Base (linear algebra)

Definition

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 $(\ vec {\ imath}, \ vec {\ jmath})$ or ( e₁ , e₂ ). In space, they are three vectors not Coplanaire S; these vectors are often noted $(\ vec {\ imath}, \ vec {\ jmath}, \ vec {K})$ or ( e₁ , e₂ , e₃ ).

We in space place. An unspecified vector $\ vec {U}$ can thus be written

\ vec {U} = x_u \ cdot \ vec {\ imath} + y_u \ cdot \ vec {\ jmath} + z_u \ cdot \ vec {K}

where x_u , y_u and z_u is realities, named “coordinated” or “components” of the vector. Let us note that this decomposition is single. One can also write

\ vec {U}

in the form of a matrix - column:

\ vec {U} = \ begin {pmatrix}

x_u \\ y_u \\ z_u \end{pmatrix} One also frequently uses another notation of the components, which often goes hand in hand with the other notation for the base:

\ mathbf {U} = u_1 \ cdot \ mathbf {e_1} + u_2 \ cdot \ mathbf {e_2} + u_3 \ cdot \ mathbf {e_3}

\ mathbf {U} = \ begin {pmatrix}

u_1 \\ u_2 \\ u_3 \end{pmatrix}

One a:

\ vec {\ imath} = \ mathbf {e_1} = \ begin {pmatrix}

1 \ \ 0 \ \ 0 \ end {pmatrix}

\ vec {\ jmath} = \ mathbf {e_2} = \ begin {pmatrix}

0 \ \ 1 \ \ 0 \ end {pmatrix}

\ vec {K} = \ mathbf {e_3} = \ begin {pmatrix}

0 \ \ 0 \ \ 1 \end{pmatrix}

Basic types

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 $(\ widehat {\ vec {\ imath}, \ vec {\ jmath}})$ - or $(\ widehat {\ mathbf {e_1}, \ mathbf {e_2}})$ - 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 ( x_u , y_u , z_u ) are the components of $\ vec {U}$ and ( x_v , y_v , z_v ) are those of $\ vec {v}$ , then one a:

a \ cdot \ vec {U} = \ begin {pmatrix}

\ cdot x_u \ \ has \ cdot y_u \ \ has \ cdot z_u \end{pmatrix} and

\ vec {U} + \ vec {v} = \ begin {pmatrix}

x_u + x_v \ \ y_u + y_v \ \ z_u + z_v \end{pmatrix} or with the other notation

a \ cdot \ mathbf {U} = \ begin {pmatrix}

\ cdot u_1 \ \ has \ cdot u_2 \ \ has \ cdot u_3 \ end {pmatrix} \ {\ rm and} \ \ mathbf {U} + \ mathbf {v} = \ begin {pmatrix} u_1 + v_1 \ \ u_2 + v_2 \ \ u_3 + v_3 \end{pmatrix}

A orthonormée base is particularly interesting to calculate the scalar Produit:

\ vec {U} \ cdot \ vec {v} = x_u \ cdot x_v + y_u \ cdot y_v + z_u \ cdot z_v

||\ vec {U}||^2 = x_u^2 + y_u^2 + z_u^2

(this can also result from the Théorème of Pythagore). With the other notation, that gives

\ mathbf {U} \ cdot \ mathbf {v} = \ sum_ {i=1} ^3 u_i \ cdot v_i

\ mathbf {U} ^2 = \ sum_ {i=1} ^3 u_i^2

So moreover the orthonormée base is direct, that simplifies the calculation of the vector Product. Indeed, there is then

\ vec {K} = \ vec {\ imath} \ wedge \ vec {\ jmath}

;

\ vec {\ imath} = \ vec {\ jmath} \ wedge \ vec {K}

;

\ vec {\ jmath} = \ vec {K} \ wedge \ vec {\ imath}

or with the other notation

\ mathbf {e_3} = \ mathbf {e_1} \ wedge \ mathbf {e_2}

;

\ mathbf {e_1} = \ mathbf {e_2} \ wedge \ mathbf {e_3}

;

\ mathbf {e_2} = \ mathbf {e_3} \ wedge \ mathbf {e_1}

It is then enough to put side by side both matrix-column representing the vectors, to add a fourth line including/understanding the first component ( x_u or U ₁), and to withdraw the products in cross of the lines two by two; the result is placed in the line not intervening in calculation. For example the result of the product in cross between the line X and (lines 1 and 2) is placed there in the line Z (line 3) of the resulting matrix-column.

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 ( x_A , y_A , z_A ) and those of the point B are ( x_B , y_B , z_B ), then the vector $\ overrightarrow {AB}$ has as components:

\ overrightarrow {AB} = \ begin {pmatrix}

x_B - x_A \ \ y_B - y_A \ \ z_B - z_A \end{pmatrix} in particular, there is

\ overrightarrow {OA} = \ begin {pmatrix}

x_A \\ y_A \\ z_A \end{pmatrix} By using the other notation, the coordinates of has are ( has ₁, has ₂, has ₃) and those of the point B are ( B ₁, B ₂, B ₃), and the vector U = AB has as components:

u_i = b_i - a_i

Transformations

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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