Computing with a vector subspace equipped with a prescribed basis over finite fields

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?