Linear Algebra

Michael E. Taylor, the author of this book, has been busy of late: in addition to this text, he is the author of three books in analysis recently published by the AMS: two of them, Introduction to Analysis in Several Variables and Introduction to Analysis in One Variable, are, like this book, apparently intended for an undergraduate audience, and the third, Introduction to Complex Analysis is intended as a graduate text. His webpage also references a fifth book, Multivariable Calculus, that has not yet been published. And these are just his recent publications: earlier ones include, for example, a three-volume magnum opus on partial differential equations. 

 

The book now under review, on linear algebra, has certain features in common with the one other book by Taylor that I am personally familiar with, the one on complex analysis. Taylor has an interesting way of looking at things, and, as will be discussed below, that occasionally nonstandard point of view results in books that are not like the competition, both in topic coverage and method of exposition.    

 

The book starts more or less from scratch with vector spaces over the real and complex numbers, and proceeds at a rapid clip to discuss (all in the first chapter) subspaces, linear independence and dependence and the existence of bases (in the finite-dimensional case only), linear transformations, determinants and row and column reduction. The fact that all of this is covered in about 45 pages should give some indication of the succinctness of Taylor’s exposition. The next chapter discusses eigentheory, and again accomplishes a lot in a small number of pages: in about 20 pages, the chapter goes from the definitions of eigenvalues and eigenvectors to the Jordan Canonical Form. 

 

The next chapter is on inner product spaces. It begins with the basic facts about orthogonality (e.g., the existence of orthonormal bases), segues into a fairly extensive discussion of linear operators on inner product spaces (self-adjoint, unitary, etc.) and culminates in a treatment of the polar decomposition, singular values, and the matrix exponential. (Interestingly, the author writes “Gramm” instead of “Gram”, a spelling that, as best as I can recall, I have not seen in any other linear algebra textbook.) This is followed by chapter 4, whose title refers to “other basic concepts”, but which includes discussions of topics not generally introduced in an introductory course—dual and quotient spaces, positive matrices (including a statement and proof of the Perron-Frobenius theorem) and convex sets (including a first look at the Krein-Millman theorem). 

 

From this point on, the book really starts to distinguish itself from competing linear algebra texts. The remaining four chapters would not look out of place in a sophisticated abstract algebra text: they cover such topics as multilinear algebra, general fields (including field extensions), modules, principal ideal domains and modules over them, polynomial rings, Noetherian rings and modules, algebraic numbers and algebraic integers, octonions, Cayley numbers, and general algebras (including Lie algebras, Jordan algebras and Clifford algebras).  The theory of modules over a PID is developed in detail and used to provide an alternative approach to the Jordan canonical form. This material is often seen in abstract algebra texts, but not so often in linear algebra ones. 

 

The book concludes with an Appendix covering further sophisticated topics: a proof of the Fundamental Theorem of Algebra, averaging rotations in the special orthogonal group, a section on groups (with applications to modular arithmetic and public key cryptography), and a section on fields (taking the reader through field extensions, up to and including the definition of the Galois group). A noted above, one would expect to find much of this material in a book on abstract algebra (or number theory) but not in a text on linear algebra. 

 

The book has a good number of exercises of reasonable difficulty. Some substantial topics are developed in these exercise sets rather than the text itself, including orthogonal complements and normal operators. Some of the exercises, as is the author’s wont, stray from linear algebra; a sequence of them, for example, discuss the use of ideal-theoretic arguments in basic number theory (greatest common divisor, Euclidean algorithm, etc.). There are no solutions to the exercises in the text, but many are accompanied by substantial hints. 

 

All of this is very nicely done, and while I am delighted to own such an interesting book, I was left wondering just who its expected readership is. Despite its placement in the AMS Pure and Applied Undergraduate Texts series (AKA the “Sally series”), this is, as the discussion above probably makes clear, really not a text for a beginning undergraduate course in linear algebra, or even, for that matter, a book for a second undergraduate course, at least not at a “normal” school like the one at which I teach.   A text for a graduate linear algebra course, then? Perhaps, but if that is the intent, why start from scratch with things like basic vector space theory and row reduction, which are often just assumed in such a course? And why include material that overlaps so significantly with graduate courses in abstract algebra? 

 

In the final analysis, this book reminded me of some of the books by another prolific author, Serge Lang, that I have read over the years: interesting, idiosyncratic, informative, and insightful, but probably more likely to be enjoyed by instructors than by their students.

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