What are some applications of elementary linear algebra outside of math? I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside of mathematics that I might talk about in discussion section.
In our class, we cover the basics (linear transformations; matrices; subspaces of $\Bbb R^n$; rank-nullity), orthogonal matrices and the dot product (incl. least squares!), diagonalization, quadratic forms, and singular-value decomposition.
Showing my ignorance, the only application of these I know is the one that was presented in the linear algebra class I took: representing dynamical systems as Markov processes, and diagonalizing the matrix involved to get a nice formula for the $n$th state of the system. But surely there are more than these.
What are some applications of the linear algebra covered in a first course that can motivate the subject for students? 

Computer science has a lot of applications!


*

*Manipulating images.

*Machine learning (everywhere).
For example: Multivariate linear regression $(X^TX)^{-1}X^{T}Y$.  Where X is an n x m matrix and Y is an N x 1 Vector.

