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The Little Book of Linear Algebra

(github.com)
468 points scapbi | 13 comments | 02 Sep 25 14:17 UTC | HN request time: 1.885s | source | bottom
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andrewla ◴[02 Sep 25 18:02 UTC] No.45106778[source]▶
It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math.

But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matrices and Gaussian elimination, all the while bored out of your skull, until you have the tools to really start to approach the interesting areas.

Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away until one day when you're looking at a chain of linear transformations you realize that everything clicks.

This "little book" seems to take a fairly standard approach, defining all the boring stuff and leading to Gaussian elimination. The other approach I've seen is to try to lead into it by talking about multi-linear functions and then deriving the notion of bases and matrices at the end. Or trying to start from an application like rotation or Markov chains.

It's funny because it's just a pedagogical nightmare to get students to care about any of this until one day two years later it all just makes sense.

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1. Sharlin ◴[02 Sep 25 18:48 UTC] No.45107368[source]▶
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.)
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2. andrewla ◴[02 Sep 25 18:58 UTC] No.45107508[source]▶
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.
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3. cassepipe ◴[02 Sep 25 19:02 UTC] No.45107556[source]▶
...

What is this incredible motivating way ? Please do tell

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4. Sharlin ◴[02 Sep 25 19:16 UTC] No.45107725[source]▶
I’m not sure what you mean. Beyond rotation matrices, there’s really only trig involved in graphics if you actively want it.
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5. andrewla ◴[02 Sep 25 19:32 UTC] No.45107935{3}[source]▶
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.
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6. Sharlin ◴[02 Sep 25 19:33 UTC] No.45107947[source]▶
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.
7. bmacho ◴[02 Sep 25 19:51 UTC] No.45108162[source]▶
There is no such thing as affine algebra: https://en.wikipedia.org/wiki/Affine_algebra
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8. bmacho ◴[02 Sep 25 20:05 UTC] No.45108315[source]▶
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.

9. Sharlin ◴[02 Sep 25 20:05 UTC] No.45108320{4}[source]▶
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.

10. Sharlin ◴[02 Sep 25 20:09 UTC] No.45108354[source]▶
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.)

11. srean ◴[02 Sep 25 20:23 UTC] No.45108501{4}[source]▶
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.

12. greymalik ◴[02 Sep 25 20:23 UTC] No.45108518[source]▶
> there’s an incredibly motivating and rewarding way to learn the basics of linear algebra

What is it?

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13. tptacek ◴[02 Sep 25 20:47 UTC] No.45108820[source]▶
They mean graphics programming; learning graphics programming will give you intuitions for a lot of linear algebra stuff.