Pretty much all of machine learning relies on linear algebra.
Most scientific computing relies on lI near algebra.
Graph algorithms often use linear algebra.
If you just want to be a Web Dev or an app developer, it probably doesn't matter, tho.
But if you just want to do Web Dev or mobile apps then you don't really need most of computer science.
1. Ranking in search engines (more generally, any kind of random walk analysis) [1]
2. Fourier analysis, and as a consequence most signal processing involves some understanding of linear algebra because integrals are linear. [2]
3. Regression [3] and more generally linear modeling of anything.
4. Facial recognition [4]
5. Community detection [5], where most leading methods analyze the spectrum of a graph to find communities. In fact, applied network science in general has a ton of linear algebra.
6. Greedy algorithms are characterized by a kind of generalization of linear systems [6]
7. Linear programming, perhaps the most applied piece of mathematics ever, needs a strong foundation of linear algebra [7]
8. All of quantum computing is literally just linear algebra [8].
9. Cryptography has a ton of linear algebra in it, and a large portion of the techniques are reasoned about with linear algebra.
10. Most of calculus relies on linear algebra, most importantly optimization [9]
11. Recent data analysis techniques based on topology do so through linear algebra [10]
12. Coding theory, including the algorithms used to correct errors on DVDs. Basically, any time you want to encode data so that you can recover from white noise, you're going to use a linear code. [11] This includes compression techniques.
13. Of course graphics.
I could go on...
[1]: http://jeremykun.com/2011/06/12/googles-pagerank-introductio...
[2]: http://jeremykun.com/2012/07/18/the-fast-fourier-transform/
[3]: http://jeremykun.com/2013/08/18/linear-regression/
[4]: http://jeremykun.com/2011/07/27/eigenfaces/
[5]: http://jeremykun.com/2014/05/19/community-detection-in-graph...
[6]: http://jeremykun.com/2014/08/26/when-greedy-algorithms-are-p...
[7]: http://jeremykun.com/2014/06/02/linear-programming-and-the-m...
[8]: http://jeremykun.com/2014/12/08/a-motivation-for-quantum-com...
[9]: http://jeremykun.com/2013/11/30/lagrangians-for-the-amnesiac...
[10]: http://jeremykun.com/2013/04/10/computing-homology/
[11]: http://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_corr...
http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebr...
0. http://en.wikipedia.org/wiki/Singular_value_decomposition
I absolutely enjoyed learning Linear Algebra from these beautiful lectures by Prof. Gilbert Strang (MIT): https://www.youtube.com/watch?v=ZK3O402wf1c&list=PLE7DDD9101...
Seriously, I gained a new found appreciation for Linear algebra after going through these lectures. You should go over some of these lectures even if you already know linear algebra - it might give you insights you never had before (it did, for me). Absolute must-watch if you are into machine learning or related areas.
