Examples of vector spaces

This page presents a list of examples of vector spaces . You can consult the article vector Space to find the definitions of the concepts employed there below.

Also see the articles on the dimension, the bases.

We will note $\backslash \; mathbb\; \{K\}$ a commutative body arbitrary such as that of the real $\backslash \; mathbb\; \{R\}$ or that of the complex $\backslash \; mathbb\; \{C\}$.

Commonplace or null vector space

See also: null Space

The simplest example of vector space is null space $\backslash \; \{0\; \backslash \}$, which contains only the null vector (see the axiom 3. vector spaces). The vectorial addition and the multiplication by a scalar are commonplace. A bases this space vector is the Empty set, thus {0} is the vector space of dimension 0 on $\backslash \; mathbb\; \{K\}$. Any vector space on $\backslash \; mathbb\; \{K\}$ contains an isomorphous vectorial subspace with this one.

The body

The next simple example is the body $\backslash \; mathbb\; \{K\}$ itself. The vectorial addition is simply the addition of the body and the multiplication by a scalar is the multiplication of the body. The neutral element of $\backslash \; mathbb\; \{K\}$ for the multiplication forms a base of $\backslash \; mathbb\; \{K\}$ and thus $\backslash \; mathbb\; \{K\}$ is a vector space of dimension 1 on itself.

$\backslash \; mathbb\; \{K\}$ has only two subspaces {0} and $\backslash \; mathbb\; \{K\}$ itself.

Space tuples

The most important example of vector space is undoubtedly that which follows. For all Whole strictly positive naturalness $n$, the whole of the $n$- uplets of elements of $\backslash \; mathbb\; \{K\}$ form a vector space of dimension $n$ on $\backslash \; mathbb\; \{K\}$ called the space of the tuples, noted $\backslash \; mathbb\; \{K\}\; ^n$. An element of $\backslash \; mathbb\; \{K\}\; ^n$ is written:

$x\; =\; (x\_1,\; x\_2,\; \backslash \; ldots,\; x\_n)\; \backslash ,$

where each $x\_i$ is an element of $\backslash \; mathbb\; \{K\}$. The operations on $\backslash \; mathbb\; \{K\}\; ^n$ are defined by:

$x\; +\; there\; =\; (x\_1\; +\; y\_1,\; x\_2\; +\; y\_2,\; \backslash \; ldots,\; x\_n\; +\; y\_n)\; \backslash ,$

$\backslash \; alpha\; X\; =\; (\backslash \; alpha\; x\_1,\; \backslash \; alpha\; x\_2,\; \backslash \; ldots,\; \backslash \; alpha\; x\_n)\; \backslash ,$

The neutral element for the addition is:

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

and opposite of an element: $x\; =\; (x\_1,\; x\_2,\; \backslash \; ldots,\; x\_n)\; \backslash ,$ is the vector

$-x\; =\; (-\; x\_1,\; -\; x\_2,\; \backslash \; ldots,\; -\; x\_n)\; \backslash ,$

The most frequent cases are those where $\backslash \; mathbb\; \{K\}$ is or the body of the real numbers giving the Euclidean Espace $\backslash \; mathbb\; \{R\}\; ^n$, or the body of the complex numbers giving $\backslash \; mathbb\; \{C\}\; ^n$.

The whole of the Quaternion S and the Octonion S is respectively vector spaces of dimension four and eight on the body of the real numbers.

The vector space $\backslash \; mathbb\; \{K\}\; ^n$ is generally provided with a bases natural called canonical Base:

$e\_1\; =\; (1,\; 0,\; \backslash \; ldots,\; 0)\; \backslash ,$

$e\_2\; =\; (0,\; 1,\; \backslash \; ldots,\; 0)\; \backslash ,$

$\backslash \; vdots\; \backslash ,$

$e\_n\; =\; (0,\; 0,\; \backslash \; ldots,\; 1)\; \backslash ,$

where 1 indicates the multiplicative neutral element of $\backslash \; mathbb\; \{K\}$.

