In geometry, a projective transformation may be a correlation or a collineation. The latter classification includes the case of a perspectivity and a projectivity which is a composition of perspectivities. Another type of projective transformation is a homography which acts on a one-dimensional projective space called a projective line.
Given a projective space, its collection of projectivities forms a group under composition of functions. In fact, such groups have historical importance. For instance, the Möbius group has become a tool of science.
Projective transformations are often expressed in terms of linear transformations on a vector space over a field F. Points of the projective space are represented using homogeneous coordinates from F. A non-singular n × n matrix corresponds to a projective transformation on n − 1 dimensional projective space (due to the homogeneous factor). This context supersedes affine transformation in a hyperplane of the projective space that can be written x ↦ Mx + v or in terms of column vectors x, y and v of length n − 1:
where M is square and non-singular. The projective linear group PGL(n, F) is formed using the general linear group GL(n, F) and its center ZGL(n, F) consisting of matrices k I where I is the identity matrix and k is in F. Then PGL(n, F) is the quotient group of GL(n, F) by its center.