I'm an undergrad taking my first linear algebra course, so bear with me.
Whenever we talk about a matrix, is is safe to say that we are discussing a vector space? Vector spaces have very clear definitions and axioms associated with them, but a matrix is just a rectangular layout of numbers. A shorthand, if you will, to make vector spaces easier to deal with. I don't think I've seen matrices used for anything else, but I'm not sure if we can talk about matrices and vector spaces interchangeably or not.
For example, I've been reading Gilbert Strang's "Linear Algebra and its Applications." Let A be a matrix. It makes sense to talk about the column space of A, or the null space of A, or the row space of A. But does it make sense to talk about the dimension of A or the basis of A?