Suppose that $U$ is a vector space with a basis $B$ of size $n$. Then \begin{align*} \vect{u}\in\spn{\set{\vectorlist{u}{k}}} \end{align*} if and only if \begin{align*} \vectrep{B}{\vect{u}}\in\spn{\set{\vectrep{B}{\vect{u}_1},\,\vectrep{B}{\vect{u}_2},\,\vectrep{B}{\vect{u}_3},\,\ldots,\,\vectrep{B}{\vect{u}_k}}} \end{align*}