In Linear algebra, a vector space is a algebraic Structure making it possible in practice to carry out linear combinations.

Being given a body (commutative) K , a vector space E on K is a commutative group (whose law is noted +) provided with a compatible action of K (see the exact definition). The elements of E are called Vecteur S, and the elements of K of the scalar .

For an introduction to the concept of vector, to see the article Vector .

Objects and morphisms

Vector space

Formal definition

That is to say K a body. One calls vectorial K-space any triplet ( E , +,•) where:

* E is a unit;

* ( E , +) is a abelian Groupe;

* • is an external law on E with scalar in K such as:

** the element unit “1” of the body K is on the left neutral for the law “ • ”:
$\backslash \; forall\; \backslash ,\; U\; \backslash \; in\; E,\; \backslash \; quad\; 1\; \backslash \; cdot\; U\; =\; U\; \backslash ,$

** the law “ • ” distributive compared to the addition + of E is on the left:
$\backslash \; forall\; \backslash ,\; \backslash \; lambda\; \backslash \; in\; \backslash \; mathbb\; \{K\},\; \backslash \; \backslash \; forall\; \backslash ,\; U\; \backslash \; in\; E,\; \backslash \; \backslash \; forall\; \backslash ,\; v\; \backslash \; in\; E,\; \backslash \; quad\; \backslash \; lambda\; \backslash \; cdot\; (U\; +\; v)\; =\; (\backslash \; lambda\; \backslash \; cdot\; U)\; +\; (\backslash \; lambda\; \backslash \; cdot\; v)\; \backslash ,$

** the law “ • ” exodistributive compared to the addition of the body K is on the right:
$\backslash \; forall\; \backslash ,\; \backslash \; lambda\; \backslash \; in\; \backslash \; mathbb\; \{K\},\; \backslash \; \backslash \; forall\; \backslash ,\; \backslash \; driven\; \backslash \; in\; \backslash \; mathbb\; \{K\},\; \backslash \; \backslash \; forall\; \backslash ,\; U\; \backslash \; in\; E,\; \backslash \; quad\; (\backslash \; lambda\; +\; \backslash \; driven)\; \backslash \; cdot\; U\; =\; (\backslash \; lambda\; \backslash \; cdot\; U)\; +\; (\backslash \; driven\; \backslash \; cdot\; U)\; \backslash ,$

** the law “ • ” exo-associative compared to the multiplication of the body K is (it “imports it” in the vector space):
$\backslash \; forall\; \backslash ,\; \backslash \; lambda\; \backslash \; in\; \backslash \; mathbb\; \{K\},\; \backslash \; \backslash \; forall\; \backslash ,\; \backslash \; driven\; \backslash \; in\; \backslash \; mathbb\; \{K\},\; \backslash \; \backslash \; forall\; \backslash ,\; U\; \backslash \; in\; E,\; \backslash \; quad\; (\backslash \; lambda.\; \backslash \; driven)\; \backslash \; cdot\; U\; =\; \backslash \; lambda\; \backslash \; cdot\; (\backslash \; driven\; \backslash \; cdot\; U)\; \backslash ,$.

If $\backslash \; mathbb\; \{K\}\; =\; \backslash \; mathbb\; \{Q\},\; \backslash ;\; \backslash \; mathbb\; \{R\}$ or $\backslash \; mathbb\; \{C\}$, one respectively speaks about vector space rational, real or complex. This terminology is used in particular in analysis.

In certain works, the vectors can be noted surmounted by an arrow or writings with letters in fat.

First properties

Axioms above rise the following properties:

• the element zero “0” of the body K is exoabsorbant for the law on the left • :
  $\backslash \; forall\; \backslash ,\; U\; \backslash \; in\; E,\; \backslash \; quad\; 0\; \backslash \; cdot\; U\; =\; 0\_\; \{E\}\; \backslash ,$
• the neutral element of the vectorial addition is absorbing for the law on the right • :
  $\backslash \; forall\; \backslash ,\; \backslash \; lambda\; \backslash \; in\; \backslash \; mathbb\; \{K\},\; \backslash \; quad\; \backslash \; lambda\; \backslash \; cdot\; 0\_\; \{E\}\; =\; 0\_\; \{E\}\; \backslash ,$.

