Algebraic reflexivity of linear transformations

Let $\mathcal{L}(U, V)$ be the set of all linear transformations from $U$ to $V$, where $U$ and $V$ are vector spaces over a field $\mathbb{F}$. We show that every $n$-dimensional subspace of $\mathcal{L}(U, V)$ is algebraically $\lfloor \sqrt {2n} \rfloor$-reflexive, where $\lfloor t \rfloor$ denotes the largest integer not exceeding $t$, provided $n$ is less than the cardinality of $\mathbb{F}$.