When, many moons ago, I was preparing for my PhD qualifying examination in algebra, and having had a very orthodox undergraduate upbringing pretty much across the mathematical spectrum (if you’ll excuse the pun), I asked UCLA’s Richard Elman for advice: what books should I look at, or rather, dissect, in order to get ready. He was particularly adamant about two books in particular, namely, Atiyah-MacDonald’s famous Introduction to Commutative Algebra and the book under review, Hoffman and Kunze’s Linear Algebra (MIT-style). I am happy to report that, contrary to my usual goofy ways in my often misspent youth, I followed Elman’s advice, and both of these books have served me very well in a number of different ways over the years even after my finishing my doctoral studies.
Regarding Hoffman-Kunze, suffice it to say that all undergraduate-level material is done the right way (and then some), meaning that everything is proved, very carefully and with no compromises, and material is dealt with that is most often introduced no earlier than in graduate algebra, or possibly in an honors course in advanced linear algebra.
I want to lament in this connection that over recent decades a shift has occurred regarding linear algebra: things once covered in lower division work are now part of the standard graduate course, and accordingly erstwhile solidly undergraduate linear algebra, e.g., the full discussion of the eigenvalue problem leading to the Jordan canonical form, have fallen off the undergraduate table. The result is that this fantastic book is not usable in most undergraduate linear algebra courses. But it is an unsurpassed text for highly gifted kids who intend to do graduate school right and then go on to do real mathematics.
(Indeed, in my Jurassic way I claim that, by and large, for genuine future theorem provers, as an MIT graduate colleague of mine calls them, the best sources are throw-backs. I admit that my views might also be somewhat parochial, given that even back in the 1970s my other choices for sources to study for the algebra “qual” from included Van der Waerden’s Modern Algebra, originally composed in Göttingen in 1931.)
All right, what else is there to say about the Hoffman-Kunze book? Well, as I already said, it’s all there. And when you need to retrieve something later, it’s a fantastic place to do it: the way things are done here resonate throughout the course of one’s mathematical maturation. Just have a look at the fantastic treatment of spectral theory in §9.5.
Qua pedagogy, there are problem sets throughout the book, and they’re carefully crafted: going through such a set typically requires the rookie reader to have a decent grasp of the attendant material, to be on target with the examples that are given in the preceding discussion, and to get his hands dirty; as is proper, the problems describe an orbit in the direction of increasing intricacy and difficulty.
I like this book for sentimental as well as mathematical reasons: it’s properly fitted in the very center of the algebra education of any serious mathematics student.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.