Linear span

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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.

1 Definition
2 Matroids
3 Examples
4 Theorems
5 References

[ 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. Conversely, S is called a spanning set of W

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

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

The span of S may also be defined as the set of all linear combinations of the elements of S, which follows from the above definition.

[ Matroids

Generalizing the definition of the span of points in space, a subset X of the ground set of a matroid is called a spanning set if the rank of X equals the rank of the entire ground set.

[ Examples

The real vector space R³ has {(2,0,0), (0,1,0), (0,0,1)} as a spanning set. This particular spanning set is also a basis. If (2,0,0) were replaced by (1,0,0), it would also form the canonical basis of R³.

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: Every spanning set S of a vector space V must contain at least as many elements as any linearly independent set of vectors from V.

Theorem 3: Let V be a finite dimensional vector space. Any set of vectors that spans V can be reduced to a basis for V by discarding vectors if necessary (i.e. if there are linearly dependent vectors in the set). If the axiom of choice holds, this is true without the assumption that V has finite dimension.

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