FACTOID # 19: Cheap sloppy joes: Looking for reduced-price lunches for schoolchildren? Head for Oklahoma!
 
 Home   Encyclopedia   Statistics   States A-Z   Flags   Maps   FAQ   About 
   
 
WHAT'S NEW
 

SEARCH ALL

FACTS & STATISTICS    Advanced view

Search encyclopedia, statistics and forums:

 

 

(* = Graphable)

 

 


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

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 (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. ... ↔ ⇔ ≡ 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.


External links

  • M.I. Voitsekhovskii (2001), “Linear hull”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
The Encyclopaedia of Mathematics is a large reference work in mathematics. ... 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. ... This article is about vectors that have a particular relation to the spatial coordinates. ... 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 linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... 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. ... The column space of an m-by-n matrix with real entries is the subspace of Rm generated by the column vectors of the matrix. ... In computer science and mathematics, the row space of an m-by-n matrix with real entries is the subspace of Rn generated by the row vectors of the matrix. ... In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ... In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. ... For more specific information regarding the eigenvalues and eigenvectors of matrices, see Eigendecomposition (matrix). ... In regression analysis, least squares, also known as ordinary least squares analysis, is a method for linear regression that determines the values of unknown quantities in a statistical model by minimizing the sum of the residuals (the difference between the predicted and observed values) squared. ... Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ... For the cross product in algebraic topology, see Künneth theorem. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or A′) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A... In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. ...

  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), (−1,1/2,3), (1,1,1)}, but this set is not a basis, because it is linearly dependent.
  More results at FactBites »

 
 

COMMENTARY     


Share your thoughts, questions and commentary here
Your name
Your comments

Want to know more?
Search encyclopedia, statistics and forums:

 


Press Releases |  Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms, 1022, m