How many books on introductory linear algebra exceed the 725 pages of this one? At a guess, none. Of such books, how many exclude coverage of standard topics such as the Gram-Schmidt process, quadratic forms, hermitian and unitary matrices and Jordan forms etc? In my estimation, only a strict minority. Is there any other book that concentrates so heavily on establishing the concepts the basic concepts of linear algebra? I very much doubt it.
By way of illustration, the first four hundred pages of Larry Knop’s book are devoted to eliciting the concept of vector space. The treatment begins with revision of elementary vector algebra in R2 and generalises such properties to Rn. In fact, up to page 146, everything is kept at an intuitively practical level prior to the emergence of vector space axioms. Subsequently, the notions of subspace, spanning sets, linear independence, and bases are carefully elicited and not until page 391 is it felt safe to introduce the ideas and methodology of linear transformations.
So, after 520 pages of lively narrative, and hundreds of exercises and examples, one encounters the two remaining chapters on Determinants and Eigenspaces respectively. The remaining sixty pages are allocated to the solutions of selected exercises.
I have to say that I’m very much in favour of this approach to the teaching of linear algebra because, recalling my own introduction to the subject (about 200 years ago, it seems), and recalling courses that I have designed and taught to undergraduates, the mistake was to proceed too rapidly through the early stages, and move on to more advanced techniques too quickly.
Consequently, the thickness of this book is partly due to the nature of its expositional narrative, which is almost conversational, and it aims to get students to read more independently and of their own volition (this always begins with an element of directed reading of course).
Another appealing feature of Larry Knop’s book is the range and quality of the illustrations and the myriad of applications that appear from the earliest chapters onwards.
To summarize, by limiting the mathematical scope of this book, and by carefully smoothing the path from the particular to the general, the author has provided an alternative approach to linear algebra that will be of interest to many of those who currently teach it.
The very first lecture on linear algebra that Peter Ruane received as a student began with a delineation of the axioms and a range of theorems with no motivational material whatever (and certainly no applications).