When Irving Kaplansky died in 2006, the mathematical community lost both a first-rate mathematician and an exceptional writer. His gifts for lucid, compelling exposition are very much evident in this superb book, which I read in its first edition about forty years ago and from which I first became aware of the extent to which linear algebra can be used as a tool to study (and in fact actually define) various topics in Euclidean and non-Euclidean geometry.

I had sat in on an undergraduate course taught by David Bloom which related the two subjects (and which later morphed into Bloom’s textbook *Linear Algebra and Geometry*) but it was the book under review that taught me for the first time about the strong and beautiful connections between projective geometry and linear algebra. The interplay between linear algebra and geometry has interested me ever since, and I have made it a hobby over the years to read as much about these ideas as I can, an activity that has been fortuitously facilitated by my association with this column. See, e.g., Tarrida’s *Affine Maps, Euclidean Motions and Quadrics*, Gallier’s *Geometric Methods and Applications for Computer Science and Engineering*, Borceux’s *An Algebraic Approach to Geometry: Geometric Trilogy II*, and *Linear Algebra and Geometry* by Shafarevich and Remizov (which, unfortunately, despite the title, tends to obscure the geometry behind the algebra).

Kaplansky’s book, however, is not *just* about geometry from the linear-algebraic viewpoint; in fact, geometry doesn’t make an appearance until the third and final chapter of the book, after some fairly sophisticated linear algebra has been discussed in the first two. The book (a Dover republication of the second edition, which as far as I can tell differs from the first only in the inclusion of a very brief Addendum mentioning a few developments that occurred between 1969 and 1974) opens with a chapter on symmetric bilinear forms over an arbitrary field; Kaplansky refers to these as “inner products”, a term which for many people has a narrower meaning. Symmetric bilinear forms can be identified with quadratic forms, a subject about which entire books have been written. (Lam’s text is the most well-known example, I think, but there is also a very nice book by Snapper and Troyer called *Metric Affine Geometry*, now unfortunately out of print, that highlights the geometric interpretation of these.) Kaplansky manages to condense the subject down to its essentials (diagonalization, the law of inertia, Witt cancellation theorem, etc.) in about 55 pages of elegantly written text. Attention is also paid to Hermitian inner products, the theory over fields of characteristic 2, and to infinite-dimensional cases.

In the next chapter (the shortest in the book), Kaplansky discusses the interplay between spaces with symmetric bilinear forms and linear mappings defined on these spaces. Focusing on real and Hermitian inner products when necessary, Kaplansky discusses the basics of orthogonal, self-adjoint, normal and positive operators; the polar decomposition is also discussed. A section on orthogonal similarity is prefaced by a quick summary, mostly without proofs, of similarity in general, including an overview of the Jordan canonical form.

The third and final chapter introduces geometry. As Kaplansky puts it in the preface of the book, this chapter “represents my answer to the problem: given a roomful of students who have been crammed full of linear algebra, what geometry should you teach them, and how?” Actually this statement is a bit misleading, since this chapter does not require a student to be “crammed full of linear algebra”; a good entry-level course in the subject (abstract vector spaces and linear transformations, and, later, dual spaces), as well as a semester of abstract algebra (including the notion of a field) should allow the student to get quite far. (A decent background in abstract algebra is also required at times in the first two chapters, but generally speaking a student without such a background can just skip over a technical point or two and continue on.)

Chapter 3 begins with the definition of an affine plane. Although current practice — as in the books referenced in the first paragraph of this review — call for affine planes to be defined in a fairly technical way in terms of a vector space operating on a set, Kaplansky keeps things nice and concrete by simply defining an affine plane to be a two-dimensional vector space over a field: the “points” are the vectors in this space, and the “lines” are the additive cosets of the one-dimensional subspaces. This captures the basic properties of incidence (including the Euclidean parallel postulate) but does not address distance-related geometric properties. However, even here, there are interesting theorems to prove, such as those of Menelaus and Ceva, which may not be as well known as they should be. Kaplansky then brings metric considerations into our geometry, via the introduction of a non-singular inner product (“non-singular” just means that only the zero vector is perpendicular to everything), and proves some results that depend on distance and angles, such as the concurrence of the altitudes, and perpendicular bisectors of the sides, of a triangle. (The author does cheat here a bit by avoiding a full discussion of angles, a concept that can be fairly subtle; instead, he just focuses on right angles.)

The bulk of chapter 3, however, is concerned with projective geometry. Kaplansky defines a projective plane to consist of points (the one-dimensional subspaces of a three-dimensional vector space) and lines (the two-dimensional subspaces). By fixing a basis for the vector space, points and lines can be expressed in terms of homogeneous coordinates. Kaplansky spends most of the rest of this chapter using this analytic approach to the subject to discuss a wide array of topics in classical projective geometry, including: cross ratio (this was, as I recall, the first book to really make the idea clear to me); projective transformations; duality; conics; and the theorems of Desargues, Pappus, Pascal, Steiner and Brianchon. The discussion of duality, not surprisingly, makes use of the dual space of a vector space.

Throughout the book Kaplansky occasionally throws in sections of an expository nature, summarizing topics without proofs, just to give the student an idea of what’s out there. These sections include, for example, one on infinite dimensional inner product spaces in the first chapter, as well as sections in chapter 3 on higher dimensional geometry (everything was planar up to this point) and the consequences of looking at division rings instead of fields in the development of geometry; it is here that Kaplansky points out the connection between commutativity and Pappus’s theorem. A final section is a masterful one on the synthetic foundations of geometry, not just projective but Euclidean as well. An Appendix discusses the topology of projective spaces. Again, to quote the preface: “As the work comes to a close the theorems fade away in favor of hand-waving on a large scale. But that is the way every course should finish.”

This is very much a textbook, and the student reader is always kept firmly in mind. Just about every section ends with an assortment of exercises, many of them pretty hard.

There aren’t very many books that convey mathematics as well as this one, and in such an insightful and pleasant way. I thought this book was exceptional when I first read it, and, looking at it again, I see nothing to make me revise that opinion.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.