# Can a matrix be transformed like a vector?

Suppose I have a vector space V and a matrix M such that multiplying every vector in V by M creates another vector space W. Now suppose I have another matrix A that I can also use to change vectors in V into other vectors. Does there exist a third matrix B such that - for any vector v1 in V - if Av1 = v2, Mv1 = w1 and Mv2 = w2 then Bw1 = w2 ? In other words, is there a way to transform matrix A into a matrix B analogous to the way M changes vectors in V into vectors in W, so that a kind of homomorphism is arrived at between the relationship between A and vectors in V and the relationship between B and the vectors in W?

mathwonk