The two operations on a vector space make it possible to define the linear Combinaison , i.e. the finished sum of affected vectors of coefficients (scalar). Linear combination of a family of vectors $(x\_i)\; \_\; \{\backslash ,\; I\; \backslash ,\; \backslash \; in\; \backslash ,\; I\}$ having for coefficients $(\backslash \; lambda\_i)\; \_\; \{\backslash ,\; I\; \backslash ,\; \backslash \; in\; \backslash ,\; I\}$ is the vector of E given by: $\backslash \; sum\_\; \{\backslash ,\; I\; \backslash ,\; \backslash \; in\; \backslash ,\; I\}\; \backslash \; lambda\_i\; \backslash ,\; x\_i$.
When the whole of indexing $\backslash \; I$ is Infini, it is necessary to suppose that the support of the family $(\backslash \; lambda\_i)\; \_\; \{\backslash ,\; I\; \backslash ,\; \backslash \; in\; \backslash ,\; I\}$ is finished.

Examples

See also: Examples of vector spaces

The class of the vector spaces is not a Ensemble. The class of the vector spaces on a body fixed, even identified except for linear isomorphism, is not either a unit. In fact, the following list of examples could not be exhaustive.

• the null Espace is the vector space on a body K comprising a single element, which is necessarily the null vector. Null space is the initial object and the final object of the category of the vector spaces on K .
• the body K is presented in the form of a vector space on itself: the addition and the multiplication respectively provide the internal law and the law externe.
  Plus generally, all Extension of body of K , i.e. any plunging of K in a body L , provides L with a structure of vector space on K .
• the whole of the translations of a Espace refines (for example the usual plan or space) is a vector space.
• the unit K ^N of the $n$- uplets of elements of K form a vector space in which the addition is made term in the long term and the multiplication by a scalar is distributed on each terme.
  This construction extends to the vector space K ^∞ from the continuations to support finished.
• For any unit I , the unspecified product K ^I is a K - vector space provided with the laws of term addition in the long term and external multiplication on each term:
• $+\backslash ;\; :\; \backslash ;\; (x\_i)\; +\; (y\_i)\; =\; (x\_i+y\_i)$
• $\backslash \; cdot\; \backslash ;:\; \backslash ;\; \backslash \; lambda\; \backslash \; cdot\; (x\_i)\; =\; (\backslash \; lambda\; x\_i)$
• : (The null vector is then the family of scalars all the terms are null.)

When I is the Cartesian product $1;\; N\; \backslash \; times\; 1;\; p$, this product is the vector space $\backslash \; mathcal\; \{M\}\; \_\; \{N,\; p\}\; (\backslash \; mathbb\; \{K\})$ of the matrices to N lines and p columns with coefficients in K .

• more generally, the unspecified product of a family ( E _I ) of vector spaces on K is a vector space on K which notes $\backslash \; prod\; E\_i$. (It is about the product of the vector spaces within the meaning of the categories.)

the whole of the families with support finished $(v\_i)\; \backslash \; in\; \backslash \; prod\; E\_i$ forms also a vector space, called the direct sum of spaces E _I and which notes $\backslash \; bigoplus\; E\_i$.

• the unit $\backslash \; mathcal\; C^0\; (X)$ of the continuous functions real or complexes defined on topological space X is a vector space.
• the whole of (germs of) the solutions of a linear differential equation homogeneous is a vector space (real or complex).
• the whole of the numerical continuations satisfying a linear Relation of recurrence is a vector space.

Vectorial subspace

See also: vectorial Subspace

A vectorial subspace of a vector space E is a sub-group additive F of E stable by the action of K , i.e. such as:

• the element zero 0 belongs to F
• the sum of two elements of F is an element of F
• the product of a scalar by a vector of F belongs to F .

In an equivalent way, a vectorial subspace is a nonempty and stable part by linear Combinaison.

For example, the whole of the functions real polynomials is a vectorial subspace of the vector space $\backslash \; mathcal\; \{C\}\; ^0\; (\backslash \; mathbb\; \{R\},\; \backslash \; mathbb\; \{R\})$ of the continuous functions on $\backslash \; mathbb\; \{R\}$.

