# Transvection

A linear mapping of a (right) vector space over a skew-field with the properties

where is the identity linear transformation. A transvection can be represented in the form

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

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.

**How to Cite This Entry:**

Transvection.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Transvection&oldid=16329