[tptacek]

As someone currently teaching themselves linear algebra (via Strang), I'm curious as to why you believe linear algebra is ridiculously important. I see its applications to graph programming and obviously to cryptography, but I've done a lot of work in both of those subjects and never strictly needed a background in linear algebra to be effective.

[cabinpark]

Linear algebra is insanely important. I would consider the most important area of numerical mathematics to know. Any sort of mathematical modeling you do is going to involve linear algebra at some point.

I have argued that almost all numerical mathematics, in some form, can be modeled as a linear algebra problem. Google's original page rank algorithm is linear algebra. Remember that Netflix challenge? All linear algebra. Optimisation? Linear algebra. Want to do any engineering of a system? Behind the scenes it is all linear algebra as every single numerical technique for solving PDEs, that I am aware of, can be thought of in a linear algebra sense. Fluid modeling is all linear algebra. A lot of the machine learning I've seen is just linear algebra. I would also that almost every simulation running on the world's supercomputers involves linear algebra.

Do you absolutely need linear algebra to do this things? No. Just like I don't need to understand how my car works to drive it. But having an understanding how these systems operate can really help you use them in a more logical way.

[platz]

`ridiculously` might be hyperbole, but the two biggest ones I can think of are computer graphics and machine learning; these of which you can't even sneeze at without getting in linear algebra.

[tptacek]

I have some sense of the linear algebra implicated in machine learning (I lived for many years with a comp. neuroscience PhD), but I have no visibility into graphics. So with my ignorance pinned to my lapel: the linear algebra involved in computer graphics is pretty simple, right? Just knowing how to manipulate vectors and matrices? Not a lot of eigenvalues, or for that matter orthogonalization?

I'm asking not to rebut but because I hope to prompt the sort of "sell us on linear algebra" statement that will make me study harder. :)

[jfarmer]

Much of computer graphics operates in projective space (http://en.wikipedia.org/wiki/Projective_space), so it's a little bit more than cookie-cutter linear algebra. This is done so that translations in 3D space — which aren't linear transformations — can be represented as 4D linear transformations in projective space via homogeneous coordinates (http://en.wikipedia.org/wiki/Homogeneous_coordinates).

Still, many folks use the APIs without really grokking that, so in practice it can be a bit cookie-cutter. I think of it similarly to how people can use crypto APIs without, say, really understanding what's going on under the hood.

BTW, projective space is also intimately related to elliptic curves (as you may or may not know — not implying anything!). So that darn linear algebra is lurking all over the place.

Likewise, any time you're talking about fields (even finite fields), vector spaces and linear algebra are right around the corner.

[tptacek]

I'm familiar with projective coordinates for elliptic curves, but the funny thing about curves is that, for high-speed software, the math is tricky enough that you don't "need" to grok it: there's an "explicit formulas database" that you can just copy from:

http://www.hyperelliptic.org/EFD/g1p/index.html

Curves are what made algebra 'click' for me, getting me from my high school understanding of "algebra is math about unknown variables" to "algebra is about sets of related objects with operators that have identities and inverses".

[platz]

Disclaimer: I'm not very experienced with CG, but I think your intuition is correct on CG, for lots of 'boilerplate' game or photoshop programming.

However, I do think you can find plenty of examples in academia. For example, applying PCA

http://cmp.felk.cvut.cz/~hlavac/TeachPresEn/11ImageProc/15PC...

(Perhaps this is just the result of crossover between CG and ML/Statistics here).

Not a good sell, but perhaps someone with more experience can chime in.