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. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...

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. In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...

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. In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

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. In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ...

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. In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...

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

• M.I. Voitsekhovskii, "Linear hull" SpringerLink Encyclopaedia of Mathematics (2001)