> The reason it's not on the list is I
just didn't see what it had to offer that
isn't covered in later books. Happy to be
convinced otherwise though.

Halmos wrote Finite Dimensional Vector
Spaces in 1942 -- shortly after he got
his Ph.D. from J. Doob at University of
Illinois -- at the Institute for Advanced
Study in Princeton as an assistant to von
Neumann.

The book is a mostly finite dimensional
introduction to von Neumann's Hilbert
space theory. Some physics profs tell
their students to read that book to learn
the Hilbert space theory they need for
quantum mechanics. Generally the book
uses Hilbert space, that is infinite
dimensional, notation and arguments when
they also work for the finite dimensional
results -- cute. This approach is easier
and cleaner when it works.

There is emphasis on duality, as is
important in functional analysis including
Hilbert space.

Some of the special emphasis is on
spectral theory, important in
transformations on Hilbert space.

Halmos includes some multi-linear algebra
(can see again in exterior algebra and
more modern approaches to general
relativity) and near the end has an
ergodic theorem!

At one time the book was one of three
basic references for Harvard's somewhat
famous or notorious Math 55 as in

http://www.american.com/archive/2008/march-april-magazine-co...

The other two references were Rudin's
Principles and Spivak's Calculus on
Manifolds. So, from Harvard get a good
recommendation for excellence for the
Halmos book.

I very much enjoyed that Halmos book and
later more of his books. He was one of
the best writers in math of the 20th
century.

His proof of the Hamilton-Cayley theorem
is a little less general than the one in,
say, E. Nering's book, and I wrote Halmos
about that. He wrote back a nice letter.

I learned linear algebra from a course in
abstract algebra, Murdoch, Linear Algebra
for Undergraduates, (about the easiest
linear algebra book anyone could read),
one by E. Nering, some work I did in
mathematical physics, the first chapters
of Nickerson, Spencer, and Steenrod's
Advanced Calculus, my ugrad math honors
paper on group representation theory,
Herstein's Topics in Algebra on abstract
algebra, Halmos, FDVS, Forsythe and
Molar on numerical linear algebra,
applications in multivariate statistics,
some work in optimization, carefully wrote
out my own notes, enough for a book, and
more.

The OP list also includes Horn's book.
After all that study, mostly independently
and part of my career in applied math and
computing, eventually I took Horn's
course, intended to be an advanced course.
As part of my application to his grad
school, I did include a copy of the Halmos
letter!

There was next to nothing in the Horn
course new to me, and I didn't try very
hard. Everything in the course was
carefully graded. I did the homework but
didn't study for the tests. On an early
homework assignment, the grader made a
mistake on my paper; I corrected him; and
he made no more mistakes. About 3/4ths
the way through the course the grader
showed me the class scores -- I was first
on everything by wide margins. Did that
again on the final. Apparently did that
again on the corresponding Ph.D.
qualifying exam.

In the course, when Horn got to the polar
decomposition, I got excited and blurted
out in class "That's my favorite theorem"
which it was and is. Horn's remark was,
"Thank you Dr. Halmos".

The polar decomposition? Go through a lot
of stuff in linear transformations and
then discover something really simple:
All a linear transformation can do is just
a rigid motion (rotations, reflections)
followed by stretching/shrinking on
orthogonal axes. A singular
transformation? One or more of the axes
go to zero. In the non-singular case, the
proof is really simple. In the singular
case the difficulty is that there are many
choices for the rigid motion part and just
have to pick one.

Horn did at times have some especially
good proofs.

Horn is very precise with high quality.

It's possible to have a quite fast
introduction to linear algebra, but then
are missing out on a lot if don't work
carefully through a classic text such as
Hoffman and Kunze. IIRC it is available
as PDF on the internet for free.

E. Nering's book is also fine except for
his material in the back on linear
programming which is easy (for a lot of
it, just slightly tweak Gauss elimination)
but he makes difficult. For linear
programming, get a good book on linear
programming, e.g., Chvatal or Bazaraa and
Jarvis. In the back, Nering also does
group representations -- very cute.

I'd recommend Halmos, FDVS as ice cream
and cake desert after something like
Hoffman and Kunze.

For the numerical stuff, should look at
the documentation for Linpack. Maybe also
look at Forsythe and Molar. Some that has
been done since then for numerical
methods, e.g., for models of oil
reservoirs, is impressive -- might at
least glance at that.

Broadly there are two directions in linear
algebra: One direction assumes that the
field is the real and/or complex numbers
and is aimed at analysis and applied
math. The other direction assumes finite
fields and is aimed at high end abstract
algebra and has applications in, say,
algebraic coding theory (high end error
correcting codes).

The direction with real and complex
numbers is, sure, a start on functional
analysis (you are in a vector space, but
each point is a function -- get to learn
how to approximate functions by other
functions, e.g., how to approximate random
variables by other random variables, e.g.,
learn Fourier theory), but for that also
should have measure theory, e.g., from
Royden, Real Analysis and/or the first
half of Rudin, Real and Complex
Analysis. For that, as in Math 55, a
prerequisite is Rudin's, Principles.

Rudin's Functional Analysis is more
advanced, still. I got accepted to grad
school at Brown's Division of Applied Math
(didn't go there), and Rudin's FA was
popular there, but I haven't seen a lot of
popularity for it elsewhere.

A huge fraction of applied math, somewhere
on the way to implementation and code,
boils down to linear algebra.