Space continuations with finished support

Either $\backslash \; mathbb\; \{K\}\; ^\; \{\backslash \; infty\}$ the whole of the infinite continuations of elements of $\backslash \; mathbb\; \{K\}$ such as only a number finished of elements or not no one. More precisely, if we write an element of $\backslash \; mathbb\; \{K\}\; ^\; \{\backslash \; infty\}$ in the form:

$x\; =\; (x\_1,\; x\_2,\; x\_3,\; \backslash \; ldots)\; \backslash ,$

then only one finished number of the $x\_i$ numbers is nonnull (in other words the coordinates of the vector $x$ become null starting from a certain row).

The addition and the Multiplication by a scalar are then defined as on the vector space of the $n$-uplets.

The unit $\backslash \; mathbb\; \{K\}\; ^\; \{\backslash \; infty\}$ is a vector space of countable size infinite. The canonical Base is that formed by the vectors $\backslash \; textbf\; \{E\}\; \_i$ which comprise one 1 with the $i$ème place and of the zeros everywhere else.

This vector space is the Coproduit of a countable number of copies of the vector space $\backslash \; mathbb\; \{K\}$.

Notice the role of the condition of finitude here. We could consider arbitrary continuations of elements of $\backslash \; mathbb\; \{K\}$, which constitute also a vector space (often noted $\backslash \; mathbb\; \{K\}\; ^\; \{\backslash \; mathbb\; \{NR\}\}$).

However the dimension of this space is infinite noncountable and there is no obvious choice of bases. Since dimensions are distinct, the vector space of the continuations of $\backslash \; mathbb\; \{K\}$ is not isomorphous with $\backslash \; mathbb\; \{K\}\; ^\; \{\backslash \; infty\}$. This vector space is the produced of a countable number of copies of $\backslash \; mathbb\; \{K\}$.

It is worth the sorrow to note that $\backslash \; mathbb\; \{K\}\; ^\; \{\backslash \; mathbb\; \{NR\}\}$ is the dual Espace of $\backslash \; mathbb\; \{K\}\; ^\; \{\backslash \; infty\}$. Thus, in the comparison with the rigid case of finished dimension, we see that a vector space of infinite size does not have no need to be isomorphous with its dual (and even less to be isomorphous with its Bidual).

Matrices

That is to say $\backslash \; mathbb\; \{K\}\; ^\; \{m\; \backslash \; times\; N\}$ the whole of the matrices to coefficients in $\backslash \; mathbb\; \{K\}$. Then $\backslash \; mathbb\; \{K\}\; ^\; \{m\; \backslash \; times\; N\}$ provided with the addition with the matrices and the multiplication by a scalar with the matrices (consisting in multiplying each coefficient by the same scalar) is a vector space on $\backslash \; mathbb\; \{K\}$. The null vector is not other than the null Matrice. The dimension of $\backslash \; mathbb\; \{K\}\; ^\; \{m\; \backslash \; times\; N\}$ is equal to $mn$.

The canonical base is the base formed by the matrices having only one coefficient equal to 1 and all the other coefficients equal to 0.

Vector space of the polynomials

with unspecified

The whole of the Polynôme S with coefficients in $\backslash \; mathbb\; \{K\}$ is a vector space on $\backslash \; mathbb\; \{K\}$ noted $\backslash \; mathbb\; \{K\}$. The vectorial addition and the multiplication by a scalar are defined in an obvious way. This space is of countable size infinite. If one keeps only the polynomials of which the degree remains lower or equal to $n$ then we obtain the vector space $\backslash \; mathbb\; \{K\}\; \_n$ which is of dimension $n+1$.

The canonical base of this space is a Base monomiale.

with several unspecified

The whole of the Polynôme S with several unspecified with coefficients in $\backslash \; mathbb\; \{K\}$ is a vector space on $\backslash \; mathbb\; \{K\}$ noted $\backslash \; mathbb\; \{K\}\; X\_2,\; \backslash \; ldots,\; X\_n$. Here $n$ is a natural entirety not no one representing the number of unspecified.

