To clarify, I mean if the columns of a 3x3 matrix A are A1, A2, and A3, and the scalar elements of vector x are <x1, x2, x3> then Ax = x1*A1 + x2*A2 + x3*A3. Each column of A is scaled by an element of x and then added together.
Is that what you had in mind by the dot-product explanation? To me, the dot product explanation is that in Ax = b, b1 = <row1 of A> dot x, b2 = <row2 of A> dot x, and b3 = <row3 of A> dot x.
Of course these (and all other valid) interpretations of matrix multiplication are "the same", but this is less geometrically intuitive to me.
Is that what you had in mind by the dot-product explanation? To me, the dot product explanation is that in Ax = b, b1 = <row1 of A> dot x, b2 = <row2 of A> dot x, and b3 = <row3 of A> dot x.
Of course these (and all other valid) interpretations of matrix multiplication are "the same", but this is less geometrically intuitive to me.