There are lots of applications of linear algebra:
- Pagerank, the algorithm behind Google's success is based on eigen values and eigen vectors

- the winning entry in the Netflix Prize was based on Singular Value Decomposition

- The winning entry in the Netflix Prize was based on Singular Value Decomposition

- 3D games use matrix multiplications for computing rotations, translations, shearing transforms

- in machine learning figuring out if the function you are learning is well conditioned for Gradient Descent corresponds to having a small ratio between the min and max eigen value of the hessian matrix
- again in machine learning figuring out if you're in a saddle point, local minima or local maxima can be done by looking at the signs of the eigen values of the hessian in the current point
- and of course they are sometime used in coding contest competitions :) (fast matrix exponentation, gaussian elimination etc)

- in machine learning figuring out if your data is well conditioned for Stochastic Gradient Descent corresponds to having a small ratio between the min and max eigen value of the hessian matrix
- and of course they are sometime used in coding contest competitions :)

Have fun!