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 qPochhammer 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.