The Linear Algebra Survival Guide

This is a peculiar book. It is an encyclopedia, in alphabetic order, of all the concepts and calculations from a first course in linear algebra, along with Mathematica code and examples for calculating them. It is not a complete course, because it does not explain why the concepts are important or how they are used. For example, the entry on eigenvalues gives their definition and some examples of calculating them, but gives no indication of why they are important. To some extent the book is a condensation and reworking of the same author’s Linear Algebra: An Introduction Using Mathematica (Academic Press, 2000).

The writing is generally reliable and clear. There are a few garbled places. For example, on p. 1 (in the Introduction) it states, “By default, Mathematica works to 19 places to the right of the decimal point.” In fact Mathematica by default works to machine precision, which is usually 16 decimal digits total, regardless of the decimal point. On p. 104 the book states (in the entry on Eigenvalue) that rotations in the plane “do not map nonzero vectors to multiples of themselves”, which is generally true, but not for rotations by a multiple of \(\pi\) radians. For another, on p. 296 (in the entry on Quintic polynomial) it says, “According to the theorem of the unsolvability of the quintic, there is no algorithm for finding the solutions of arbitrary polynomials of degree 5 or higher consisting only of the four arithmetic operations together with the extraction of roots.” In fact the theorem states that there are quintics whose roots cannot be expressed in terms of radicals; it’s not a theorem about what can be done with algorithms. The definition of Toeplitz matrix (p. 378) is completely wrong; it leaves out the key property, which is that the descending diagonals are constant.

One of my rules of thumb is that every non-fiction book should have an index (some fiction books should too), but I make an exception in this case. The book does have an index, but it is like indexing the dictionary: apart from a few cross-references, the index is a list of all the headwords and which page the entry starts on. It would have been more useful to put the cross-references in their alphabetical place in the body and omit the index.

The body text is set in tiny type, about 7.5 points on 10-point line spacing, in a sans-serif face. The headwords and subheads are printed in a dithered gray rather than color or solid gray, but happily are large enough that they are still easy to read.

The book refers (p. 10) to a companion web site, but I was not able to locate this. The book makes use of a custom Mathematica package, ClassroomUtilities, that is supposed to be on this site but is not explained in the text.

Bottom line: not a bad book, but most useful for the Mathematica aspects, and that for tyros or occasional users. It is not actually a Survival Guide, because it omits how linear algebra concepts are used. It might be a Survival Guide for a course where all the exercises and exam questions are drill.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.