Let $$B = \{b_1, \ldots , b_n\}$$ be a basis of $V$
Suppose now that there is a set $M$ such that $U \subset M \subset V$ properly. We now select $b_i, b_j$ whereby $b_i \in M\setminus U$ and $b_j \in V \setminus M$. Now let
$$v = (f(b_i))^{-1}b_i - (f(b_j))^{-1}b_j.$$
Clearly $v \notin U$...