Let $U$ be a subspace of the finite dimensional vector space $V$ over a field $\mathbb{k}$. Let $B_V$ and $B_U$ be fixed bases for $V$ and $U$ respectively. Let $u \in U$ and let's give ourselves $[u]_V$, the vector representing $u$ with respect to $B_V$.

**How do we effectively compute $[u]_U$ when $\mathbb{k}$ is a finite field, say the one with two elements?**