Transvection
A linear mapping of a (right) vector space
over a skew-field
with the properties
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where is the identity linear transformation. A transvection can be represented in the form
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where ,
and
.
The transvections of a vector space generate the special linear, or unimodular, group
. It coincides with the commutator subgroup of
, with the exception of the cases when
or
and
is the field of two elements. If
is a field, then
is the group of matrices with determinant 1. In the general case (provided that
),
is the kernel of the epimorphism
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which is called the Dieudonné determinant (cf. Determinant).
References
[1] | J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) |
Comments
In the projective space , whose points are the
-dimensional subspaces of
, a transvection
as above induces a (projective) transvection with
as centre and
as axis. If one takes
to be a hyperplane at infinity in
, such a transvection induces a translation
in the remaining affine space (interpreted as a linear space). See also Shear.
Transvection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transvection&oldid=16329