Span (mathematics)

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In algebra, the span of a set of elements of a module or vector space is the set of all finite linear combinations of that set: it may equivalently be defined as the intersection of all submodules or subspaces containing the given set.

For S a subset of an R-module M we have

⟨S⟩={∑i=1nrisi:ri∈R,si∈S}=⋂S⊆N;N≤MN.

We say that S spans, or is a spanning set for ⟨S⟩.

A basis is a linearly independent spanning set.

If S is itself a submodule then S=⟨S⟩.

The equivalence of the two definitions follows from the property of the submodules forming a closure system for which ⟨⋅⟩ is the corresponding closure operator.