Linear isomorphism $f:U_1\to U_2$ with $U_2\subset U_1$ Let $V$ be a finite-dimensional vector space and $U_1,U_2\subset V$ be subspaces of $V$ such that $U_2\subset U_1$ and let $f:U_1\to U_2$ be a linear isomorphism. Prove (or give a counterexample) whether the following holds: $$U_1=U_2$$
I can't think of a counterexample but showing this also seems difficult since if $B_1$ is a Basis of $U_1$, all I know is that $f(B_1)$ is a basis of $U_2\subset U_1$ but that doesn't seem to suffice.
Thank you very much in advance.

Hint:
Since we have finite  dimensional spaces,
$$\dim (U_1/U_2)=\dim U_1 -\dim U_2. $$
Now, since $U_1$ and $U_2$ are isomorphic, they have the same dimension.
