Euclidean Space

A Euclidean space , in the current design, is a vector Space or refines real dimension finished provided with a scalar Produit. In such a space, one can treat questions length or orthogonality. In Physical traditional, the space used by the physicists is usually modelled by a space refines Euclidean dimension 3.

First approach

Euclidean space draws its name from the Greek mathematician Euclide. Historically, Euclidean space is only the physical space of dimension 2 or 3 (plan or space) in which were defined the points, the lines, the distances and the angles. With these objects properties were affected as “by two distinct points only one line passes”, or “the sum of the angles of a triangle is worth two rights”.

The transformations characteristic of these Euclidean spaces are the Isométrie S: they transform geometrical figures into other geometrical figures of the same dimension. They are in the beginning for example Congruence of triangles. The fundamental tools for work in Euclidean space are the rule and the compass. These Euclidean spaces natural are the universes where all the great theorems of the plane geometry or the Solid geometry are shown. They are the objects of study of all the geometricians before Euclide to the mathematicians of the 19th century.

On this date however, this vision of Euclidean space natural starts to show its limits. It is then necessary to give a more formal and more general definition of it. The definition of the Produces scalar of two Vecteur S by

\ langle \ vec U|\ vec v \ rangle = \|\ vec U \| \ cdot \|\ vec v \| \ cdot \ cos (\ widehat {\ vec U, \ vec v})

will allow this change.

Mathematical definitions

Euclidean vector space

An Euclidean vector space is a vector space on

\ R

, of size finished and provided with a scalar Produit :

\ langle \ vec U|\ vec v \ rangle

. One can then define in it a standard , called " euclidienne" normalizes; :

\|\ vec U \|= \ sqrt {\ langle \ vec U|\ vec U \ rangle}

and a concept of Angle : the geometrical angle (U, v) of two nonnull vectors is reality

\ theta

ranging between 0 and

\ pi

such as

\ cos (\ theta) = \ frac {\ langle \ vec U|\ vec v \ rangle} {\|\ vec U \|\ cdot \|\ vec v \|}

(this quotient lies between -1 and +1 according to the Inégalité of Cauchy-Schwarz).

The transformations characteristic of these Euclidean spaces are the transformations preserving the scalar product and the standard. They are the orthogonal automorphisms (see orthogonal Groupe).

Space refines Euclidean

A space refines Euclidean is a Espace closely connected associated with an Euclidean vector space. One can define a distance in it, concepts of geometrical angle and one finds in particular the property of Pythagore and his reciprocal like that of the sum of the angles of a triangle.

The fundamental transformations of spaces closely connected Euclidean are the isométries, transformations preserving the distances, one shows that they are applications closely connected whose associated linear application is an orthogonal automorphism.

Examples of spaces vector Euclidean

space $\ R^n$ , provided with the scalar product canonical

\ langle (x_1, x_2, \ cdots, x_n) | (y_1, y_2, \ cdots, y_n) \ rangle = x_1y_1 + x_2y_2 + \ cdots + x_ny_n = \ sum_ {i=1} ^n x_iy_i

is an Euclidean vector space (known as canonical ) of dimension N

the vector space of the Polynôme S realities of degree lower or equal to N

provided with the canonical scalar product
: $\ left \ langle \ sum_ {i=0} ^ {N} a_iX^i \ Bigg | \ sum_ {i=0} ^ {N} b_iX^i \ right \ rangle = \ sum_ {i=0} ^ {N} a_ib_i$
: of dimension N + 1
is an Euclidean space provided with the scalar product

\ langle P | Q \ rangle = \ int_0^1P (T) Q (T) \ {\ rm D} t

is also an Euclidean space from which the standard is different from the preceding one.

Properties of Euclidean spaces

In addition to the properties inherent in the Standard and the scalar Produces (see also bilinear Forme), Euclidean space has additional properties due to its character of space of finished size.

Base

Any Euclidean space has an orthonormal base (each vector of the base is of standard 1, the scalar product of two different vectors is always null). More precisely: if $(u_1, u_2, \ cdots, u_n)$ is a base of E, there exists a base $(v_1, v_2, \ cdots, v_n)$ orthonormal and such as, for entire K of 1 to N, $\ operatorname {Vec} (\ {u_1, u_2, \ cdots, u_k \}) = \ operatorname {Vec} (\ {v_1, v_2, \ cdots, v_k \})$

Orthogonality

Associated vector and linear form

Assistant of a endomorphism

If U is a endomorphism of E , there exists a single endomorphism noted

u^*

and called assistant of U , such as

\ forall X \ in E, there \ in E, \ qquad \ langle U (X)|there \ rangle= \ langle X|u^* (there) \ rangle

One defines the concepts of endomorphism autoadjoint or symmetrical ( $u=u^*$ ), anti-autoadjoint or antisymmetric ( $u=-u^*$ ).

In an orthonormal or orthonormée base, the matrix representative of $u^*$ is the transposed of that of U .

Related concepts

a vector space on $\ R$ or $\ mathbb C$ provided with a scalar product and of infinite size is a Espace préhilbertien. The absence of the finitude of the base makes him lose the list of above mentioned properties. If space is complete it is a Espace of Hilbert.
a vector space on $\ mathbb C$ provided with a scalar product and finished size is called square Espace

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