matrices – Computing with a subvector subspace equipped with a prescribed base on finite files

Leave $ U $ Be a subspace of the vector space of finite dimensions. $ V $ over a field $ mathbb {k} $. Leave $ B_V $ Y $ B_U $ be fixed bases for $ V $ Y $ U $ respectively. Leave $ u in U $ and we are going to give us $[u]_V $, the vector that represents $ u $ with respect to $ B_V $.

How do we calculate effectively? $[u]_U $ when $ mathbb {k} $ Is it a finite field, let's say that of two elements?