One way would be to take a course on a topic that uses linear algebra heavily, rather than taking a generic linear algebra course. Since all the ideas from linear algebra are then being used to actually do something, you'll get at least one example for why you would use whatever it is.

I would suggest trying these videos: http://www.stanford.edu/~boyd/ee263/videos.html. The prerequisites are very low and a main focus is on interpreting the abstract concepts in applications.