Vector Space (original) (raw)
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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
and the number of distinct 
-dimensional subspaces of 
is
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.
See also
Banach Space, Field, Function Space, Hilbert Space, Inner Product Space, Module,Quotient Vector Space, Ring,Symplectic Space, Topological Vector Space, Vector, Vector Basis Explore this topic in the MathWorld classroom
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References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 530-534, 1985.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Vector Space." FromMathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VectorSpace.html