Linear span

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". "W" is referred to as the subspace "spanned" by "S", or by the vectors in "S".

If $S\; =\; \{v\_1,...,v\_r\},$ is a finite subset of "V", then the span is

:$\{\; m\; span\; \}\; left(S\; ight)\; =\; \{\; m\; span\; \}\; left(v\_1,...,v\_r\; ight)\; =\; left\{\; \{lambda\; \_1\; v\_1\; +\; cdots\; +\; lambda\; \_r\; v\_r\; |lambda\; \_1\; ,\; ldots\; ,lambda\; \_r\; in\; mathbb\; K\}\; ight\}.$

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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