Base (linear algebra)
See also: Base
In Linear algebra, a bases is a family of Vecteur S, which, in a simplistic way, can be seen like a manner of locating itself in space by defining graduated axes. In a more rigorous way, it is a free family of vectors and generating. See the articles vectorial Geometry and vector Space.
A family of vectors forms a bases if none of these vectors can result from the others by a linear combination (such a family is known as “free”), and if any vector of space can be expressed as a linear combination of the vectors of the base (such a family is known as generator).
In the plan, a base is thus made of two vectors not colinéaires (nonproportional); these vectors are often noted or ( e1 , e2 ). In space, they are three vectors not Coplanaire S; these vectors are often noted or ( e1 , e2 , e3 ).
We in space place. An unspecified vector can thus be written
The base is known as orthonormée if the three vectors are orthogonal between them and if their standard is worth 1. It is known as direct if
- in the plan, the angle - or - is positive
- in space, the trihedron formed by the three vectors of the base is direct.
The direct orthonormées bases are the bases generally used. However, it is interesting in certain cases to have an “unspecified” base. For example, in a Crystal, the organization of the atoms defines a “natural” base (the mesh) which is not necessarily orthonormée. In the same way, when the deformation of the solids is studied, the axes making it possible to express the equations in the simplest way are not always orthogonal.
direct orthonormée Base and bases unspecified, of the plan and space
Vectorial and component operations
The first considerations relate to all the bases including the bases which are not orthonormées direct.
If ( xu , yu , zu ) are the components of and ( xv , yv , zv ) are those of , then one a:
A orthonormée base is particularly interesting to calculate the scalar Produit:
So moreover the orthonormée base is direct, that simplifies the calculation of the vector Product. Indeed, there is then
- ; ;
- ; ;
Algorithm allowing to calculate simply the vector product in the case of a direct orthonormée base
See also: analytical Geometry.
Base and locates
A reference mark (of the plan or space) is the data of a base and a point of reference, in general noted O . We will suppose here that the base used for the vectors is the same one as that used for the reference mark. If the coordinates of the point has are ( xA , yA , zA ) and those of the point B are ( xB , yB , zB ), then the vector has as components:
- ui = bi - ai
The transformations of a base are always the reverse of those of the functions which are described there. For example, the rotation of a reference mark in the trigonometrical direction is equivalent to the rotation of the vectors clockwise.
Simple: Basis (linear will algebra)
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