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 h

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.