Difference between revisions of "Vector space"
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* 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>. | * 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>. | ||
| − | * if <math>x \in V</math>, <math>1.x=x</math> | + | * if <math>x \in V</math>, <math>1.(\mathbf x)=(\mathbf x)</math> |
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===Examples of vector spaces=== | ===Examples of vector spaces=== | ||
Revision as of 13:04, 4 November 2006
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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 
.- if
, 
, 
