differential geometry, once described the difficulties he had reading a derivation of the wave equation, and I remember thinking how elegantly he had described my own feelings on that subject:

I’ve never really been able to understand which parts of the standard derivations are supposed to be obvious, which are mathematically simplifying assumptions, which steps are supposed to correspond to empirically discovered physical laws, or even what all the words are supposed to mean.

Thus, it was with a small sense of trepidation that I approached the book under review, which, although teaching mathematical topics, is part of *Springer’s Undergraduate Lecture Notes in Physics* series. However, I needn’t have worried. There are lots of physics examples in this book (including rotations, Minkowski spacetime and electromagnetism, Kepler motions and angular velocity) but these enhanced the mathematical discussions rather than detracted from them.

There were some occasional instances of strange terminology. I don’t believe, for example, that I have ever seen the term “abelian ring” in any abstract algebra textbook. I think that the authors’ use of the two phrases “linearly independent”, referring to vectors, and “free”, referring to sets of vectors, was unnecessary and unconventional. But these were few, minor, and non-disruptive. Also, unnecessary superscripts and the Einstein summation convention are avoided, so in a discussion of linear independence, for example, we get to see sensible expressions like \[\lambda_1v_1+ \dots +\lambda_n v_n=0_V\] rather than horrendous ones like \[\lambda_i v^i = 0_V.\]

The first six chapters concern basic sophomore-level linear algebra. The book starts with a very physics-oriented look at vectors (i.e., arrows) and basic vector calculus, but then quickly gets more algebraic, discussing vector spaces, inner products, matrices, determinants, linear transformations (and their relationship to matrices), and systems of linear equations. Vector spaces are mostly over the field of real numbers, although complex ones are mentioned as appropriate; brief reference is made to arbitrary fields but nothing much is done with them. The focus is on finite-dimensional spaces, and the existence of a basis is established only for these (by pruning down a finite spanning set to a basis).

Starting with chapter 7, the topics become somewhat more sophisticated. There are chapters on the dual space of a vector space, operators on inner product spaces (real and complex), diagonalization and (briefly and without proof) the Jordan canonical form, spectral theory, and quadratic forms. There is also a nice chapter that detours from standard linear algebra to discuss linear rotations in two and three dimensional Euclidean space, with discussions also of the exponential of a matrix, matrix groups, and a glimpse at Lie theory.

The last three chapters of the book discuss geometry from the point of view of linear algebra. The first two of these chapters address, respectively, affine geometry and Euclidean affine geometry, which is basically affine geometry with an inner product. (For a very nice book-length treatment of these interesting subjects, see *Metric Affine Geometry* by Snapper and Troyer.) The third chapter is on conic sections, with applications to physics.

I have some quibbles with the approach that is taken to affine and Euclidean affine geometry in this book. The general precise mathematical approach starts with a vector space \(V\) and an arbitrary set \(A\) on which \(V\) acts (satisfying some axioms); intuitively, we can think of the elements of \(V\) as “arrows” which move an element of \(A\) from the base of the arrow to the tip. The authors adopt a variation of this approach, limited to a very specific situation: as both \(V\) and \(A\), they take \(\mathbb{R}^n\), although in the case of the set \(A\) they use the notation \(\mathbb{A}^n\) to distinguish it from \(\mathbb{R}^n\) (even though it’s the same set). For two elements \(P\) and \(Q\) in \(A\), they define \(\alpha(P,Q)\) to be the vector \(Q – P\), which acts on \(A\) by ordinary vector addition. Using this formalism, the authors define lines and planes and discuss some basic facts about them.

This all seemed to me like a lot of fuss over nothing. Why not just define “point” to be an element of \(\mathbb{R}^n\) (or any vector space, for that matter), define line to be a coset of a one-dimensional subspace, “plane” to be a coset of a two-dimensional space, and so on? (And, for that matter, why use the totally unintuitive notation \(\alpha(P,Q)\) instead of the more geometrically compelling notation \(\vec{PQ}\)?) There are in fact some good reasons for the abstract approach, but the authors don’t make any kind of case for it; also, using only one specific example detracts from the benefits of the more general approach. Also, having gone to all this trouble, it seems like a missed opportunity to not use this machinery to prove actual geometric theorems (such as the concurrency of the medians of a triangle).

The writing throughout is generally clear, but sometimes a bit hurried; for example, the few pages devoted to the Jordan canonical form would likely, I think, give most undergraduates some trouble. Also, the fact that both authors teach at universities in Italy (and therefore presumably do not have English as their native language) is occasionally evident (e.g., exercise 10.1.4 begins “For the matrix \(A\) considered in the exercise 10.1.2 one has easily compute that \(A\)….” ) but these lapses are neither frequent nor severe enough to seriously interfere with the overall quality of exposition.

Although this issue is not a very big impediment to the use of this book as a text, something else is, and that is the omission of exercises. It is true that things that are called “exercises” are embedded throughout the book, but these are really worked examples rather than traditional exercises. As a very typical example, exercise 5.3.5 provides a \(3\times 3\) matrix \(A\), begins “Let us compute the inverse of the matrix \(A\)”, goes through a calculation, and then writes down the inverse explicitly. There is thus literally nothing for the student to do here other than just read the book.

I’ve never been a fan of back-of-the-book solutions to the exercises, but at least in that case, one can pretend that some students will spend time thinking about the problem before peeking. Providing the solution as part of the exercise itself, especially accompanied by phrases like “Let *us*…”, does not even *encourage* the students to attempt the problem on their own. And of course it goes without saying that any instructor who wants to assign graded homework will have to create his or her own problems.

There is another, less serious, omission: there is no bibliography. I think that encouraging upper-level students to use the library and become familiar with other books (or, even better, accessible articles) is an important part of their mathematical upbringing.

To summarize and conclude: Because of the problem with exercises, I wouldn’t use this book as a text for a course, but people who teach linear algebra might enjoy keeping a copy nearby for the occasional physics discussions.

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