Let $V$ be a finite-dimensional [vector space](/page/Vector%20Space) over a field $k$, and let $U \subset V$ be a subspace. Then there exists an ordered basis $\mathcal B=(v_1,\ldots,v_n)$ of $V$ adapted to $U$: if $m=\dim U$, then the initial segment $(v_1,\ldots,v_m)$ is an ordered basis of $U$, with the convention that the empty initial segment is a basis of the zero subspace when $m=0$.