Linear span

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the linear span (also called the linear hull or just span) of a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> 
 
 
 
 S
 
 
 {\displaystyle S}
 
 of elements of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is the smallest <a href="/facts/Linear_subspace/2wtKTgHL">linear subspace</a> of 
 
 
 
 V
 
 
 {\displaystyle V}
 
 that contains 
 
 
 
 S
 .
 
 
 {\displaystyle S.}
 
 It is the set of all finite <a href="/facts/Linear_combination/CVjL6kuN">linear combinations</a> of the elements of S, and the intersection of all linear subspaces that contain 
 
 
 
 S
 .
 
 
 {\displaystyle S.}
 
 It often denoted span(S) or 
 
 
 
 ⟨
 S
 ⟩
 .
 
 
 {\displaystyle \langle S\rangle .}
 
 
For example, in <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, two <a href="/facts/Linearly_independent/8JrHfIIa">linearly independent</a> <a href="/facts/Vector_(geometry)/bCLrdksy">vectors</a> span a <a href="/facts/Plane_(geometry)/tAUtRFcn">plane</a>.
To express that a vector space V is a linear span of a subset S, one commonly uses one of the following phrases: S spans V; S is a spanning set of V; V is spanned or <a href="/facts/Generator_(mathematics)/Jo1eBRbD">generated</a> by S; S is a generator set or a generating set of V.
Spans can be generalized to many <a href="/facts/Mathematical_structure/6CXReUNv">mathematical structures</a>, in which case, the smallest substructure containing 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is generally called the substructure generated by 
 
 
 
 S
 .
 
 
 {\displaystyle S.}

Linear span open-in-new

Linear span