Editing Linear span (section)

{{Short description|In linear algebra, generated subspace}}
[[File:Basis for a plane.svg|thumb|280px|right|The cross-hatched plane is the linear span of '''u''' and '''v''' in both '''R'''<sup>2</sup> and '''R'''<sup>3</sup>, here shown in [[Perspective (graphical)|perspective]].]]

In [[mathematics]], the '''linear span''' (also called the '''linear hull'''<ref>{{Harvard citation text|Encyclopedia of Mathematics|2020}}. Linear Hull.</ref> or just '''span''') of a [[Set (mathematics)|set]] <math>S</math> of elements of a [[vector space]] <math>V</math> is the smallest [[linear subspace]] of <math>V</math> that contains <math>S.</math> It is the set of all finite [[linear combination]]s of the elements of {{mvar|S}},<ref>{{Harvard citation text|Axler|2015}} p. 29, § 2.7</ref> and the intersection of all linear subspaces that contain <math>S.</math> It is often denoted {{math|span(''S'')}}<ref name=":0">{{Harvard citation text|Axler|2015}} pp. 29-30, §§ 2.5, 2.8</ref> or <math>\langle S \rangle.</math> 

For example, in [[geometry]], two [[linearly independent]] [[vector (geometry)|vectors]] span a [[plane (geometry)|plane]].

To express that a vector space {{mvar|V}} is a linear span of a subset {{mvar|S}}, one commonly uses one of the following phrases: {{mvar|S}} spans {{mvar|V}}; {{mvar|S}} is a '''spanning set''' of {{mvar|V}}; {{mvar|V}} is spanned or [[generator (mathematics)|generated]] by {{mvar|S}};  {{mvar|S}} is a generator set or a generating set of {{mvar|V}}.

Spans can be generalized to many [[mathematical structure]]s, in which case, the smallest substructure containing <math>S</math> is generally called the substructure ''generated'' by <math>S.</math>