Let A be a 4x3 matrix and let B be a 4x4 matrix. Then, BA is a 4x3 matrix. Let L be the linear

transformation from R4 to R3 defined by L(x) = xA, and let M be the

linear transformation from R4 to R3 defined by M(x) = xBA. Now, assume that B is an invertible matrix so that C is a

4x4 matrix and CB=BC=Identity matrix.

Show that the image of L is the same as the image of M.

Attempt-

Let y be in the image of L so that y = xA.

Now, to get to y = xBA, I have to assume that x has an inverse, invert both sides, multiply each side by B, then C, and then move the x vector back over? I'm stuck.

Thanks