Linear span

Given a vector space V over a field K, the span of a set S (not necessarily finite) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S.
If is a finite subset of V, then the span is

The span of S may also be defined as the set of all linear combinations of the elements of S, which follows from the above definition.