Part I

So, Princeton did a really bad job
teaching a first course in linear algebra
for non-STEM majors.

I'll try to make some sense out of that
and outline how a student might defend
themselves.

First, as a ugrad, I looked at college
education as career preparation,
essentially trade school and had no
understanding of the issues of status,
prestige, new research results, etc.

Point: A lot of high end US research
universities concentrate on the status,
prestige, bright students, and financial
support they can get from new research
results, etc. They can regard teaching as
a bit silly: The research results are in
the library and available to anyone who
wants them.

Second I had been influenced by the US NSF
propaganda that STEM would make a good
career so wanted to major in physics. Uh,
in simple terms, the NSF was trying to
create a labor force for US national
security.

Soon I saw some really sloppy math in
physics classes, e.g., a just awful
attempt, basically 100% wrong, to prove
Stokes theorem, guessed that the physics
profs were so bad at math that if I stayed
a physics major I would be so bad at the
relevant math -- for Maxwell's equations,
quantum mechanics, general relativity --
that my STEM career direction would be in
trouble. So, I majored in math intending
to return to physics.

I went to grad school in math to get the
rest of the math I needed for physics so I
could switch to physics. That math
department was not much interested in
teaching me the math for physics.

I got recruited to work around DC, mostly
on US national security, and there studied
math on my own. How: Get a highly
regarded text, one section at a time,
study the material, study any example
problems, work the exercises. Lesson:
That worked pretty well, and I recommend
that students consider it.

Soon, on both my job and my independent
study, I ran into quite a lot of linear
algebra and learned a lot of it.

Then I returned to grad school. I got
accepted to Princeton, Cornell, and Brown
but sensed the contempt issue and went
elsewhere.

Presto, bingo, the department Chair taught
an advanced course in linear algebra;
yup, it was a flunk out course. I told
the faculty that I didn't think I needed
more linear algebra and wanted to get on
with material I didn't already know. The
faculty just gave me a patronizing smile.
So, I took the course. I didn't intend to
embarrass the faculty and/or the
department Chair but, in the end, I did:
The course had a lot of graded homework
and tests. Early on the homework grader
made a mistake on one of my solutions; I
corrected him and he made no more
mistakes. Unintentionally, I was totally
blowing away all the other students on
homework, tests, and the midterm. When
the prof got to the polar decomposition
result, I blurted out "That's my favorite
theorem!". The prof was so flustered he
didn't complete the proof.

Lessons: (A) There is a strong propensity
among US math profs to look for
essentially superstar students, to have
contempt for everyone else, and, in
particular, to look for any flaws and,
seeing one, to have contempt for the
student. A remark from WWII was that the
German military had a big weakness, that
the officers had to prove themselves
everyday. Well, in US college and grad
school math, there is a strong propensity
to do that to the students. There is a
grand, Kryptonite-strong, better than
anything even Spiderman has, way around
that -- will mention that below. (B) The
Princeton math department, partly due to
its neighbor the Institute for Advanced
Study where at one time were Einstein and
von Neumann, has a reputation as the best
pure math department in the world. That
Princeton prof A. Wiles solved Fermat's
last theorem likely plays a big role.
Then it is easy to guess that there is
high propensity to look for superstar
performance and to have contempt for any
students who hint about anything else.
(C) There really are flunk out courses.
(D) It's possible defend yourself from the
flunk out courses and to blow away the
other students and fluster the prof --
just learn the material before taking the
course. (E) The really difficult math
courses are theorem proving courses, and
there the exercises and the tests are to
prove theorems. There a student doesn't
really need answers or "101 Solved
Problems" because soon it can be clear
when do have a proof: The difficulty is
finding the ideas for a proof; checking
the correctness of a proof tends to be
relatively easy.

Before the linear algebra course, I'd
studied linear algebra and closely related
topics from several texts, some that
likely remain standard and good and some
that were more advanced and specialized.
So, here I'll outline what worked for me:

As a ugrad, I had taken a course in
abstract algebra. So, that was about
sets, groups, rings, fields, Galois
theory, vector spaces, quaternions, basic
number theory, the fundamental theorem of
arithmetic (each positive whole number can
be written in exactly one way as a product
of prime numbers) and the fundamental
theorem of algebra (the complex numbers
are algebraically closed, that is,
each polynomial of degree n has n roots
and, thus, can be factored into a product
of n linear terms). Quite a lot of this
material now gets used in cryptography and
error correcting codes.

For a while, there were some influential,
maybe popular, texts on advanced
calculus that did quite a lot of linear
algebra. One of these was Nickerson,
Spencer, Steenrod and from Princeton, and
there in the early chapters can learn
about vector spaces and subspaces, linear
independence, dimension, linear
transformations, inner products,
orthogonality, the Gram-Schmidt process,
etc. Another was from W. Fleming from
Brown and there can also learn about
convexity. Both of these texts then
continue on to the exterior algebra of
differential forms and, in particular,
careful proofs of Stokes theorem.

So, I got both of these texts and a few
others and dug in. Actually that list of
topics -- vector spaces and subspaces,
linear independence, dimension, linear
transformations, inner products,
orthogonality, the Gram-Schmidt process --
has good intuitive explanations where can
draw nice, helpful pictures, can be
covered in not many pages, and, really,
are, say, over 60% of what should get from
a first course in linear algebra.

I heard the linear algebra book by E.
Nering was good, got a copy, and worked
quite carefully through it. It was good;
likely still should be considered good.
Nering was a student of E. Artin, right,
at Princeton.

The main examples of fields are the
rationals, the reals, and the complex
numbers. But there are some more fields,
e.g., integers modulo a given prime
number. Linear algebra and its vector
spaces need a field, and Nering does
nearly the whole book assuming any field
and not just the rationals, reals, or
complex. So, Nering does a little extra
generality -- that approach makes some
arguments a little more delicate but
otherwise causes no trouble. Error
correcting codes has some applications of
linear algebra using finite fields.

Then I heard that the P. Halmos, Finite
Dimensional Vector Spaces was a good book
on linear algebra and a also a good
introduction to the math of quantum
mechanics. Right, that book was written
by Halmos when he was an assistant to J.
von Neumann at the Institute for Advanced
Study at Princeton (the town, not really
the university). The book can be regarded
as a baby step into more general Hilbert
space theory. So, I dug in and studied
carefully. It's a good book! Halmos is
one of my favorite authors. When I
noticed that his proof of the
Hamilton-Cayley theorem didn't work for
finite fields, I wrote him a letter. Got
back a really nice, enthusiastic response!
Apparently at least then writing a really
good book on linear algebra does not make
one a rock star -- don't get a lot of
mail!

Then did more in linear algebra --
numerical linear algebra, applications to
statistics, the fast Fourier transform,
optimization, computational geometry,
error correcting codes, and more.

Lesson: Those efforts in learning linear
algebra are why effortlessly and
unintentionally I blew away the other
students in that flunk out linear algebra
course.