Editing Linear span (section)

== Definition ==
Given a [[vector space]] {{mvar|V}} over a [[field (mathematics)|field]] {{mvar|K}}, the span of a [[Set (mathematics)|set]] {{mvar|S}} of vectors (not necessarily finite) is defined to be the intersection {{mvar|W}} of all [[linear subspace|subspaces]] of {{mvar|V}} that contain {{mvar|S}}. It is thus the smallest (for [[set inclusion]]) subspace containing  {{mvar|S}}. It is referred to as the subspace ''spanned by'' {{mvar|S}}, or by the vectors in {{mvar|S}}. Conversely, {{mvar|S}} is called a ''spanning set'' of {{mvar|W}}, and we say that {{mvar|S}} ''spans'' {{mvar|W}}.

It follows from this definition that the span of {{mvar|S}} is the set of all finite [[linear combinations]] of elements (vectors) of {{mvar|S}}, and can be defined as such.<ref>{{Harvard citation text|Hefferon|2020}} p. 100, ch. 2, Definition 2.13</ref><ref name=":02">{{Harvard citation text|Axler|2015}} pp. 29-30, §§ 2.5, 2.8</ref><ref>{{Harvard citation text|Roman|2005}} pp. 41-42</ref> That is, <math display="block"> \operatorname{span}(S) =  \biggl \{ \lambda_1 \mathbf v_1 + \lambda_2 \mathbf v_2 + \cdots + \lambda_n \mathbf v_n \mid n \in \N,\; \mathbf v_1,...\mathbf v_n   \in S, \; \lambda_1,...\lambda_n  \in K \biggr \}</math> 

When {{mvar|S}} is [[empty set|empty]], the only possibility is {{math|1=''n'' = 0}}, and the previous expression for <math>\operatorname{span}(S)</math> reduces to the [[empty sum]].{{efn| This is logically valid as when {{math|1= ''n'' = 0}}, the conditions for the vectors and constants are empty, and therefore [[vacuously]] satisfied.}} The standard convention for the empty sum implies thus <math>\text{span}(\empty) = \{\mathbf 0\}, </math> a property that is immediate with the other definitions. However, many introductory textbooks simply include this fact as part of the definition.

When <math>S=\{\mathbf v_1,\ldots, \mathbf v_n\}</math> is [[finite set|finite]], one has
<math display="block"> \operatorname{span}(S) =  \{ \lambda_1 \mathbf v_1 + \lambda_2 \mathbf v_2 + \cdots + \lambda_n \mathbf v_n \mid \lambda_1,...\lambda_n  \in K  \}</math>