Can coordinates be inverted to get back the original vector/transformation?

Question

For example, take a Transformation T: X->Y

If $x\in X$ and $y \in Y$ and I know that $[T(x)]_B = [y]_B$, is it true that $T(x) = y$, and how can I show this?

Answer 1

Yes. The coordinate transformation is an isomorphism.