Binius: Highly efficient proofs over binary fields

Later, after Nering and Halmos, which I read on my own, carefully, one word at a time, chewing on all the proofs, working nearly all the exercises, trying to understand the subject more broadly, I got pushed into an "advanced" course in linear algebra: Before the class started, to brush up a little, I wrote out 80 pages of the subject. During the course I blew away all the other students and did it again on the linear algebra part of the qualifying exams. The course was a waste, but maybe I "proved myself". There was a sense that the prof resented me, that is, wanted to see me struggle, expected that I would struggle and suffer like the other students. Silly nonsense.

But, beyond that silly course, I ran off into some other directions: (1) Took some course in optimization. So, covered some graph algorithms that ran in (n)log(n) instead of n^2 -- there was some linear algebra there. (2) Then there was a lot of linear programming -- mostly linear algebra. (3) Then there was nonlinear programming, e.g., Kuhn-Tucker conditions where linear algebra was the local linearization of the non-linear problems. (4) For more, there was some multi-objective linear programming. (5) Some chemists had some questions about group theory and quantum mechanics, so I wrote a paper on group representations, nearly all linear algebra. (6) And got to use some of Nering's use of finite fields and some abstract algebra with finite fields, groups, etc. in a course in coding theory. (7) Outside of school, did some with numerical linear algebra and iterative solution methods -- e.g., covered the M. Newman work applying some number theory and the Chinese remainder theorem to finite precision arithmetic to numerically exact matrix inversion. That exact matrix inversion is cute, and maybe has some really important applications, but the motivation at the time was ill-conditioned linear problems for the normal equations of regression analysis where the variables were nearly dependent, and that exactness is polishing a ...: The response is to return to the statistics problem and see what can be done there or with accepting and working with the ill-conditioned matrix. (8) I fell into the big pool of the FFT (fast Fourier transform) when it was a hot topic. So, the FFT can be regarded as mostly just more linear algebra. (9) There are the connections with Markov processes, and once I took a WWII paper on search and encounter rates at sea and used it with some Markov and Poisson math and some linear algebra to do a Monte Carlo of war at sea, in particular to see how long some submarines would last.

But, all that aside, after as much as Halmos and/or Nering, for more pure math, just take linear algebra as baby functional analysis, e.g., Banach space and, especially, Hilbert space. That's what Halmos wrote his book for, for a finite dimensional version of von Neumann's work -- Halmos was a post-doc under von Neumann at the time. E.g., get to address the issue of being locally compact, i.e., are the usual closed and bounded enough to be compact where every infinite subset has a limit point?

Point: Once you have worked carefully through Halmos, Nering, or anything else competitive, MOVE ON to applications, deeper theory, etc.

Warning: If you have a "job" and are working for someone, some boss, manager, know more math than they do, and use it for some work that is clearly really good for the work of the manager's group, WATCH OUT! The manager may get really, seriously torqued and work to get you fired. Similarly, a resume that shows some good work in pure/applied math, say, along with a lot in computing, can be career poison: E.g., (1) -- (9) above ended my career. With an honest resume, I couldn't get hired even to sweep floors, mow grass, etc., LITERALLY. The costs were high. The math background was worse that, I would guess, than a felony conviction. So, I'm doing a startup, a Web site. Buried deeply, there is some original pure/applied math, crucial, but won't be visible to the users or advertisers. The math might be quite valuable, but basically just have to keep it and its applications a secret.

    >>> x = 113338949109682836687814795709948365013
    >>> n = 1
    >>> y = 80490928931346377909947075573303248723 + 170141183460469231731687303715884105727 * n
    >>> y**2 % (2**127 - 1) == x
    True