Elementary Linear Algebra. By Evar D. Nering. Saunders, Philadelphia, Pennsylvania. 1974 ix + 375 pp. $12.50. (Telegraphic Review, March 1974.)

Can linear algebra wander far from its geometric moorings and still be successful at the sophomore level? Those that are convinced that the answer must be "no" have heretofore been left with two choices--insubstantial books firmly rooted in geometry, and substantial books inadequately rooted in motivation. As an example of the first, Introduction to Modern Algebra, by John Kelly, D. Van Nostrand Co., 1960, has accompanying pictures for almost every topic, but he never gets past R³. On the other hand, which is the easier way to remember Bessel's inequality, as

and equality holds if and only if

or as the geometric statement: "a vector is at least as long as its projection on any subspace, and equal to that projection in length if and only if it is in the subspace"? I claim that omitting the latter as do Hoffman and Kunze, Linear Algebra, Prentice Hall, Inc., 1961, p. 232 is inexcusable. A textbook should not be a list of facts.

Nering's book is a partially successful attempt to bridge that gap. For example, determinants are treated geometrically but axiomatically. Projections, rotations, reflections and shear transformations are illustrated geometrically in the text. Then an abstract definition of projection in the exercises (p. 183) paves the way for the spectral resolution theorem two chapters later (p. 268). Most books either play down numerical examples or present them mechanically. Nering does a good job of letting numerical examples expose the theory.

There is no discouraging introduction to set theory. (The discouragement of the first chapter is that matrix operations are unmotivated.) Instead, set theory is relegated to an appendix for reference. I introduced what notation I needed as I went along. Occasionally Nering, in his attempt to keep the notation uncluttered, uses a symbol both for a set of vectors and for the space they span (p. 102) or for a basis and the natural map from Rⁿ to the space with that basis (consistently; see, e.g., p. 111). This confused the students, but helped them to develop a healthy criticism of material they read. A list of errors for the first five chapters (11 of them) is available from this reviewer. [*] Only one of them is conceptual: a coset is not a subspace (p. 203).

The students were pleased that current journal articles are accessible to them, as illustrated by the parallel between pp. 102-103 and W. S. Ericksen's article The intersection of subspaces, this Monthly, 81(1974) 159-60. Unfortunately that section is one of the most uneven expositions of the book. His comments range from fatherly advice to obscurities. The important concept of isomorphism receives only passing mention there, and one preparatory problem before that (12, p. 75) prior to its formal introduction on p. 177. The lack of problems on isomorphism is typical of the computational orientation of the problems in general. If you plan to use the book, plan to supplement it with problems of a theoretical nature. What little emphasis there is on proofs in the problems is for very special settings. Since one of the primary goals of linear algebra at the sophomore level is to increase the students' ability to do proofs, I feel that his failure here is Nering's major weakness.

The consistent use of the pivot operation unifies the treatment. A gentle introduction to mapping diagrams proves to be illuminating. The selection of topics is predictably standard -- similar up to permutation to the rest of the deluge of linear algebra books. (Cf. reviews in this Monthly by D. E. Christie, June-July, 1973, and by David E. Kullman, March, 1974.) Student reaction uniformly applauded the pictures, criticized complicated notation, and wished that the text were more suitable for self-study. I too felt that lectures were necessary to fill in the gaps in the text.

Since the same students were taking advanced calculus from me, it was with some satisfaction that I was able to present the chain rule for derivatives from R^m ->Rⁿ in the "clean" matrix form because of their exposure to this concrete but modern introduction to linear algebra. I especially recommend the book to those instructors who like a text which is different from the way they will present the material in class. A stereoscopic view is more illuminating than a monocular view.

[*] Some but not all of the errors mentioned in the above review were corrected in a later printing, according to the editor of the Monthly.