Transvection

This article is with being read in parallel with that on the dilations.

Vectorial Transvection

A transvection of a vector space $E \,$ is either the identity, or a endomorphism $f \,$ of $E \,$ such as the whole of the vectors invariants $H= \ mathrm {Ker} (F \ mathrm {id}) \,$ is a hyperplane of $E \,$ ( bases transvection) and $D= \ mathrm {Im} (F \ mathrm {id}) \,$ ( direction of the transvection) is included in $H \,$ (i.e. for all $x \,$ of $E \,$ , $f (X)- X \,$ belongs to $H \,$ ).

Equivalent condition 1: $f \,$ is linear and $(F \ mathrm {id}) ^2=0 \,$ .

Equivalent condition 2: there exists a linear form $h \,$ on $E \,$ and a vector invariant $u \,$ such as for all $x \,$ of $E \,$ :

f (X) =x+h (X) U \,

The transvections are bijective ( $f^ {- 1} (X) =x-h (X) U \,$ ) and, in finished dimension, are of determinant 1; they generate the linear special Groupe of $E \,$ : $SL (E) \,$ . The whole of the basic transvections $H \,$ forms of it a sub-group, isomorph with the additive group $H \,$ (with $u \,$ of $H \,$ , to make correspond the transvection $x \ mapsto x+h (X) U \,$ ).

Stamp transvection

In a base of $E \,$ containing a base of $H \,$ of which one of the vectors is a directing vector of $D \,$ , the transvection has as a matrix a matrix of the type $\ begin {bmatrix}
1 & & & & \ \
& 1 & & \ lambda & \ \
& &. & & \ \
& & & 1 & \ \
& & & & 1
\end{bmatrix}=I_n+\lambda E_{ij}$ avec $i \ j$ . These matrices are called matrices of transvection ; they generate the linear special Groupe $SL_n (K) \,$ .

The most reduced shape, which is its form of Jordan, of the matrix of a transvection different from the identity is $\ begin {bmatrix}
1 & & & & \ \
& 1 & & & \ \
& &. & & \ \
& & & 1 & 1 \ \
& & & & 1
\end{bmatrix}$

Transvection closely connected

A transvection of a space refines $E \,$ is either the identity, or a Application refines $E \,$ in $E \,$ whose whole of the points invariants is a hyperplane of $H \,$ of $E \,$ ( bases transvection) and such as for any point $M \,$ the vector $\ overrightarrow {ME}$ remains parallel to $H \,$ . The vectors $\ overrightarrow {ME}$ form a vectorial line then $\ overrightarrow {D}$ (direction of the transvection).

A transvection closely connected has to some extent linear a vectorial transvection, but the applications closely connected having to some extent linear a vectorial transvection are the slipped transvections , composed of a transvection and a translation of vector parallel with the base.

Being given two points $A \,$ and $A' \,$ such as the line $(AA') \,$ parallel with a hyperplane $H \,$ , but is not included in this hyperplane, it exists a single basic transvection $H \,$ sending $A \,$ on $A' \,$ ; one obtains easily the image $M' \,$ of a point $M \,$ by construction:

Projective Transvection

If one plunges space refines $E \,$ in its supplemented projective, by associating a hyperplane to him ad infinitum $H' \,$ , one knows that one can provide complementary the $E' \,$ of the hyperplane $H \,$ of a structure of space refines (the lines which is secant in a point of $H \,$ in $E \,$ become parallel in $E' \,$ and those which are parallel in $E \,$ become secant in a point of $H' \,$ ).

Any transvection of hyperplane $H \,$ of $E \,$ is then associated an application closely connected with $E' \,$ which is not other than a translation!

So now another hyperplane is sent that $H \,$ and $H' \,$ ad infinitum, the transvection becomes a special homology.

In short, there is, in projective geometry, identity between the translations, the transvections, and the homologies special.

Euclidean Transvection

Realization of a transvection per parallel prospect

Dives Euclidean space $E_n \,$ of dimension N like hyperplane of a space $E_ {n+1} \,$ of dimension N +1 and let us make turn $E_n \,$ around its hyperplane $H \,$ , in order to obtain from it a copy $\ tilde E_n \,$ .

Any point $M \,$ of $E_n \,$ has a copy $\ M$ tilde in $\ tilde E_n \,$ , therefore also the image $M' \,$ of $M \,$ by a basic transvection $H \,$ .

One shows that the line $(M \ tilde Me)$ guard a direction fixes $D \,$ , which shows that $\ M'$ tilde is obtained by projection of $M \,$ in $E_ {n+1} \,$ (projection basic $\ tilde E_n \,$ and of direction $D \,$ ).

See here a realization concretes of this process.

Internal bonds

Transformation
Homology

Sources

Source for the projective part: Alain Bigard, Geometry, Courses and exercises corrected for the Capes and aggregation, Masson, 1998

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