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?