The projective space to a - vector space is the set of all straight lines through the zero point in , that is, the quotient space with respect to the equivalence relation .
Now let and be vector spaces and and the associated projective spaces, then a mapping is called
The following (non-equivalent) definition can also be found with individual authors:
Let and be projective spaces and a projective subspace of , then a mapping is called
The invertible projective images of a projective space on themselves form a group which is referred to as a projective linear group . The elements of this group are especially true to the line, i.e. collineations .
The projective linear group over a vector space over a field is the factor group , with the normal (even central ) subgroup of the scalar multiples of identity being with off . The names etc. correspond to those of the general linear group. If a finite body are and equally powerful but generally not isomorphic.
Projective images preserve the incidence structure.
The name comes from projective geometry , where the analogue to the general linear group is the projective linear group , while the group belongs to the -dimensional projective space , it is the group of all projectivities of space.
Fractional linear transformations
In the case of the projective straight line , the projective images are precisely the fractional-linear transformations.
After identifying with (through ) works on through
.
Projective images get the double ratio of 4-tuples of collinear points. This property can be seen as a distinctive feature of projective geometry . See also: Erlanger program . These connections were already known in antiquity and can be found e.g. B. at Pappos . They are the main reason why the term double ratio was even developed.