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

$\langle S \rangle = \left\lbrace \sum_{i=1}^n r_i s_i : r_i \in R,~ s_i \in S \right\rbrace = \bigcap_{S \subseteq N; N \le M} N .\,$

We say that S spans, or is a spanning set for $\langle S \rangle$ .

A basis is a linearly independent spanning set.

If S is itself a submodule then $S = \langle S \rangle$ .

The equivalence of the two definitions follows from the property of the submodules forming a closure system for which $\langle \cdot \rangle$ is the corresponding closure operator.