A vector space 
is a set that is closed under finite vector addition
and scalar multiplication. The basic example
is 
-dimensional Euclidean
space 
,
where every element is represented by a list of 
real numbers, scalars are real numbers, addition is componentwise,
and scalar multiplication is multiplication on each term separately.
For a general vector space, the scalars are members of a field 
, in which case 
is called a vector space over 
.
Euclidean 
-space
is called a real
vector space, and 
is called a complex vector space.
In order for 
to be a vector space, the following conditions must hold for all elements 
and any scalars 
:
1. Commutativity:
 |
(1)
|
2. Associativity of vector addition:
 |
(2)
|
3. Additive identity: For all 
,
 |
(3)
|
4. Existence of additive inverse: For any 
, there exists a 
such that
 |
(4)
|
5. Associativity of scalar multiplication:
 |
(5)
|
6. Distributivity of scalar sums:
 |
(6)
|
7. Distributivity of vector sums:
 |
(7)
|
8. Scalar multiplication identity:
 |
(8)
|
Let 
be a vector space of dimension 
over the field
of 
elements (where 
is necessarily a power of a prime number). Then the number
of distinct nonsingular linear operators on 
is
 |  |  |
(9)
|
 |  |  |
(10)
|
and the number of distinct 
-dimensional subspaces of 
is
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  |
(13)
|
where 
is a q-Pochhammer symbol.
A consequence of the axiom of choice is that every vector space has a vector basis.
A module is abstractly similar to a vector space, but it uses a ring to define coefficients instead of the field used for vector spaces. Modules have coefficients in much more general algebraic objects.
