Examples of vector spaces

This page lists some examples of vector spaces. See vector space for a definition and more discussion.

Notation. We let F denote an arbitray field such as the real numbers R or the complex numbers C. See also: table of mathematical symbols.

Trivial vector space

The simplest example of a vector space is the trivial one. Let 0 denote the space which contains only the zero element of F. Then 0 is 0-dimensional vector space over F. Every vector space over F contains a subspace isomporphic to this one.

The field itself

The field F is a 1-dimensional vector space over itself. Vector addition is just field addition and scalar multiplication is just field multiplication.

Coordinate space

For any positive integer n, the space of all n-tuples of elements of F forms an n-dimensional vector space over F sometimes called coordinate space and denoted Fⁿ. An element of Fⁿ is written

$x=(x_{1},x_{2},\ldots ,x_{n})$

where each x_i is an element of F. The operations on Fⁿ are defined by

$x+y=(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n})$

$\alpha x=(\alpha x_{1},\alpha x_{2},\ldots ,\alpha x_{n})$

$0=(0,0,\ldots ,0)$

$-x=(-x_{1},-x_{2},\ldots ,-x_{n})$

The most common cases are where F is the field of real numbers giving the real coordinate space Rⁿ, or the field of complex numbers giving the complex coordinate space Cⁿ.

The vector space Fⁿ comes with a standard basis:

$e_{1}=(1,0,\ldots ,0)$

$e_{2}=(0,1,\ldots ,0)$

$\vdots$

$e_{n}=(0,0,\ldots ,1)$

where 1 denotes the multiplicative identity in F.

Infinite coordinate space

Let F^∞ denote the space of N-indexed tuples of elements of F such that only finitely many elements are nonzero. That is, if we write an element of F^∞ as

$x=(x_{1},x_{2},x_{3},\ldots )$

only a finite number of the x_i are nonzero. Addition and scalar multiplication are given as in finite coordinate space.

The dimensionality of F^∞ is countably infinite. A standard basis consists of the vectors e_i which contain a 1 in the i-th slot and zeros elsewhere.

Polynomial vector spaces

One variable

The set of polynomials with coefficients in F is vector space over F denoted F[x]. Vector addition and scalar multiplication are defined in the obvious manner. If the degree of the polynomials is unrestricted then the dimension of F[x] is countably infinite. If one restricts to polynomials with degree strictly less than n then we have a vector space with dimension n.

One possible basis for this vector space is a monomial basis.

Several variables

The set of polynomials in several variables with coefficients in F is vector space over F denoted F[x₁, x₂, …, x_r]. Here r is the number of variables.

See also: polynomial ring

Function space

Let X be an arbitrary set and V an arbitrary vector space over F. The space of all functions from X to V is a vector space over F with coordinate-wise addition and multiplication. That is, let f : X → V and g : X → V denote two functions. We define

$(f+g)(x)=f(x)+g(x)$

$(\alpha f)(x)=\alpha f(x)$

where the operations on the right hand side are those in V. The zero vector is given by the constant function sending everything to the zero vector in V.

Coordinate space is a special case of this example where X is a finite set with n elements and V is just F.