The easiest way to look at this is to go from first principles.
Now in one basis we have some vector v in Basis B, and in another basis we have v' in Basis B'. Let x be the true vector that is invariant to both bases: then this implies
Bx = v and
B'x = v'. If B and B' are both basis then B and B' are invertible giving
x = B^(-1)v = B'^(-1)v'. Writing these equations to get v in terms of v' (and vice versa) gives:
B*B'^(-1)v' = v and
B'*B^(-1)v = v'
Now a linear transformation acting on R^2 is simply Ax = b taking x and mapping it to b, and the above can relate Av to Av' in a straight-forward manner.