story
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.
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. :)
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.
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".
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.
