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

Notation. LetF denote an arbitraryfield such as thereal numbersR or thecomplex numbersC.

The simplest example of a vector space is the trivial one: {0}, which contains only the zero vector (see the third axiom in theVector space article). Both vector addition and scalar multiplication are trivial. Abasis for this vector space is theempty set, so that {0} is the 0-dimensional vector space overF. Every vector space overF contains asubspaceisomorphic to this one.

The zero vector space is conceptually different from thenull space of a linear operatorL, which is thekernel ofL. (Incidentally, the null space ofL is a zero space if and only ifL isinjective.)

The next simplest example is the fieldF itself. Vector addition is just field addition, and scalar multiplication is just field multiplication. This property can be used to prove that a field is a vector space. Any non-zero element ofF serves as a basis soF is a 1-dimensional vector space over itself.

The field is a rather special vector space; in fact it is the simplest example of acommutative algebra overF. Also,F has just twosubspaces: {0} andF itself.

A basic example of a vector space is the following. For anypositiveintegern, theset of alln-tuples of elements ofF forms ann-dimensional vector space overF sometimes calledcoordinate space and denotedFⁿ.^[1] An element ofFⁿ is written

${\displaystyle x=(x_1,x_2,\dots ,x_n)}$

where eachx_i is an element ofF. The operations onFⁿ are defined by

${\displaystyle x+y=(x_1+y_1,x_2+y_2,\dots ,x_n+y_n)}$

${\displaystyle \alpha x=(\alpha x_1,\alpha x_2,\dots ,\alpha x_n)}$

${\displaystyle 0=(0,0,\dots ,0)}$

${\displaystyle -x=(-x_1,-x_2,\dots ,-x_n)}$

Commonly,F is the field ofreal numbers, in which case we obtainreal coordinate spaceRⁿ. The field ofcomplex numbers givescomplex coordinate spaceCⁿ. Thea + bi form of a complex number shows thatC itself is a two-dimensional real vector space with coordinates (a,b). Similarly, thequaternions and theoctonions are respectively four- and eight-dimensional real vector spaces, andCⁿ is a2n-dimensional real vector space.

The vector spaceFⁿ has astandard basis:

${\displaystyle e_1=(1,0,\dots ,0)}$

${\displaystyle e_2=(0,1,\dots ,0)}$

${\displaystyle ⋮}$

${\displaystyle e_n=(0,0,\dots ,1)}$

where 1 denotes the multiplicative identity inF.

LetF^∞ denote the space ofinfinite sequences of elements fromF such that onlyfinitely many elements are nonzero. That is, if we write an element ofF^∞ as

${\displaystyle x=(x_1,x_2,x_3,\dots )}$

then only a finite number of thex_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 ofF^∞ iscountably infinite. A standard basis consists of the vectorse_i which contain a 1 in thei-th slot and zeros elsewhere. This vector space is thecoproduct (ordirect sum) of countably many copies of the vector spaceF.

Note the role of the finiteness condition here. One could consider arbitrary sequences of elements inF, which also constitute a vector space with the same operations, often denoted byF^N - seebelow.F^N is theproduct of countably many copies ofF.

ByZorn's lemma,F^N has a basis (there is no obvious basis). There areuncountably infinite elements in the basis. Since the dimensions are different,F^N isnot isomorphic toF^∞. It is worth noting thatF^N is (isomorphic to) thedual space ofF^∞, because alinear mapT fromF^∞ toF is determined uniquely by its valuesT(e_i) on the basis elements ofF^∞, and these values can be arbitrary. Thus one sees that a vector space need not be isomorphic to its double dual if it is infinite dimensional, in contrast to the finite dimensional case.

Starting fromn vector spaces, or a countably infinite collection of them, each with the same field, we can define the product space like above.

LetF^m×n denote the set ofm×nmatrices with entries inF. ThenF^m×n is a vector space overF. 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 thezero matrix. Thedimension ofF^m×n ismn. One possible choice of basis is the matrices with a single entry equal to 1 and all other entries 0.

Whenm =n the matrix issquare andmatrix multiplication of two such matrices produces a third. This vector space of dimensionn² forms analgebra over a field.

The set ofpolynomials with coefficients inF is a vector space overF, denotedF[x]. Vector addition and scalar multiplication are defined in the obvious manner. If thedegree of the polynomials is unrestricted then the dimension ofF[x] iscountably infinite. If instead one restricts to polynomials with degree less than or equal ton, then we have a vector space with dimensionn + 1.

