Introduction to Modern Algebra and Matrix Theory

Unfortunately this reprint of a classic is misrepresented in the way it’s marketed: the dust jacket and the blurbs one reads on-line have it that “[t]his unique text provides students with a single-volume treatment of the basics of calculus and analytic geometry…” This characterization misses the mark by a mile. It were far more accurate to characterize this wonderful book as a thoroughgoing rendering of linear algebra at a concrete level (with a huge emphasis on determinants and therefore on, for lack of a better word, calculational aspects) and some “higher” algebra centered on group and field theory. The only place in the book where the word “derivative” appears, for example, is in an exercise on p. 243, dealing with the derivative of a polynomial (only): this is not a calculus book!

I guess that all this trouble arises due to an evident misinterpretation of the third paragraph of Sperner’s “Preface to Volume I of the German Edition,” where we read that “[t]his textbook is motivated by the idea of offering the student, in two basic courses on Calculus and Analytic Geometry, all that he needs for a profitable continuation of his studies adapted to modern requirements.” It looks like the putative course on calculus involves another book.

Regarding the advertised analytic geometry, it appears that if we take it in its broadest sense, we are much closer to the mark. Says Sperner: “a stronger emphasis than has been customary [is placed] on algebra, in line with recent developments in that subject” (written ca. 1930). Thus, we find in the pages of the book a superb discussion of affine space, Euclidean space, and linear transformations, and a wonderful development of the theory of determinants. This stuff, and its presentation in the book under review, may be old fashioned, but it is obviously critical and the authors’ discussion is quite elegant and thorough.

Furthermore, with the added coverage of some serious algebra, e.g., group theory (the authors go all the way up to the fundamental theorem of finitely generated abelian groups), this book serves as a good aid in the study of matrix groups, for example. The book’s fifth and last chapter, containing a wonderful discussion of invariant subspaces, as well as treatments of unitary, orthogonal, Hermitian and symmetric matrices, is worth the price of admission by itself. The chapter (and the book) ends with coverage of the Jordan normal form.

The book also contains a decent number of problems.

It is my view that Introduction to Modern Algebra and Matrix Theory would make a good supplementary text for courses in linear algebra and, possibly, something more advanced, say, along the lines of algebraic groups. Additionally, the book is a pleasure to read, especially if you’re disposed to explicit calculations.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.