Linear span

In mathematics, the linear span (also called the linear hull^[1] or just span) of a set ${\displaystyle S}$ of elements of a vector space ${\displaystyle V}$ is the smallest linear subspace of ${\displaystyle V}$ that contains ${\displaystyle S.}$ It is the set of all finite linear combinations of the elements of S,^[2] and the intersection of all linear subspaces that contain ${\displaystyle S.}$ It often denoted span(S)^[3] or ${\displaystyle \langle S\rangle .}$

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 ${\displaystyle S}$ is generally called the substructure generated by ${\displaystyle S.}$