The intersection of two vectorial subspaces is a vectorial subspace but the union is not one in general. This is why one defines the nap of two vectorial subspaces F and G like the unit

$\backslash \; F+G=\; \backslash \; left\; \backslash \; \{x+y\; \backslash /\backslash \; (X,\; there)\; \backslash \; in\; F\; \backslash \; times\; G\; \backslash \; right\; \backslash \}$.

This sum is a vectorial subspace. It is even the smallest vectorial subspace (within the meaning of inclusion) containing the two initial vectorial subspaces.

Two vectorial subspaces F and G of E are known as in direct Somme when their intersection is null space and that their nap is space E . One notes $E\; then\; =\; F\; \backslash \; oplus\; G$.
In this case, the vectorial subspaces F and G are known as additional one of the other in E . The axiom of the choice makes it possible to ensure the existence of additional any vectorial subspace, but it there forever unicity (except in the case of the null subspace or of total space). For example, in $\backslash \; mathbb\; \{R\}\; ^3$, the additional ones of an unspecified vectorial plan are all the vectorial lines not contained in this plan. There thus exists here an infinity of additional different.

If E is the direct sum of F and G , any vector of E then breaks up in a single way into a sum of two vectors, one pertaining to F and the other with G . More generally, a family of vectorial subspaces $(F\_i)$ is known as all in all direct in E if any vector of E is written in a single way like a linear combination $\backslash \; sum\; \backslash \; lambda\_i\; x\_i$ with for all I , $x\_i\; \backslash \; in\; F\_i$. This definition implies that the vectorial subspaces $F\_i$ are of null intersection two to two and that their amount is equal to E but the reciprocal one is false. It is enough to take as counterexample in the plan $\backslash \; \backslash \; mathbb\; \{R\}\; ^2$ the three lines directed by $\backslash \; (0,1)$, $\backslash \; (1,0)$ and $\backslash \; (1,1)$.

Linear application

See also: Linear application

Are $\backslash \; E$ and $\backslash \; F$ two spaces vector. An application $\backslash \; f$ of $\backslash \; E$ in $\backslash \; F$ is known as linear if it preserves the linear combinations, i.e.:

$\backslash \; forall\; (X,\; there)\; \backslash \; in\; E^2,\; \backslash \; forall\; \backslash \; lambda\; \backslash \; in\; \backslash \; mathbb\; \{K\},\; \backslash \; F\; (\backslash \; lambda\; x+y)\; =\; \backslash \; lambda\; F\; (X)\; +f\; (there)$.

The whole of the linear applications of E in F is noted L (E, F) . When $\backslash \; F=E$, these applications are called endomorphisms E and one notes their unit L (E) .
A isomorphism of vector spaces is a linear application bijective. A automorphism is a bijective endomorphism. The whole of the automorphisms of E is the linear group noted $\backslash \; GL\; (E)$.

One can quote the following examples of linear applications.

• the null application is the neutral for the vectorial addition in L (E, F) .
• the identity application and more generally all the vectorial Homothétie S (nonnull) are automorphisms.
• the symmetries, rotations and vectorial similarities (nonnull) are automorphisms of the plan or usual space.
• the projectors are endomorphisms.
• the derivation is a endomorphism of the vector space of the polynomials.

A linear Form on a K - vector space E is a linear application of E in K .
The whole of the linear forms on E is called the dual Espace of E and it is noted E ^*.

That is to say $\backslash \; F\; \backslash \; in\; L\; (E,\; F)$.

• One calls core of $\backslash \; f$ and one notes $\backslash \; \{\backslash \; rm\; Ker\}\; (F)$ the unit $\backslash \; \backslash \; left\; \backslash \; \{X\; \backslash \; in\; E\; \backslash /F\; (X)\; =0\_F\; \backslash \; right\; \backslash \}$.
• One calls image $\backslash \; f$ and one notes $\backslash \; \{\backslash \; rm\; Im\}\; (F)$ the unit $\backslash \; \backslash \; left\; \backslash \; \{F\; (X),\; X\; \backslash \; in\; E\; \backslash \; right\; \backslash \}$.

