Examples of vector spaces

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This page lists some examples of vector spaces. See vector space for the definitions of terms used on this page. See also: dimension, basis.

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

Contents 1 Trivial or zero vector space 2 The field 3 Coordinate space 4 Infinite coordinate space 5 Matrices 6 Polynomial vector spaces 6.1 One variable 6.2 Several variables 7 Function spaces 7.1 Generalized coordinate space 7.2 Linear transformations 7.3 Continuous functions 7.4 Differential equations 8 Field extensions 9 Finite vector spaces

[edit] Trivial or zero vector space

The simplest example of a vector space is the trivial one: {0}, which contains only the zero vector (see axiom 3. of vector spaces). Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one.

Do not confuse this space with the null space of a linear operator F, which is the kernel of F.

[edit] The field

The next simplest example is the field F itself. Vector addition is just field addition and scalar multiplication is just field multiplication. The identity element of F serves as a basis so that F is a 1-dimensional vector space over itself.

The field is a special vector space, because commutativity is not only meaningful for scalar multiplication, but holds. Also, F has just two subspaces: {0} and F itself.

[edit] Coordinate space

Perhaps the most important example of a vector space is the following. 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

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

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 quaternions and the octonions are respectively four- and eight- dimensional vector spaces over the reals.

The vector space Fⁿ comes with a standard basis:

where 1 denotes the multiplicative identity in F.

[edit] Infinite coordinate space

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

only a finite number of the x_i are nonzero (i.e. the coordinates become all zero after a certain point). 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. This vector space is the coproduct of countably many copies of the vector space F.

Note the role of the finiteness condition here. One could consider arbitrary sequences of elements in F, which also constitute a vector space with the same operations (often denoted by F^N). However the dimension of this space is uncountably infinite and there is no obvious choice of basis. Since the dimensions are different, the space of unbounded sequences is not isomorphic to F^∞. This vector space is the product of countably many copies of F. It is worth noting that F^N is the dual space of F^∞. Thus one sees that a vector space need not be isomorphic to its dual (and hence also not isomorphic to its double dual) if it is infinite dimension, in stark comparison to the finite-dimensional case.

[edit] Matrices

Let F^m×n denote the set of matrices with entries in F. Then F^m×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). The zero vector is just the zero matrix. The dimension of F^m×n is mn. One possible choice of basis is the matrices with a single entry equal to 1 and all other entries 0.

[edit] Polynomial vector spaces

[edit] 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.

[edit] 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

[edit] Function spaces

See main article at Function space, especially the functional analysis section.

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

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.

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

Many of the vector spaces that arise in mathematics are subspaces of some function space. We give some further examples.

[edit] Generalized coordinate space

Let X be an arbitrary set. Consider the space V of all functions from X to F which vanish on all but a finite number of points in X. Then V is a vector subspace of all possible functions from X to F. To see this note that the union of two finite sets is finite so that the sum of two functions will still vanish on a finite set.

We call this space of functions generalized coordinate space for the following reason. If X is the set of numbers between 1 and n then this space is easily seen to be equivalent to the coordinate space Fⁿ. Likewise, if X is the set of natural numbers, N, then this space is just F^∞.

A preferred basis for V is the set of functions f_x for each x in X which are given by

The dimension of V is therefore equal to the cardinality of X. In this manner we can construct a vector space of any dimension over any field. Furthermore, every vector space is isomorphic to one of this form. Any choice of basis determines an isomorphism by sending the basis onto the preferred one for V.

[edit] Linear transformations

An important example arising in the context of linear algebra itself is the vector space of linear transformations. Let L(V,W) denote the set of all linear transformations from V to W (both of which are vector spaces over F). Then L(V,W) is a subspace of all possible functions from V to W since it is closed under addition and scalar multiplication.

Note that L(Fⁿ,F^m) can be identified with the space of matrices F^m×n in a natural way. In fact, by choosing appropriate bases for finite-dimensional spaces V and W, L(V,W) can also be identified with F^m×n. This identification normally depends on the choice of basis.

[edit] Continuous functions

If X is some topological space, such as the unit interval [0,1], we can consider the space of all continuous functions from X to R. This is a vector subspace of all possible real-valued functions on X since the sum of any two continuous functions is continuous and scalar multiplication is continuous.

[edit] Differential equations

The subset of the space of all functions from R to R consisting of functions that satisfy a certain differential equation is also a subspace if the equation is linear. This is because differentiation is a linear operation, i.e. (a f + b g)' = a f' + b g', where ' is the differentiation operator.

[edit] Field extensions

Suppose K is a subfield of F (cf. field extension). Then F can be regarded as a vector space over K by restricting scalar multiplication to elements in K (vector addition is defined as normal). The dimension of this vector space is called the degree of the extension. For example the complex numbers C form a two dimensional vector space over the real numbers R. Likewise, the real numbers R form an (uncountably) infinite-dimensional vector space over the rational numbers Q.

If V is a vector space over F it may also be regarded as vector space over K. The dimensions are related by the formula

dim_KV = (dim_FV)(dim_KF)

For example Cⁿ, regarded as a vector space over the reals, has dimension 2n.

[edit] Finite vector spaces

Apart from the trivial case of a zero-dimensional space over any field, a vector space over a field F has a finite number of elements if and only if F is a finite field and the vector space has a finite dimension. Thus we have F_q, the unique finite field with q elements. Here q must be a power of a prime (q = p^m with p prime). Then any n-dimensional vector space V over F_q will have qⁿ elements. Note that the number of elements in V is also the power of a prime. The primary example of such a space is the coordinate space (F_q)ⁿ.