This is how I learned linear algebra, as a series of algorithmic steps to perform to reach the solution. Only a handful of homework examples even hinted that matrices were powerful and useful. I learned the semester before that class how matrices can be used to represent physical locations and how multiplying these matrices can move you along the coordinate frames for the joints in a robot arm all the way to the end effector. In fact, you can take your final transformation matrix after you multiply each joint together and use that to translate your points in your native/world coordinate system into coordinates in your tool coordinate system. I later learned that matrices can be used for quantum computation/pyhsics and all sorts of other things like weather prediction, physics, graph theory, Bayesian networks, and so on.

Now that I'm out of school, I really want to learn how to properly use matrix algebra. I don't care about HOW to find a determinant or HOW LU decomposition works as I've already done those, but rather why I would want to perform that operation on the matrix in the first place.

The ability for matrices to model real processes is something that really fascinates me.

Do you have any other good reading material for someone like myself?