Editing Linear span (section)

== Generalizations ==
Generalizing the definition of the span of points in space, a subset {{mvar|X}} of the ground set of a [[matroid]] is called a spanning set if the rank of {{mvar|X}} equals the rank of the entire ground set{{sfnp|Oxley|2011|p=28}}

The vector space definition can also be generalized to modules.<ref>{{Harvard citation text|Roman|2005}} p. 96, ch. 4</ref><ref>{{Harvard citation text|Mac Lane|Birkhoff|1999}} p. 193, ch. 6</ref> Given an {{mvar|R}}-module {{mvar|A}} and a collection of elements {{math|''a''<sub>1</sub>}}, ..., {{math|''a<sub>n</sub>''}} of {{mvar|A}}, the [[submodule]] of {{mvar|A}} spanned by {{math|''a''<sub>1</sub>}}, ..., {{math|''a<sub>n</sub>''}} is the sum of [[cyclic module]]s
<math display="block">Ra_1 + \cdots + Ra_n = \left\{ \sum_{k=1}^n r_k a_k \bigg| r_k \in R \right\}</math>
consisting of all ''R''-linear combinations of the elements {{math|''a<sub>i</sub>''}}. As with the case of vector spaces, the submodule of ''A'' spanned by any subset of ''A'' is the intersection of all submodules containing that subset.