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.

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. When S is a finite set, then W is referred to as the subspace spanned by the vectors in S.

Notes

A spanning set is usually not a basis for S as the spanning vectors need not be linearly independent. On the other hand a minimal spanning set for a given vector space S is a basis in a finite dimensional space. In other words in a finite dimensional space a spanning set is a basis for S if and only if every vector in S 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. However it does not indicate what will be the span of the empty set. (That is, in fact, the trivial vector space {0}.)

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.

External links

Abstract algebra | Linear algebra

Lineare Hülle | Sous-espace vectoriel engendré | Span lineare | קבוצה פורשת | Lineair omhulsel | Lin (matematyka) | Линейная оболочка | Linearna ogrinjača