One possible basis forF[x] is amonomial basis: the coordinates of a polynomial with respect to this basis are itscoefficients, and the map sending a polynomial to the sequence of its coefficients is alinear isomorphism fromF[x] to the infinite coordinate spaceF^∞.

The vector space of polynomials with real coefficients and degree less than or equal ton is often denoted byP_n.

The set ofpolynomials in several variables with coefficients inF is vector space overF denotedF[x₁,x₂, ...,x_r]. Herer is the number of variables.

See main article atFunction space, especially the functional analysis section.

LetX be a non-empty arbitrary set andV an arbitrary vector space overF. The space of allfunctions fromX toV is a vector space overF underpointwise addition and multiplication. That is, letf :X →V andg :X →V denote two functions, and letα inF. We define

${\displaystyle (f+g)(x)=f(x)+g(x)}$

${\displaystyle (\alpha f)(x)=\alpha f(x)}$

where the operations on the right hand side are those inV. The zero vector is given by the constant function sending everything to the zero vector inV. The space of all functions fromX toV is commonly denotedV^X.

IfX is finite andV is finite-dimensional thenV^X has dimension |X|(dimV), otherwise the space is infinite-dimensional (uncountably so ifX is infinite).

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

LetX be an arbitrary set. Consider the space of all functions fromX toF which vanish on all but a finite number of points inX. This space is a vector subspace ofF^X, the space of all possible functions fromX toF. To see this, note that the union of two finite sets is finite, so that the sum of two functions in this space will still vanish outside a finite set.

The space described above is commonly denoted (F^X)₀ and is calledgeneralized coordinate space for the following reason. IfX is the set of numbers between 1 andn then this space is easily seen to be equivalent to the coordinate spaceFⁿ. Likewise, ifX is the set ofnatural numbers,N, then this space is justF^∞.

A canonical basis for (F^X)₀ is the set of functions {δ_x |x ∈X} defined by

${\displaystyle {\delta }_x(y)=\{\begin{array}{l}1x=y\\ 0x\ne y\end{array}}$

The dimension of (F^X)₀ is therefore equal to thecardinality ofX. 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 canonical one for (F^X)₀.

Generalized coordinate space may also be understood as thedirect sum of |X| copies ofF (i.e. one for each point inX):

${\displaystyle ({\mathbf{F}}^X{)}_0=\underset{x\in X}{⨁}\mathbf{F}.}$

The finiteness condition is built into the definition of the direct sum. Contrast this with thedirect product of |X| copies ofF which would give the full function spaceF^X.

An important example arising in the context oflinear algebra itself is the vector space oflinear maps. LetL(V,W) denote the set of all linear maps fromV toW (both of which are vector spaces overF). ThenL(V,W) is a subspace ofW^V since it is closed under addition and scalar multiplication.

Note that L(Fⁿ,F^m) can be identified with the space of matricesF^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 withF^m×n. This identification normally depends on the choice of basis.

IfX is sometopological space, such as theunit interval [0,1], we can consider the space of allcontinuous functions fromX toR. This is a vector subspace ofR^X since the sum of any two continuous functions is continuous and scalar multiplication is continuous.

The subset of the space of all functions fromR toR consisting of (sufficiently differentiable) functions that satisfy a certaindifferential equation is a subspace ofR^R if the equation is linear. This is becausedifferentiation is a linear operation, i.e., (af +bg)′ =af ′ +bg′, where ′ is the differentiation operator.

SupposeK is asubfield ofF (cf.field extension). ThenF can be regarded as a vector space overK by restricting scalar multiplication to elements inK (vector addition is defined as normal). The dimension of this vector space, if it exists,^[a] is called thedegree of the extension. For example, thecomplex numbersC form a two-dimensional vector space over the real numbersR. Likewise, thereal numbersR form a vector space over therational numbersQ which has (uncountably) infinite dimension, if a Hamel basis exists.^[b]

IfV is a vector space overF it may also be regarded as vector space overK. 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.

Apart from the trivial case of a zero-dimensional space over any field, a vector space over a fieldF has a finite number of elements if and only ifF is afinite field and the vector space has a finite dimension. Thus we haveF_q, the unique finite field (up toisomorphism) withq elements. Hereq must be a power of aprime (q =p^m withp prime). Then anyn-dimensional vector spaceV overF_q will haveqⁿ elements. Note that the number of elements inV is also the power of a prime (because a power of a prime power is again a prime power). The primary example of such a space is the coordinate space (F_q)ⁿ.

These vector spaces are of critical importance in therepresentation theory offinite groups,number theory, andcryptography.

• Lang, Serge (1987).Linear Algebra. Berlin, New York:Springer-Verlag.ISBN 978-0-387-96412-6.

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