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.