In mathematics, the linear span (also called the linear hull or just span) of a set S {\displaystyle S} of elements of a vector space V {\displaystyle V} is the smallest linear subspace of V {\displaystyle V} that contains S . {\displaystyle S.} It is the set of all finite linear combinations of the elements of S, and the intersection of all linear subspaces that contain S . {\displaystyle S.} It is often denoted span(S) or ⟨ S ⟩ . {\displaystyle \langle S\rangle .} For example, in geometry, two linearly independent vectors span a plane. 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 generated by S; S is a generator set or a generating set of V. Spans can be generalized to many mathematical structures, in which case, the smallest substructure containing S {\displaystyle S} is generally called the substructure generated by S . {\displaystyle S.}
Definition Given a vector space V over a field K, the span of a set S of vectors (not necessarily finite) is defined to be the intersection W of all subspaces of V that contain S. It is thus the smallest (for set inclusion) subspace containing S. It is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W, and we say that S spans W. It follows from this definition that the span of S is the set of all finite linear combinations of elements (vectors) of S, and can be defined as such. That is, span ( S ) = { λ 1 v 1 + λ 2 v 2 + ⋯ + λ n v n ∣ n ∈ N , v 1 , . . . , v n ∈ S , λ 1 , . . . , λ n ∈ K } {\displaystyle \operatorname {span} (S)={\biggl \{}\lambda _{1}\mathbf {v} _{1}+\lambda _{2}\mathbf {v} _{2}+\cdots +\lambda _{n}\mathbf {v} _{n}\mid n\in \mathbb {N} ,\;\mathbf {v} _{1},...,\mathbf {v} _{n}\in S,\;\lambda _{1},...,\lambda _{n}\in K{\biggr \}}} When S is empty, the only possibility is n = 0, and the previous expression for span ( S ) {\displaystyle \operatorname {span} (S)} reduces to the empty sum. The standard convention for the empty sum implies thus span ( ∅ ) = { 0 } , {\displaystyle {\text{span}}(\emptyset )=\{\mathbf {0} \},} a property that is immediate with the other definitions. However, many introductory textbooks simply include this fact as part of the definition. When S = { v 1 , … , v n } {\displaystyle S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}\}} is finite, one has
span ( S ) = { λ 1 v 1 + λ 2 v 2 + ⋯ + λ n v n ∣ λ 1 , . . . , λ n ∈ K } {\displaystyle \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\}}
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