Linear span

In mathematics, the linear span (also called the linear hull or just span) of a set S {\displaystyle S} of elements of a vector space V {\displaystyle V} is the smallest linear subspace of V {\displaystyle V} that contains S . {\displaystyle S.} It is the set of all finite linear combinations of the elements of S, and the intersection of all linear subspaces that contain S . {\displaystyle S.} It is often denoted span(S) or ⟨ S ⟩ . {\displaystyle \langle S\rangle .} For example, in geometry, two linearly independent vectors span a plane. To express that a vector space V is a linear span of a subset S, one commonly uses one of the following phrases: S spans V; S is a spanning set of V; V is spanned or generated by S; S is a generator set or a generating set of V. Spans can be generalized to many mathematical structures, in which case, the smallest substructure containing S {\displaystyle S} is generally called the substructure generated by S . {\displaystyle S.}