An Intuitive Guide to Linear Algebra (2012)

Now that the ideas of things like vector spaces, norms, orthogonality, rank, basis, etc are nearly second nature, the concepts are useful as I study other branches of math which would feel impenetrable otherwise.

YMMV, and if you can learn from condensed materials go for it, but I might be too dumb for it work lol. I think the real benefit accrues to the author who had to work out how to teach these concepts to others.

snicker7 said it very succinctly:

> However, in mathematics proper, it is absolutely the case that linear algebra is about linear transformations. Indeed, this is the only interpretation that remains meaningful when trying to generalize (e.g. to functional analysis / multilinear algebra).

If you're point is that I failed to comprehend matrices, then I don't think you have enough data to make that claim since I don't really talk about matrices. I kind of address that in my other comment [2].

I don't follow your point around "a wide array of problems and models are the same thing". That's a very vague general statement that I certainly comprehend (not sure how you inferred otherwise). Specifically, I don't see how that point relates at all to the claim I made about linear algebra.

[0] https://news.ycombinator.com/item?id=22419018

[1] https://news.ycombinator.com/item?id=22417764

[2] https://news.ycombinator.com/item?id=22417595

Geometric algebra seems more practical than most subjects but even its introductions suffer from this.

W. Wesley Peterson and E. J. Weldon, Jr., Error-Correcting Codes, Second Edition, The MIT Press.

-- gee, people are still studying/learning that?

The prof knew the material really well, but to up my game in the finite field theory from other courses, I used

Oscar Zariski and Pierre Samuel, Commutative Algebra, Volume I, Van Nostand, Princeton.

which did have a lot more than I needed!

My 50,000 foot overview of linear algebra is that the subject still rests on the apparently very old problem of the numerical solution of systems of simultaneous (same unknowns) linear equations, e.g., via Gauss elimination (it's really easy, intuitive, powerful, and clever, surprisingly stable numerically, and is fast and easy to program; someone might want to type in, say, just an English language description!). Since such the subject of linear equations significantly pre-dates matrix theory, the start of matrix theory was maybe just easier notation for working with systems of linear equations. In principle, everything done with matrix theory could have been with just systems of linear equations although often at a price of a mess notationally. In particular, as I outline below, now there are lots of generalizations of systems of linear equations that use different notation and not much matrix theory.

What's amazing are the generalizations, all the way to linear systems (e.g., their ringing) in mechanical engineering, radio astronomy, molecular spectroscopy, frequencies in radio broadcasting, stochastic processes, music, mixing animal feed, linear programming, oil refinery operation optimization, min-cost network flows, non-linear optimization, Fourier theory, Banach space, oil prospecting, phased array sonar, radar, and radio astronomy, seismology, quantum mechanics, yes, error correcting codes, linear ordinary and partial differential equations, ..., and then

Nelson Dunford and Jacob T. Schwartz, Linear Operators Part I: General Theory, ISBN 0-470-22605-6, Interscience, New York.

An interactive Linear Algebra course to complement the article.