There are the following properties:

• the image and the core are vectorial subspaces respectively of space drank and space source.
• a linear application is injective if and only if its core is null space.
• the unit L (E, F) is a vector space.
• the vector space $L\; (\backslash \; mathbb\; \{K\}\; ^p,\; \backslash \; mathbb\; \{K\}\; ^n)$ is identified with the space of the matrices $\backslash \; mathcal\; \{M\}\; \_\; \{N,\; p\}\; (\backslash \; mathbb\; \{K\})$.

Vector space quotient

That is to say F a vectorial subspace of E . The space quotient E/F is a vector space such as projection $E\; \backslash \; rightarrow\; E/F$ is linear of core F .
A vectorial subspace G of E is additional of F if and only if the restriction of projection induces an isomorphism of G on E/F .

Family of vectors and dimension

Free families, generating families, bases

A family $\backslash \; mathcal\; \{L\}$ of elements of E is known as free (on $\backslash \; mathbb\; \{K\}$) when any linear combination of elements of $\backslash \; mathcal\; \{L\}$ with null coefficients not all is nonnull, in other words when the only null linear combination of elements of $\backslash \; mathcal\; \{L\}$ is that of which all the coefficients are null. In this case the vectors of this family are known as linearly independent (from/to each other).
On the contrary, a family of elements of E is known as dependant when it is not free, i.e. if there exists a null linear combination of the elements of this family with null coefficients not all (it is what is called a relation of linear dependence ).

For example, a family of only one vector not no one is always free. A contrario , an unspecified family comprising the null vector is always dependant.

It is shown that a family $\backslash \; (u\_1,\; \backslash ,\; u\_2)$ of two vectors of E is dependant if and only if there exists a scalar $\backslash \; \backslash \; alpha$ such as $\backslash \; u\_2\; =\; \backslash \; alpha\; \backslash ,\; u\_1$ or a scalar $\backslash \; \backslash \; beta$ such as $\backslash \; u\_1\; =\; \backslash \; beta\; \backslash ,\; u\_2$. One says in this case that the two vectors are colinéaires . On the other hand, nothing ensures that a dependant family comprising at least three vectors contains two vectors colinéaires inevitably.

A family of elements of E is known as generating (of E ) when any element of E can be expressed at least manner in the form of a linear combination of the elements of this family.

One calls bases vector space E any family of elements of E free and generating.

The interest of the bases lies in the following properties:

• a family $\backslash \; mathcal\; \{B\}$ of elements of E is a base if and only if any element U of E is expressed in a single way like linear combination of the elements of $\backslash \; mathcal\; \{B\}$.
• the choice of a base for a K - vector space makes it possible to identify this one with a power of K and, if this base is finished, to use the matrices to represent linear applications.

For a given vector space, the existence of at least a base is ensured in any general information by the Axiome of the choice. However, one can exempt this assumption when there exists a finished generating family (or subscripted by a Bon order).

Definition of dimension

If a K - vector space E admits a base having a number finished D of elements, then any base of E has this same cardinal D .

The entirety D is called the dimension of E , noted $\{\backslash \; rm\; dim\}\; \_\; \{\backslash \; mathbb\; \{K\}\}\; E$ or, if there is no ambiguity, $\{\backslash \; rm\; dim\}\; \backslash \; E$. It is said whereas E is a vector space of finished size (on K ), equal to D .

In particular null space, having an empty base, is of finished size equalizes to 0.

One calls right vectorial any space vector of finished size equal to 1 and plane vectorial any space vector of finished size equalizes to 2. A hyperplane is a vectorial subspace of dimension n-1 in a vector space of dimension N .

If a linear application F has an image of finished size, this dimension is called row of F and notes $\backslash \; \{\backslash \; rm\; rg\}\; (F)$.

A vector space E is of finished size if and only if he admits a generating family having a finished number of elements.

The vector spaces which are not of finished size is known as of infinite size It is the case in particular vector spaces of polynomials, spaces of continuations and more generally spaces of function.

So that a vector space E is of infinite size, it is necessary and it is enough that there exists an infinite free family of elements of E .

Properties of the vector spaces of finished size

That is to say E a vector space of finished size (nonnull) equal to N .

