Let x be the vector in the standard basis. Then given x_a in basis a and x_b in basis b we have the property:
x_a = A*x
x_b = B*x
where A and B are the basis operators for each space.
So lets look at going between bases: we know that A^(-1)*x_a = B^(-1)*x_b which means that:
x_a = A*B^(-1)*x_b and
x_b = B*A^(-1)*x_a