Vector space
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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 
.
