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In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. The linear span of a set of vectors is therefore a vector space.

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Definition

Given a vector space V over a field K, the span of a set S (not necessarily finite) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S.

If S = \{v_1,\dots,v_r\}\, is a finite subset of V, then the span is

\operatorname{span}(S) = \operatorname{span}(v_1,\dots,v_r) = \{ {\lambda _1 v_1 + \dots + \lambda _r v_r \mid \lambda _1 , \dots ,\lambda _r \in \mathbf{K}} \}.

Notes

The span of S may also be defined as the collection of all (finite) linear combinations of the elements of S.

If the span of S is V, then S is said to be a spanning set of V. A spanning set of V is not necessarily a basis for V, as it need not be linearly independent. However, a minimal spanning set for a given vector space is necessarily a basis. In other words, a spanning set is a basis for V if and only if every vector in V can be written as a unique linear combination of elements in the spanning set.√

Examples

The real vector space R3 has {(1,0,0), (0,1,0), (0,0,1)} as a spanning set. This spanning set is actually a basis.

Another spanning set for the same space is given by {(1,2,3), (0,1,2), (−1,1/2,3), (1,1,1)}, but this set is not a basis, because it is linearly dependent.

The set {(1,0,0), (0,1,0), (1,1,0)} is not a spanning set of R3; instead its span is the space of all vectors in R3 whose last component is zero.

Theorems

Theorem 1: The subspace spanned by a non-empty subset S of a vector space V is the set of all linear combinations of vectors in S.

This theorem is so well known that at times it is referred to as the definition of span of a set.

Theorem 2: Let V be a finite dimensional vector space. Any set of vectors that spans V can be reduced to a basis by discarding vectors if necessary.

This also indicates that a basis is a minimal spanning set when V is finite dimensional.

References








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