The span of a set $S\subseteq V$ is the set of all finite linear combinations of elements of $S$. It is the smallest subspace containing $S$, and $\operatorname{span}(\varnothing)=\{0\}$. For a finite list $v_1,\ldots,v_k$, membership means the matrix equation $[v_1\ \cdots\ v_k]c=b$ is consistent.
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