• Any family generating of E has at least N elements. If a generating family of E has N elements, it is a base of E (it is said that the bases are the “minimal generating families”).
• Any family free of E has with more N elements. If a free family of E has N elements, it is a base of E (it is said that the bases are the “maximum free families”).
• Theorem of base incomplete stipulates that if $\backslash \; (u\_1,\; \backslash \; dowries,\; u\_p)$ is a free family of vectors of E such as (in other words, a free family which is not a base since it is not maximum), then it exists N - p vectors of E , that one can note $\backslash \; (u\_\; \{p\; +\; 1\},\; \backslash \; dowries,\; u\_n)$, such as the family $\backslash \; (u\_1,\; \backslash \; dowries,\; u\_n)$ is a base of E .

• Any subspace vectorial F of E is of finished size, and dim F ≤ dim E .

• If F is a vectorial subspace of E such as dim F = dim E , then F = E .
• If F ₁ and F ₂ is two vectorial subspaces of E , then

$\backslash \; dim\; (F\_1\; +\; F\_2)\; +\; \backslash \; dim\; (F\_1\; \backslash \; course\; F\_2)\; =\; \backslash \; dim\; F\_1\; +\; \backslash \; dim\; F\_2$.

This relation is known under the name of formula of Grassmann.

• All the additional ones of a vectorial subspace F of E have the same dimension, which is called Codimension of F in E .

• the dual space of E is also of finished size and $\backslash \; \{\backslash \; rm\; dim\}\; (E^\; \{*\})\; =\; \{\backslash \; rm\; dim\}\; (E)$.

• Any hyperplane of E is the core of a nonnull linear form on E .
• Is $\backslash \; \backslash \; mathcal\; \{B\}\; =\; \backslash \; left\; (e\_1,\dots ,\; e\_n\; \backslash \; right)$ a base of E .

There exists a single base $\backslash \; \backslash \; mathcal\; \{B\}\; ^\; \{*\}\; =\; \backslash \; left\; (e^*\_1,\dots ,\; e^*\_n\; \backslash \; right)$ of $\backslash \; E^*$ such as $\backslash \; forall\; (I,\; J)\; \backslash \; in\; \{\backslash \; left\; \backslash \; \{1,\dots ,\; N\; \backslash \; right\; \backslash \}\}\; ^2,\; \backslash \; e^*\_i\; (e\_j)\; =\; \backslash \; delta\_\; \{ij\}$,

where $\backslash \; \{\backslash \; delta\}\; \_\; \{ij\}$ is the Symbole of Kronecker.

One says whereas $\backslash \; \backslash \; mathcal\; \{B\}\; ^*$ is the bases dual associated with $\backslash \; \backslash \; mathcal\; \{B\}$.

• the whole of the alternate N-linear forms on a vector space of dimension N is a vector space of dimension 1. This result is at the base of the theory of the determinant.
• If F is a linear application F of E in a vector space F (not necessarily of finished size), then F induces a Isomorphisme of very additional of $\backslash \; \{\backslash \; rm\; Ker\}\; (F)$ in $\backslash \; \{\backslash \; rm\; Im\}\; (F)$.

This property makes it possible to show the Théorème of the row :

$\backslash \; \{\backslash \; rm\; dim\}\; (E)\; =\; \{\backslash \; rm\; dim\}\; (\{\backslash \; rm\; Im\}\; (F))+\; \{\backslash \; rm\; dim\}\; (\{\backslash \; rm\; Ker\}\; (F))$, in other words $\backslash \; \{\backslash \; rm\; rg\}\; (F)\; =\; \{\backslash \; rm\; dim\}\; (E)\; -\; \{\backslash \; rm\; dim\}\; (\{\backslash \; rm\; Ker\}\; (F))$.

• the row is the dimension of the space of the columns, which is equal to that of the space of the lines

$\backslash \; rm\; row\; (A)=\; \backslash \; dim\; (\backslash \; mathcal\; \{C\}\; (A))\; =\; \backslash \; dim\; (\backslash \; mathcal\; \{L\}\; (A))$

Related structures

Relative structures

• an even of vector spaces is the data of a vector space and a vectorial subspace of this one.
• more generally, a vector space can be filtered by the data of a family of vectorial subspaces increasing or decreasing.
• a flag on a vector space of dimension N is the data of N encased vectorial subspaces, of increasing size of 1 into 1.
• a finished vector space of dimension can be directed by the choice of a orientation on its bases.
• a vector space graduated is a family of vector spaces, generally subscripted by $\backslash \; mathbb\; \{NR\}$, $\backslash \; mathbb\; \{Z\}$ or $\backslash \; mathbb\; \{Z\}\; /2$. A morphism between two such graduated vector spaces is then a family of linear applications which respects the graduation.

