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 
.- if
, 
, 
Subspaces
If 
, and 
is a vector space itself, then it is called a subspace of
.
Independent Subsets
Let be a vector space over the complex field. Let 
be a subset of such that no linear combination of elements of with coefficients not all zero gives the null vector. Then is said to be a linearly independent subset of .
Linear Manifolds
Let 
be a subset of some vector space . Then it can be proved that the set of all linear combinations of the elements of forms a vector space. This space is said to have been generated by , and is called the linear manifold 
of .
