# Span (mathematics)

From Citizendium

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

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

A basis is a linearly independent spanning set.

If *S* is itself a submodule then .

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