Add me to the list of people who have the same question, but with a slightly different perspective. I understand the interpretation of matrix multiplication as

Add me to the list of people who have the same question, but with a slightly different perspective. I understand the interpretation of matrix multiplication as a linear combinations, but I would also like to understand what is occurring geometrically. If I matrix multiply a 2D vector <a, b> by the basis vectors <<0, 1>, <1, 0>>, then I understand I am "weighting" an x, and y unit vector (as defined by my basis) by the x-component and y-component of my <a, b> vector, and then adding them together. So geometrically, I'm stretching by a factor of a, and b, then summing to get my resulting vector. So, the way I'm trying to think of it geometrically is: any matrix is just a representation of some kind of basis (i.e some kind of linear system). I picture this as a sort of orthogonal cartesian grid that we squish and stretch while still being a series of intersecting parallel lines. Matrix multiplication, is therefore taking in some vector, and "weighting" the basis vectors represented by our matrix by the components of that vector, and then summing them. To put it another way, we're essentially mapping a vector into that new basis system represented by our matrix. Is that correct? Is there a better way to try and think of matrix multiplication geometrically?