A nonempty subset W of a vector space V that is closed under addition and scalar multiplication (and therefore contains the 0 -vector of V ) is called a linear subspace of V , or simply a subspace of V , when the ambient space is unambiguously a vector space. Subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set S of vectors is called its span, and it is the smallest subspace of V containing the set S . Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of S .