presumably, you mean , that is, . suppose we have a vector in ginve in C-coordinates, that is:
or . in other words,
now if we had such a matrix A. then:
would be its i-th column. but that would just be the i-th basis vector of C in B-coordinates. so, for example,
convince yourself that a linear combination of the basis vectors in C is taken to the same linear combination of the images of the basis vectors in B-coordinates, because a matrix is a linear mapping.