Also see: the Ring of the polynomials

Functional spaces

See the principal article in the page functional Espace, and more particularly the section entitled analyzes functional.

That is to say $X$ an unspecified unit and $E$ an arbitrary vector space on $\backslash \; mathbb\; \{K\}$. The whole of all the applications of $X$ in $E$ is a vector space on $\backslash \; mathbb\; \{K\}$ with the addition and the multiplication by a scalar of the functions.

These laws is in the following way defined: let us consider $f:\; X\; \backslash \; rightarrow\; E$ and $g:\; X\; \backslash \; rightarrow\; E$ two functions, and $\backslash \; alpha\; \backslash \; in\; \backslash \; mathbb\; \{K\}$ one has

$\backslash \; forall\; X\; \backslash \; in\; X,\; (F\; +\; G)\; (X)\; =\; F\; (X)\; +\; G\; (X)\; \backslash ,$

$\backslash \; forall\; X\; \backslash \; in\; X,\; (\backslash \; alpha\; F)\; (X)\; =\; \backslash \; alpha\; F\; (X)\; \backslash ,$

where the laws $+$ and $.$ appearing in the second member are that of $E$. The null vector is the null constant function sending all the elements of $X$ on the null vector of $E$.

If $X$ is finished and $E$ is a vector space of size finished then the vector space of the functions of $X$ in $E$ is of dimension $|X|\backslash \; times\; \{\backslash \; rm\; dim\}\; E$, if not the vector space is of infinite size (noncountable if $X$ is infinite).

Many vector spaces considered in mathematics are subspaces of functional spaces. Let us give other examples.

Generalization of spaces of continuations to finished support

That is to say $X$ an unspecified unit. Let us consider the vector space $F$ of all the applications of $X$ in $\backslash \; mathbb\; \{K\}$ which cancel everywhere safe of finished number of points of $X$.

Then $F$ is a vectorial subspace of the vector space of all the applications of $X$ towards $\backslash \; mathbb\; \{K\}$. To see it, notice that the meeting of two finished units is finished and thus the sum of two applications of $F$ will be still cancelled in a finished number of points.

If $X$ is the whole of the entireties ranging between 1 and $n$ then this space can easily be comparable with the space of the $n$-uplets $\backslash \; mathbb\; \{K\}\; ^n$. In a similar way, if $X$ is the whole of the natural whole $\backslash \; mathbb\; \{NR\}$, then this space is not other than $\backslash \; mathbb\; \{K\}\; ^\; \{\backslash \; infty\}$.

A natural base of $F$ is the whole of the functions $f\_x$ where $x$ belongs to $X$, such as

$f\_x\; (there)\; =\; \backslash \; begin\; \{boxes\}\; 1\; \backslash \; quad\; X\; =\; there\; \backslash \; \backslash \; 0\; \backslash \; quad\; X\; \backslash \; neq\; there\; \backslash \; end\; \{boxes\}$

The dimension of $F$ is thus equal to the cardinal of $X$. In this way, we can build a vector space of any dimension on any body. Moreover any vector space is isomorphous with a vector space of this form. Any choice of a base determines an isomorphism by sending this base on a given basis of $F$.

Linear applications

An important example resulting from the Linear algebra is the vector space of the linear applications. That is to say $\backslash \; mathcal\; \{L\}\; (E,\; F)$ the whole of the linear applications of $E$ in $F$ ($E$ and $F$ being vector spaces on the same commutative body $\backslash \; mathbb\; \{K\}$). Then $\backslash \; mathcal\; \{L\}\; (E,\; F)$ is a vectorial subspace of the space of the applications of $E$ towards $F$ since it is stable for the addition and the multiplication by a scalar.

