Linear Algebra: A Pure Mathematical Approach

The title and the cover make the nature of this book fairly clear. The title specifies that linear algebra will be treated as a part of pure mathematics. On the cover, one sees the equations defining a linear transformation, as follows:

(v + w)φ = vφ + wφ
(av)φ = a(vφ)

This highlights the author's decision that functions should be written to the right of their arguments: vφ, not φ(v). The preface argues that such a choice is more logical (which it is) and easier to read (which it probably isn't). It also indicates that this notation will not be used consistently; when applications to analysis are considered, functions will go back to acting on the left. The author, then, is writing for students with the mental agility to handle such notational acrobatics, i.e., for students are ready to go beyong the notation to interact directly with the concepts.

The preface makes it clear that the author thinks of this as a kind of "second course." He says students will be well-served to have had a course on "matrix algebra", at least in two and three dimensions, though this is not critical since all the necessary concepts are introduced afresh. And indeed, such a second course on linear algebra may well be a good idea for many undergraduate math majors.

The first chapter sets the tone by doing some algebraic preliminaries: it covers groups, rings, and fields and reviews some basics on permutation groups (presumably with a view to using them to define the determinant later). After that, the book covers the basic theory of linear spaces and linear transformations, including the Cayley-Hamilton theorem and the Jordan form. After a (rather strange) detour on finite fields, the book treats hermitian and inner-product spaces, and concludes with brief treatments of a few further topics.

Notably missing are quotient spaces, duality, and multilinear algebra. This puzzles me. Given a reader who is interested in a treatment of linear algebra at this level, it seems absurd to exclude this material. Yes, no book can include everything. But a book that invests this much in an abstract treatment might as well show how that approach pays off. The definition of the determinant via exterior powers, for example, would demonstrate the advantages of going abstract. 

Furthermore, duality and quotient spaces are such fundamental ideas that not treating them seems very strange. And since the chapter on finite fields uses the usual argument (constructing quotients of polynomial rings) to prove that fields exist, what's the point of avoiding quotients of vector spaces?

There is certainly room for a clean, well-written exposition of linear algebra as pure mathematics. Rose's attempt does not quite do it for me.

Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME.