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? 

Affine Geometry works well to smoothly fit curves and surfaces to a set of desired data points.  Affine Geometry builds on top of Linear Algebra and is essential to any engineering that combines absolute and relative for both direction and displacement.  Engineering the moving parts of a vehicle combine absolute and relative motion with both rotational and non-rotational movement.
