I believe I've had the "clicked" moment. It's all about basis vectors, right?
I.e., What would these given vectors mean if their basis vectors were these; what are their vectors in terms of these basic vectors. Matrices can be seen as simply groups of basis vectors. That is why multiplying two matrices together corresponds to composition of their respective transformations; because, when doing AxB, you are rewriting the B's basis vectors in terms of A, and the and thus this new matrix has the basis vectors that are exactly what you would get if you had done Av Bv (v is a vector) separately. And matrix by vector multiplication is rewriting with the Matrix' basis vectors.
E.g., Iterating with Markov matrices is simply generating new points in space as if their components were weighted by the probabilities in the markov matrix. So the markov matrix is also a list of basis vectors. Also the iterative power method for finding eigvenvectors converges because we keep generating vectors that approach the analogues [1, 1, 1] vector.
Also gaussian elim./equation stuff is simply a special case of all this? It was first motivated by those problems but then we figured out it was all about linear mappings with these basis vectors, right?
Moderators, please don't close this post if it doesn't conform to the math exchange decorum/etc. I need to know if my insight is right.
And vector spaces are simply generalizations of the spaces of number vectors. Anything linear with an analogous plus/scale is amenable to this theory - all the stuff proved about plain old number vector spaces.
EDIT: my god, it all makes sense now. the stretching, rotating, shrinking, composition, relationship to systems of linear equations.
EDIT 2: Without linear independence in vectors, we don't gain extra dimensions in our transformation.
And is the eigenvalue basically a descriptor of what happens to area before/after transformation? Is it the analogous 1x1 square in the new space? I know this is a ramble, but can anybody understand anything I'm trying to convey ?