Examples of vector spaces

The field is a rather special vector space; in fact it is the simplest example of a commutative algebra over F. Also, F has just two subspaces: {0} and F itself.

[1] An element of Fn is written where each xi is an element of F. The operations on Fn are defined by Commonly, F is the field of real numbers, in which case we obtain real coordinate space Rn.

The a + bi form of a complex number shows that C itself is a two-dimensional real vector space with coordinates (a,b).

The vector space Fn has a standard basis: where 1 denotes the multiplicative identity in F. Let F∞ denote the space of infinite sequences of elements from F such that only finitely many elements are nonzero.

That is, if we write an element of F∞ as then only a finite number of the xi are nonzero (i.e., the coordinates become all zero after a certain point).

A standard basis consists of the vectors ei which contain a 1 in the i-th slot and zeros elsewhere.

One could consider arbitrary sequences of elements in F, which also constitute a vector space with the same operations, often denoted by FN - see below.

Let Fm×n denote the set of m×n matrices with entries in F. Then Fm×n is a vector space over F. Vector addition is just matrix addition and scalar multiplication is defined in the obvious way (by multiplying each entry by the same scalar).

Vector addition and scalar multiplication are defined in the obvious manner.

If instead one restricts to polynomials with degree less than or equal to n, then we have a vector space with dimension n + 1.

The vector space of polynomials with real coefficients and degree less than or equal to n is often denoted by Pn.

The set of polynomials in several variables with coefficients in F is vector space over F denoted F[x1, x2, ..., xr].

If X is finite and V is finite-dimensional then VX has dimension |X|(dim V), otherwise the space is infinite-dimensional (uncountably so if X is infinite).

Contrast this with the direct product of |X| copies of F which would give the full function space FX.

Note that L(Fn,Fm) can be identified with the space of matrices Fm×n in a natural way.

The dimension of this vector space, if it exists,[a] is called the degree of the extension.