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Encyclopedia > 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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... 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. ... This article is about linear subspaces of an abstract vector space. ...

## Contents

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 (called vectors) that, informally speaking, may be scaled and added. ... 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 objects considered as a whole. ... This article is about linear subspaces of an abstract vector space. ... 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. ...

Let $v_1,...,v_r in V$. The span of the set of these vectors is

${ rm span } left(v_1,...,v_rright) = left{ {lambda _1 v_1 + cdots + lambda _r v_r |lambda _1 , ldots ,lambda _r in mathbb K} right}.$

## 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. In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ... 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. ... â†” â‡” â‰¡ logical symbols representing iff. ... 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 real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...

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 objects considered as a whole. ...

The set {(0.3,4.5,0), (5.7,17,0), (1.59,1.87,0)} 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.

Results from FactBites:

 PlanetMath: span (300 words) Span is both a noun and a verb; a set of vectors can span a vector space, and a vector can be in the span of a set of vectors. To see whether a set of vectors spans a vector space, you need to check that there are at least as many linearly independent vectors as the dimension of the space. This is version 17 of span, born on 2001-11-13, modified 2008-05-26.
 Linear span (471 words) 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. Another spanning set for the same space is given by {(1,2,3), (0,1,2), (â1,1/2,3), (1,1,1)}, but this set is not a basis, because it is linearly dependent.
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