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Linear Algebra and Geometry

Igor R. Shafarevich and Alexey O. Remizov
Publication Date: 
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[Reviewed by
Mark Hunacek
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Linear algebra and geometry are such nicely interconnected subjects — each one illuminating and enhancing the other — that I can’t help but wish that there were more undergraduate-level books devoted to an in-tandem development of both. Of course, many linear algebra books use geometry to illustrate the concepts discussed in the text — after all, it would be pedagogical malpractice to discuss subspaces without drawing lines and planes, or to discuss orthogonal matrices and transformations without discussing rotations and reflections in the plane — but the number of books which make a serious attempt to use the linear algebra to actually define and develop the geometrical ideas is somewhat smaller.

There are some, of course. Irving Kaplansky’s beautiful little book, Linear Algebra and Geometry: A Second Course, rescued from extinction by Dover, spends about two-thirds of the book discussing, from a fairly sophisticated standpoint, topics in linear algebra that lend themselves to geometry (inner product spaces, bilinear forms, orthogonal transformations) and in the last third defines affine and projective spaces in terms of linear algebra. A good prior background in the rudiments of linear algebra is necessary for this book.

The connections between bilinear forms and geometry are discussed in another Dover paperback, Snapper and Troyer’s Metric Affine Geometry, which also has a lot of material on affine spaces, but which does practically nothing with projective geometry. Gruenberg and Weir’s Linear Geometry also discusses affine and projective geometry, and transformations, from a linear algebraic point of view. Originally published by Van Nostrand, it now appears in the Springer Graduate Texts in Mathematics series (and came out in a paperback format a few years ago) — but it is probably too sophisticated for most undergraduates. Ryan’s Euclidean and non-Euclidean Geometry: An Analytic Approach makes extensive use of linear algebra, knowledge of which is assumed. Finally, a sentimental favorite of mine — the author was a former professor of mine when I was an undergraduate, and I sat in on a course he taught that seems to have influenced the book — is David Bloom’s Linear Algebra and Geometry, which is quite accessible but also concentrates only on affine and Euclidean spaces and, unfortunately, seems to be out of print.

All of these books are at least thirty, and in some cases more than forty, years old. The exception is Ryan’s book, which was published in 1986. Some newer ones include Tarrida’s Affine Maps, Euclidean Motions and Quadrics, and Gallier’s Geometric Methods and Applications for Computer Science and Engineering. Tarrida’s book is also limited to affine and Euclidean spaces and does not discuss projective geometry; Gallier’s book contains chapters on affine, Euclidean and projective geometry developed from linear algebraic principles, but does so within the framework of quite a lot of other topics, resulting in a large and dense book that may be somewhat intimidating to students. Both are fairly demanding and are likely outside the comfort zone of many undergraduates.

Given the relative paucity of books, especially recent books, explaining the connections between linear algebra and geometry at a level accessible to most undergraduates, I was quite pleased to have the opportunity to obtain the book under review, which appeared from the table of contents to be addressed to an undergraduate audience. It develops linear algebra from the beginning and also contains chapters on (among other things) Euclidean geometry, affine geometry, projective geometry, and hyperbolic geometry. Particularly given the fact that I had previously looked at and been impressed with Shafarevich’s Basic Algebraic Geometry, I had high hopes for this book. I was (with one exception, discussed later) not disappointed.

The first seven chapters of this book constitute a rigorous, sophisticated, but nonetheless still introductory course on undergraduate linear algebra, starting with linear equations and proceeding through matrices and determinants, vector spaces, linear transformations, Jordan form, bilinear and quadratic forms, and inner products.

The authors are initially a little vague about the underlying field of scalars: in the first two chapters (on linear equations and matrices) the generic term “number” is used to refer to the entries of a matrix or the coefficients of a linear equation, and it is not until the beginning of chapter 3, which introduces the notion of a vector space, that matters are made more precise, with the statement that, depending on the background of the reader, the word “number” may be thought of as referring to a real number, a complex number, or an element of an arbitrary field; the word “field”, though, is not formally defined until much later in the text.

That is a minor quibble, easily rectified in lecture by any professor who wishes to make matters more precise initially, and is certainly outweighed by some of the interesting material found in the first six chapters. Much of it is not often found in the elementary linear algebra literature. For example, there is a section (1.3) discussing nontrivial applications of linear equations (including a discussion of the discrete Laplace equation) and another section gives an extended discussion of applications of the Jordan form to differential equations (section 5.5).

