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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

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.

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