Let us notice that $\backslash \; mathcal\; \{L\}\; (\backslash \; mathbb\; \{K\}\; ^n,\; \backslash \; mathbb\; \{K\}\; ^m)$ can be identified with the vector space of the matrices $\backslash \; mathbb\; \{K\}\; ^\; \{m\; \backslash \; times\; N\}$ in a natural way. In fact, by choosing a suitable base of the vector spaces of finished size $E$ and $F$, $\backslash \; mathcal\; \{L\}\; (E,\; F)$ can also be identified with $\backslash \; mathbb\; \{K\}\; ^\; \{m\; \backslash \; times\; N\}$. This identification depends naturally on the choice of the bases.

Continuous applications

If $X$ is a topological Espace, such as the Intervalle unit $$, then we can consider the vector space of the continuous applications of $X$ in $\backslash \; mathbb\; \{R\}$. It is a vectorial subspace of all the real functions definite on $X$ since the sum of two unspecified continuous applications is continuous and produces it by a scalar of a continuous application is continuous.

Differential equations

The subset of the vector space of the applications of $\backslash \; mathbb\; \{R\}$ in $\backslash \; mathbb\; \{R\}$ formed of applications satisfying of the linear differential equations is also a vectorial subspace of this last. That comes owing to the fact that the Dérivation is a linear application, i.e. $(F\; +\; B\; G\; has)\; “=\; has\; F”\; +\; B\; g\text{'}$ where $\text{'}$ indicates this linear application (also called linear operator).

Extensions of body

Let us suppose that $\backslash \; mathbb\; \{K\}$ is a subfield of $\backslash \; mathbb\; \{L\}$ (see Extension of body). Then $\backslash \; mathbb\; \{L\}$ can be seen like a vector space on $\backslash \; mathbb\; \{K\}$ by restricting the multiplication by a scalar with the unit $\backslash \; mathbb\; \{K\}$ (vectorial addition being normally defined). The dimension of this vector space is called degree extension. For example the whole of the complex numbers $\backslash \; mathbb\; \{C\}$ form a vector space of dimension two on the body of realities $\backslash \; mathbb\; \{R\}$. However, the unit $\backslash \; mathbb\; \{R\}$ of the real numbers form a vector space (noncountable) of infinite size on the body of the rational $\backslash \; mathbb\; \{Q\}$.

If $E$ is a vector space on $\backslash \; mathbb\; \{L\}$, then $E$ can be also seen like a vector space on $K$. Dimensions are bound by the formula:

$\{\backslash \; rm\; dim\}\; \_\; \{\backslash \; mathbb\; \{K\}\}\; E=\; \backslash \; left\; (\{\backslash \; rm\; dim\}\; \_\; \{\backslash \; mathbb\; \{L\}\}\; E\; \backslash \; right)\; \backslash \; left\; (\{\backslash \; rm\; dim\}\; \_\; \{\backslash \; mathbb\; \{K\}\}\; \backslash \; mathbb\; \{L\}\; \backslash \; right)$

For example $\backslash \; mathbb\; \{C\}\; ^n$, can be regarded as a vector space on the body of realities, of dimension $2n$.

Finished vector spaces

Separately the null space which is of dimension zero on any body, a vector space on a body $\backslash \; mathbb\; \{K\}$ has a finished number of elements if and only if $\backslash \; mathbb\; \{K\}$ is a Corps finished and the vector space is of finished size.

Thus $\backslash \; mathbf\; \{F\}\; \_q$ is the single finished body of cardinal $q$, where $q$ is an entirety which must be a power of a Prime number ($q=p^m$, $p$ being first). Then any vector space $E$ of dimension $n$ on $\backslash \; mathbf\; \{F\}\; \_q$ will have $q^n$ elements. Notice that the number of elements of $E$ is also a power of a prime number. The first example of such a space, is that of the $n$-uplets $\backslash \; left\; (\backslash \; mathbf\; \{F\}\; \_q\; \backslash \; right)\; ^n$.

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