Difference between revisions of "Vector space"
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| − | A '''vector space''' over a [[field]] (frequently the [[real number]]s) is an object which arises in [[linear algebra]] and [[abstract algebra]]. | + | A '''vector space''' over a [[field]] (frequently the [[real number]]s) is an object which arises in [[linear algebra]] and [[abstract algebra]]. A vector space <math>V</math> over a field <math>F</math> consists of a [[set]] (of [[vector]]s) and two operations, vector addition and [[scalar]] multiplication, which obey the following rules: |
===Axioms of a vector space=== | ===Axioms of a vector space=== | ||
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| + | * Under vector addition, the set of vectors forms an [[abelian group]]. Thus, addition is [[associative]] and [[commutative]] and there is an additive [[identity]] and additive [[inverse with respect to an operation | inverses]]. | ||
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| + | * Scalar multiplication is associative, so if <math>r, s \in F</math> and <math>{\bf v} \in V</math> then <math>(rs){\bf v} = r(s{\bf v})</math>. | ||
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| + | * Scalar multiplication is [[distributive]], so if <math>r \in F</math> and <math>{\bf v, w} \in V</math> then <math>r({\bf v + w}) = r{\bf v} + r{\bf w}</math>. | ||
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===Examples of vector spaces=== | ===Examples of vector spaces=== | ||
Revision as of 11:15, 9 October 2006
This article is a stub. Help us out by expanding it.
A vector space over a field (frequently the real numbers) is an object which arises in linear algebra and abstract algebra. A vector space 
over a field 
consists of a set (of vectors) and two operations, vector addition and scalar multiplication, which obey the following rules:
Axioms of a vector space
- Under vector addition, the set of vectors forms an abelian group. Thus, addition is associative and commutative and there is an additive identity and additive inverses.
- Scalar multiplication is associative, so if
and 
then 
.
and 
then 
.- Scalar multiplication is distributive, so if
and 
then 
.
and 
then 
.
