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 = 1 n r i s i : r i ∈ R , s i ∈ S } = ⋂ S ⊆ N ; N ≤ M N . {\displaystyle \langle S\rangle =\left\lbrace \sum _{i=1}^{n}r_{i}s_{i}:r_{i}\in R,~s_{i}\in S\right\rbrace =\bigcap _{S\subseteq N;N\leq M}N.\,}

We say that S spans, or is a spanning set for  ⟨ S ⟩ {\displaystyle \langle S\rangle } .

A basis is a linearly independent spanning set.

If S is itself a submodule then  S = ⟨ S ⟩ {\displaystyle S=\langle S\rangle } .

The equivalence of the two definitions follows from the property of the submodules forming a closure system for which  ⟨ ⋅ ⟩ {\displaystyle \langle \cdot \rangle } is the corresponding closure operator.