Unfortunately, none of these chapters (or any of the chapters in the book, for that matter) contain exercises, so an instructor thinking about using this book for a course will have to invest some time in preparing suitable homework problems on his or her own.

It may seem strange that the Jordan form (chapter 5) is introduced before inner products (chapter 7), which is certainly not the way the subject is taught in most American linear algebra courses, but my guess is that the authors wanted chapter 7 (“Euclidean Spaces”) to be the first of a sequence of geometry-oriented chapters of the book. It is followed immediately by chapters on affine spaces, projective spaces, exterior products (including Plucker coordinates for a subspace), quadrics, and hyperbolic geometry.

All of the various geometries involved here are defined in terms of linear algebra. The development varies with the geometry, however, so a brief summary seems warranted. A Euclidean space, discussed in chapter 7, is a real vector space with a positive definite inner product. The authors use this chapter to not only discuss the basic facts about inner product spaces (orthonormal bases, orthogonal transformations, etc.) but also provide interesting applications to mechanics and, in a few final sections, extend the ideas discussed previously to bilinear forms that may not be positive definite. Ideas relevant to relativity theory (such as Lorentz transformations) make an appearance here. In chapter 8, affine spaces are defined (as in the books of Snapper and Troyer and Tarrida) in terms of a vector space operating on a set, with the operation satisfying certain properties. Due attention is paid to affine transformations, and also to affine spaces whose underlying vector space is also a Euclidean space.

Chapter 9 defines projective spaces: the “points” of the space are one-dimensional subspaces of a vector space, and the “lines” two-dimensional subspaces; the relationship of incidence between a point and a line is just set inclusion. Again, attention is paid to projective transformations, and a final (optional) section of the chapter addresses topological properties of real and complex projective spaces. The next chapter continues the study of the ideas of chapter 9 by explaining how coordinates can be attached to subspaces of a vector space, a subject that leads to consideration of the exterior powers of a vector space.

Chapter 11 discusses the classification of quadrics in affine, projective and Euclidean spaces, and the final geometry chapter, chapter 12, discusses a model of hyperbolic space that emphasizes linear algebra: the points of hyperbolic n-dimensional space are vectors x in Rn that satisfy q(x) = –1, where q is the Lorentz quadratic form. This chapter also contains a nice section on the axiomatic development of both Euclidean and non-Euclidean geometry.

The discussion in these chapters is very algebraic in nature — so algebraic, in fact, that at times it seems that there is very little of what a beginning student would consider “geometry”. For example, it would have been nice in the chapter on affine geometry to see statements and proofs of the theorems of Ceva and Menelaus, or even more elementary results like “the diagonals of a parallelogram bisect each other” or “the medians of a triangle are concurrent.” No such theorems appear. The word “centroid,” for example, appears neither in the index nor, as far as I could see, in the chapter itself. And how can one have a chapter on projective geometry without even mentioning the theorems of Desargues or Pappus? This is the one disappointment I mentioned. Contrast this, for example, with Tarrida’s book, which in the first chapter proves not only the concurrence of the medians but also more sophisticated geometric results such as the theorems of Thales, Ceva and Menelaus, and which also provides statements of the affine versions of Desargues’ and Pappus’ theorem in the exercises.

Following these chapters on geometry, there are two additional chapters that end the book: one on groups, rings and modules and the other on group representation theory. The first of these proves the fundamental theorem of finite abelian groups and then points out the similarities between the proof of this result and the Jordan form; this leads the authors to develop the theory of modules over a Euclidean ring to put things in their proper perspective. The next (and final) chapter, on group representations, introduces the reader to the basics of the subject: irreducible representations, characters, and so on. While the connections between these topics and linear algebra is certainly clear, these chapters struck me as a bit out of place, perhaps more appropriately put in a book on abstract algebra; nevertheless, I am loathe to criticize any mathematics text for putting in too much interesting material.

All in all, this is an interesting and valuable book, and while some features of the text (such as the lack of exercises) may deter some from using it as a text, it is well worth looking at by anybody interested in either linear algebra or geometry.

Mark Hunacek ( teaches mathematics at Iowa State University.