[Sharlin]

For anyone who’s interested in graphics programming and/or is a visual learner/thinker, there’s an incredibly motivating and rewarding way to learn the basics of linear algebra. (And affine algebra, which tends to be handwaved away, unfortunately. I’m writing a MSc thesis about this and related topics.)

[andrewla]

To a degree I think this is true, but it requires (at least in my experience) that you have an intrinsic grasp of trigonometry for it to make sense. If you have some complex function analysis and e^itheta then you can skirt the problem for a bit, but if you're like me and have to break out soh-cah-toa whenever you break down a triangle then this method ends up being pretty tedious too.

[Sharlin]

I’m not sure what you mean. Beyond rotation matrices, there’s really only trig involved in graphics if you actively want it.

[andrewla]

Maybe I was making unwarranted assumptions about the nature of your way to learn linear algebra. The approaches that I've seen invariably have to produce a sample matrix, and rotation is really the best example. The rotation matrix is going to have sines and cosines, and understanding what that means is not trivial; and even now if you asked me to write a rotation matrix I would have to work it out from scratch. Easy enough to do mechanically but I have no intuitions here even now.

[Sharlin]

Rotation matrices are somewhat mysterious to the uninitiated, but so is matrix multiplication until it "clicks". Whether it ever clicks is a function of the quality of the learning resource (I certainly do not recommend trying to learn linalg via 3D graphics by just dabbling without a good graphics-oriented textbook or tutorial – that usually doesn’t end well).

Anyway, I believe that it's perfectly possible to explain rotation matrices so that it "clicks" with a high probability, as long as you understand the basic fact that (cos a, sin a) is the point that you get when you rotate the point (1, 0) by angle a counter-clockwise about the origin (that's basically their definition!) Involving triangles at all is fully optional.

[srean]

In 2D there's an alternative. One can rotate purely synthetically, by that I mean with compass and straight edge. This avoids getting into transcendentals.

Of course I am not suggesting building synthetic graphics engines :) but the synthetic approach is sufficient to show that the operation is linear.

[greymalik]

> there’s an incredibly motivating and rewarding way to learn the basics of linear algebra

What is it?

[tptacek]

They mean graphics programming; learning graphics programming will give you intuitions for a lot of linear algebra stuff.

[bmacho]

There is no such thing as affine algebra: https://en.wikipedia.org/wiki/Affine_algebra

[Sharlin]

There are affine spaces, and there is an algebra of the elements of affine spaces. That is, rules that describe how the elements can be manipulated. There are affine transforms, affine combinations, affine bases, and so on, all of them analogous to the corresponding concepts in linear algebra.

(The term "algebra" can also refer to a particular type of algebraic structure in math, but that’s not what I meant.)

[cassepipe]

...

What is this incredible motivating way ? Please do tell

[bmacho]

Linear algebra has motivations and applications everywhere, since its main defining concepts, 'addition' and 'transformation that keeps sums' are everywhere. So a linear algebra curse is a huge pile of disjointed facts. It is not such a set of material that can have motivations behind it.

But the good news is that if you are only interested in for example geometry, game theory, systems of linear equations, polynomials, statistics, etc, then you can skip 80% of the content of linear algebra books. You don't have to read them, understand them, memorize them. You'll interact with a tiny part of linear algebra anyway, and you don't have to do that upfront.

[Sharlin]

Well, graphics programming itself. Learning while doing, preferably from some good resource written with graphics in mind. 2D is fine for the basics, 3D is more challenging and potentially confusing but also more rewarding.