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Vector Space


A vector space V is a set that is closed under finite vector addition and scalar multiplication. The basic example is n-dimensional Euclidean space R^n, where every element is represented by a list of n 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 F, in which case V is called a vector space over F.

Euclidean n-space R^n is called a real vector space, and C^n is called a complex vector space.

In order for V to be a vector space, the following conditions must hold for all elements X,Y,Z in V and any scalars r,s in F:

1. Commutativity:

 X+Y=Y+X.
(1)

2. Associativity of vector addition:

 (X+Y)+Z=X+(Y+Z).
(2)

3. Additive identity: For all X,

 0+X=X+0=X.
(3)

4. Existence of additive inverse: For any X, there exists a -X such that

 X+(-X)=0.
(4)

5. Associativity of scalar multiplication:

 r(sX)=(rs)X.
(5)

6. Distributivity of scalar sums:

 (r+s)X=rX+sX.
(6)

7. Distributivity of vector sums:

 r(X+Y)=rX+rY.
(7)

8. Scalar multiplication identity:

 1X=X.
(8)

Let V be a vector space of dimension n over the field of q elements (where q is necessarily a power of a prime number). Then the number of distinct nonsingular linear operators on V is

M(n,q)=(q^n-q^0)(q^n-q^1)(q^n-q^2)...(q^n-q^(n-1))
(9)
=q^(n^2)(q^(-n);q)_n
(10)

and the number of distinct k-dimensional subspaces of V is

S(k,n,q)=((q^n-q^0)(q^n-q^1)(q^n-q^2)...(q^n-q^(k-1)))/(M(k,q))
(11)
=((q^n-1)(q^(n-1)-1)(q^(n-2)-1)...(q^(n-k+1)-1))/((q^k-1)(q^(k-1)-1)(q^(k-2)-1)...(q-1))
(12)
=(q^((k-n)n)(q^(-n);q)_k)/((q^(-n),q)_n),
(13)

where (q;a)_n 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

Vector Space

Cite this as:

Weisstein, Eric W. "Vector Space." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VectorSpace.html

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