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:
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(1) |
2. Associativity of vector addition:
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(2) |
3. Additive identity: For all ,
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(3) |
4. Existence of additive inverse: For any , there exists a such that
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(4) |
5. Associativity of scalar multiplication:
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(5) |
6. Distributivity of scalar sums:
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(6) |
7. Distributivity of vector sums:
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(7) |
8. Scalar multiplication identity:
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(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
and the number of distinct -dimensional subspaces of is
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(11) | |||
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(12) | |||
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(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.