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. Ea

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.