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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. Jump to: navigation, search A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
In mathematics, the dimension of a vector space V is the cardinality (i. ...
In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...
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. Jump to: navigation, search In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
Jump to: navigation, search In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
In mathematics, a set of symbols is frequently used in mathematical expressions. ...
Trivial vector space
The simplest example of a vector space is the trivial one: {0} which contains only the zero element of F. 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. In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...
In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...
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.
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 In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ...
Jump to: navigation, search The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
In mathematics, specifically in linear algebra, the coordinate space, Fn, is the prototypical example of an n-dimensional vector space over a field F. Definition Let F denote an arbitrary field (such as the real numbers R or the complex numbers C). ...
where each xi 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. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
Jump to: navigation, search In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
The quaternions and the octonions are respectively four- and eight- dimensional vector spaces over the reals. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
In mathematics, the octonions are a nonassociative extension of the quaternions. ...
The vector space Fn comes with a standard basis: In linear algebra, the standard basis for an -dimensional vector space is the basis obtained by taking the basis vectors where is the vector with a in the th coordinate and elsewhere. ...
where 1 denotes the multiplicative identity in F.
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 This is a page about mathematics. ...
only a finite number of the xi 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 ei which contain a 1 in the i-th slot and zeros elsewhere. In mathematics the term countable set is used to describe the size of a set, e. ...
Note the role of the finiteness condition here. One could consider arbitrary (unbounded) sequences of elements in F, which is also a vector space with the same operations. 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∞. In mathematics, an uncountable set is a set which is not countable. ...
Matrices Let Fm×n denote the set of 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). The zero vector is just the zero matrix. The dimensionality of Fm×n is mn. One possible choice of basis is the matrices with a single entry equal to 1 and all other entries 0. Jump to: navigation, search For the square matrix section, see square matrix. ...
In mathematics, a zero matrix is a matrix with all its entries being zero. ...
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. In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...
In mathematics the term countable set is used to describe the size of a set, e. ...
This article is about the term degree as used in mathematics. ...
One possible basis for this vector space is a monomial basis. In mathematics a monomial basis is a way to uniquely describe a polynomial using a linear combination of monomials. ...
Several variables The set of polynomials in several variables with coefficients in F is vector space over F denoted F[x1, x2, …, xr]. Here r is the number of variables. In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...
See also: polynomial ring In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ...
Function spaces 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 In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...
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.
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∞. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
A preferred basis for V is the set of functions fx 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. In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ...
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. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
Note that L(Fn,Fm) can be identified with the space of matrices Fm×n in a natural way.
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. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...
In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...
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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ...
In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ...
Jump to: navigation, search In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
Please refer to Real vs. ...
Jump to: navigation, search In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
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
- dimKV = (dimFV)(dimKF)
For example Cn, regarded as a vector space over the reals, has dimension 2n.
Finite vector spaces There are some vector spaces which actually have a finite number of elements. Let Fq denote the unique finite field with q elements. Here q must be a power of a prime (q = pm with p prime). Then any n-dimensional vector space V over Fq 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 (Fq)n. In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...
Jump to: navigation, search In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ...
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