Linear Algebra, 2nd ed., by Friedberg, Insel and Spence, Prentice Hall, 1997(?)

Reviewed by Jane M. Day.

I have used this book for a second semester linear algebra course for 10 years at least, and like it very much. I've not seen any other text that is better for that course, for our students. They are a mix of juniors and seniors, most of whom plan to be high school teachers, and graduate students preparing for our qualifying exam in algebra. The first linear algebra courses these people have had emphasized the spaces of real n-tuples, and usually they have fairly clear geometric understanding of what matrix transformation, subspace, basis and dimension mean in that setting.

Theoretically this text could be used for a first course, but it is written at a fairly sophisticated level. There are nice appendices on Fields and the Fundamental Theorem of Algebra. The book includes matrix arithmetic, Gaussian elimination and determinants, but goes on to present the theory of subspace, basis, dimension, change of basis, linear transformation, eigenvalues, diagonalization, inner product, etc. for general finite dimensional spaces. Topics are presented succinctly but in more depth than most elementary books do — for example, intersection of subspaces and general projections are there. In the second course, we focus on complex spaces only and I skip the material students have studied before, but I think it is reassuring to them that the more concrete material is there if they want to review. There are elementary exercises throughout, plus more sophisticated ones that are just the right level for the second course.

My most important goal is for students in this course is to become able to read the text and write proofs they believe, so we spend a lot of time unpacking notation and discussing subtleties like what well defined means, what the difference is between union and ordered union, how to prove a theorem of the form "P implies (Q or R)," etc.

We cover more material some semesters than others. I try to set the pace so that students who have the promise of being good high school math teachers should be able to earn a B at least. We always manage to learn the Jordan form thoroughly, discuss minimal polynomial some, and get an introduction to inner product spaces. I encourage students to discuss assignments with each other and me before turning them in, so they will be as correct as possible the first time; but I do allow them to rewrite problems one time.