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.
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 as a spanning set. This spanning set is actually a basis.
Another spanning set for the same space is given by , but this set is not a basis, because it is linearly dependent.
The set 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 .)
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
- M.I. Voitsekhovskii, "[Linear hull]" SpringerLink Encyclopaedia of Mathematics (2001)
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