Difference between revisions of "Vector space"
m |
m |
||
| Line 5: | Line 5: | ||
===Axioms of a vector space=== | ===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 [[inverse with respect to an operation | inverses]]. | + | * Under vector addition, the set of vectors forms an [[abelian group]]. Thus, addition is [[associative]] and [[commutative]] and there is an additive [[identity]] (usually denoted <math>\mathbf 0</math>) and additive [[inverse with respect to an operation | inverses]]. |
| − | * Scalar multiplication is associative, so if <math>r, s \in F</math> and <math>{\ | + | * Scalar multiplication is associative, so if <math>r, s \in F</math> and <math>{\mathbf v} \in V</math> then <math>(rs){\mathbf v} = r(s{\mathbf v})</math>. |
| − | * Scalar multiplication is [[distributive]], so if <math>r \in F</math> and <math>{\ | + | * Scalar multiplication is [[distributive property | distributive]] over both vector and scalar addition, so if <math>r \in F</math> and <math>{\mathbf v, w} \in V</math> then <math>r({\mathbf v + w}) = r{\mathbf v} + r{\mathbf w}</math>. |
* | * | ||
Revision as of 11:17, 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 (usually denoted
) and additive inverses.
) and additive - Scalar multiplication is associative, so if
and 
then 
.
and 
then 
.- Scalar multiplication is distributive over both vector and scalar addition, so if
and 
then 
.
and 
then 
.
