I'll outline what came to my mind here; hopefully it will get things going in the right direction.
I think your idea of trying to express elements as images is the right one, we just need to apply it in a slightly different way. Since B is invertible, we know it is a matrix of full rank, i.e. rank(B)=4; this comes from the rank-nullity theorem (because an invertible matrix has only 0 in its kernel, i.e. its nullity is 0). Since B has full rank, for any , there is a unique such that . Then and the images are the same.
Does this help? Let me know if anything is unclear or in error.