Algebraic structures

• a module M on a ring has is an additive group provided with an external law on M with coefficients in has , compatible with the addition on M and with the operations on has. But it in general lays out neither basic nor the additional ones.
• a algebra is a vector space provided with a multiplication distributive compared to the addition and compatible with the external law of composition.
• a Algèbre of Dregs is a vector space provided with a Crochet of Dregs compatible with the external law of composition.

Topological and geometrical structures

• a Espace refines is a unit provided with a action free and transitive of a vector space.
• a Euclidean vector space is a real vector space of finished size provided with a scalar Produit.
• a real or complex vector space is known as normalized when it is provided with a standard. For example, the spaces of Banach, whose spaces of Hilbert which generalize the concept of Euclidean vector space, are normalized vector spaces.
• If K is a body provided with a topology, a topological vector Space on K is a K - vector space provided with a compatible topology, i.e. the addition and the multiplication by a scalar must be continuous. It is the case inter alia normalized vector spaces and spaces of Fréchet.
• a vectorial Fibré is a surjection of a topological space on another, such as the préimage of each point is provided with a structure of vector space compatible continuously with the structures of the préimages of the close points.

History

The concept of vector space is born conceptually from the Géométrie closely connected with the introduction of the coordinated into a reference mark of the plan or usual space. About 1636, the French mathematicians Descartes and Fermat gave the bases of the analytical Geometry by associating the solution of an equation with two unknown factors with the graphic determination of a Courbe of the plan.

In order to arrive to a geometrical resolution without using the concept of coordinates, the mathematician Bolzano introduced into 1804 of the operations on the points, right-hand sides and plans, which are the precursors of the vectors. This work finds an echo in the design of the barycentric coordinated by Möbius in 1827. The stage founder of the definition of the vectors was the definition by Bellavitis of the bipoint, which is a directed segment (an end is an origin and the other a goal). Relation of équipollence, which makes equivalent two bipoints when they determine a parallelogram, thus completes to define the vectors.

The notion of vector is taken again with the presentation of the complex numbers by Argand and Hamilton, then that of the Quaternion S by this last, like elements of respective spaces $\backslash \; mathbb\; \{R\}\; ^2$ and $\backslash \; mathbb\; \{R\}\; ^4$. The treatment by linear combination finds in the linear systems of equations, defined by Laguerre since 1867.

In 1857, Cayley introduced the matric notation, which made it possible to harmonize the notations and to simplify the writing of the linear applications between vector spaces. It also outlined the operations on these objects.

About the same time, Grassmann took again the barycentric calculation initiated by Möbius by considering whole of abstract objects provided with operations. Its work exceeded the framework of the vector spaces because, by also defining the multiplication, it led to the concept of algebra. One finds there nevertheless the concepts of dimension and of linear Indépendance, as well as the scalar Produit appeared in 1844. The primacy of these discoveries is disputed with Cauchy with the publication of On the keys algebraic in the Comptes-rendus .

The Italian mathematician Peano, whose important contribution was the rigorous axiomatization of the existing concepts - in particular the construction of the usual units - was one of the first to give a contemporary definition of the concept of vector space towards the end of the XIXe century.

A significant development of this concept is due to the construction of spaces of functions by Lebesgue, construction which was formalized during the XXe century by Hilbert and Banach, at the time of its thesis of doctorate in 1920.

It is at that time that the interaction between the analyzes functional being born and the Algèbre is felt, in particular with the introduction of key concepts such as the spaces of functions '' p '' - integrable or the spaces of Hilbert. It is at that time that the first studies on the vector spaces of infinite size appear.

Sources and references

Sources

• MathPhysics
• Year Introduction to Linear Algebra , Leonid Mirsky - 1990
• Introduction to Linear Algebra and Differential Equations , John Warren Dettman - 1986

References

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