I'd go so far as to say if your LA class did not instill that one value it was more a matrix algebra and methods (nothing wrong w/ that....v useful) class.

The matrix is simply the model of the linear space. Coordinate free-ness lets you get away from coefficient chasing and tedium to focus on the real stuff.

Esp important in infinite dimensional vector spaces where many theorems cannot rely on a basis existing as it is equivalent to the axiom of choice.