It was as an upper-level undergraduate, more than half a century ago, that I first became aware of the beautiful connections between linear algebra and geometry. A professor of mine, David Bloom, called my attention to Kaplansky’s wonderful book Linear Algebra and Geometry: A Second Course, and another professor, James Jantosciak, taught a senior seminar on bilinear forms and geometry, using as a text Snapper and Troyer’s Metric Affine Geometry, which in turn was based in part on Emil Artin’s classic Geometric Algebra. Bloom himself was so interested in the connection between these subjects that he later wrote his own book on the subject, Linear Algebra and Geometry, published by Cambridge University Press but now, apparently, out of print.

A lot of time has passed since those days, but my interest in using linear algebra to study geometry has not abated; I even wrote a chapter on the subject for the second edition of Handbook of Linear Algebra, edited by Leslie Hogben and published by CRC Press. Other authors apparently share my interest in this topic, as is evidenced by such books as Linear Geometry by Artzy, Linear Algebra and Geometry by Kostrikin and Manin, An Algebraic Approach to Geometry: Geometric Trilogy II by Borceux, Linear Algebra and Geometry by Shafarevich and Remizov, and Tarrida’s Affine Maps, Euclidean Motions and Quadrics.

And then there is, perhaps most recently, the book now under review, which, although it does not mention linear algebra in the title, uses that subject constantly throughout the text to develop the basics of affine, Euclidean, projective and hyperbolic geometry. Indeed, the first two chapters of the book are more algebra than geometry: Chapter 1 reviews linear algebra (“review” is perhaps a misnomer, because this chapter goes beyond what is typically taught in an undergraduate course—it discusses linear algebra over arbitrary fields and occasionally rings rather than just the real and complex numbers), and Chapter 2 discusses bilinear and quadratic forms (again, over arbitrary fields whose characteristic is not 2). Other topics discussed in Chapter 2 include the Gram determinant and the cross-product, defined on spaces of dimension greater than or equal to 3.

Geometry makes its first real appearance in Chapter 3, which introduces the notion of an affine space (defined linear algebraically) over an arbitrary field. A number of familiar theorems of affine geometry (for example, the theorems of Menelaus and Ceva) are established. In Chapter 4, the field is the real numbers, and an inner product is introduced, thus giving us Euclidean space.  Isometries of Euclidean space are discussed in some detail.

Chapter 5 returns to affine geometry, discussing hyperquadrics in affine n-dimensional space; this chapter can be thought of as an introduction to algebraic geometry.

The next six chapters (6 – 11) all involve projective geometry. In Chapter 6, projective planes and higher-dimensional spaces are first defined axiomatically; then, the projective spaces associated with a vector space are defined. Desargues’s Axiom, which is not true in all axiomatically defined projective planes but is true in projective spaces defined by a vector space over a field, is studied in some detail in Chapter 7. Pappus’s Axiom makes an appearance in Chapter 8, which also studies projective automorphisms of the projective space defined by an n-dimensional vector space over a field. Chapter 9 discusses the connections between affine and projective geometry and introduces the cross-ratio.

Chapters 10 and 11 touch on algebraic geometry. Chapter 10 discusses projective hyperquadrics, and Chapter 11 offers a proof of Bezout’s Theorem for curves in the projective two-dimensional space defined by a field. Proofs are also given of classical results such as Pascal’s Theorem.

In Chapter 12, the notion of “absolute geometry” (often referred to in other texts as “neutral geometry”) is introduced. This refers to the geometry that is obtained by taking Euclidean geometry without the parallel postulate. The authors give a long series of axioms for absolute geometry and discuss the Poincare half-plane as an example of a non-Euclidean absolute geometry. In contrast to projective geometry, where there are no parallel lines, in the half-plane model (which is an example of hyperbolic geometry) there are lots of them: through a point P not on a line l, there are infinitely many lines parallel to l.

In Chapter 13, the final chapter of the book, it is shown that Euclidean and some non-Euclidean geometries (such as hyperbolic geometry) can be thought of as living in projective spaces.

As can be seen from this summary of the contents of the book, there is a lot of interesting mathematics to be found here. A reader-friendly account of this material would be a valuable thing to have. Unfortunately, this book is not all that reader-friendly. Although it claims to be suitable as a text for undergraduates, it does not appear to me that American undergraduates (at least those I am familiar with) could get much out of this book.

For one thing, this is a book that often relies on symbols and notation rather than explanation. For example, in the author’s definition of both the Poincare half-plane and (in Exercise 7.2) the Moulton plane, the “lines” of those planes are defined by equations, with no verbal description of what they look like. (A drawing of the Poincare half plane is provided, but this is one place where a picture is not worth a thousand words.) The excessive use of notation and symbols might, it seems to me, cause an undergraduate reader to easily become lost.

Similarly, there is not much in the way of intuitive explanations or motivation offered. Example: at the start of Chapter 3, affine n-space is given a fairly complicated half-page long definition of a vector space acting on a set, but the author just plunges into the definition without first attempting to explain what is going on.  The definition of a projective plane in Chapter 4 is similarly unmotivated; there is, for example, no reference to perspective drawing. A similar thing is true of Chapter 11, in which absolute geometry is defined with no prefatory discussion of the history of the parallel postulate, the attempts to prove that it is independent of the rest of Euclid’s geometry, and the historical significance of the discovery of non-Euclidean geometries. Even the simple pedagogical tool of beginning a chapter by explaining what is going to be covered, and its significance, is frequently not employed.

Bottom line: faculty members and other professionals might find this book valuable as a reference, but a student would likely not want to learn this material for the first time from it.

Mark Hunacek (mhunacek@iastate.edu) is a Teaching Professor Emeritus at Iowa State University.