Projective transformation

Summary

A projective transformation, or homography, is a map between projective spaces induced by an invertible linear map of the underlying vector spaces. In homogeneous coordinates it is $[x] \mapsto [Ax]$ for an invertible matrix $A$, defined up to a nonzero factor. The projective transformations of $\mathbb{P}^n(k)$ form the group $\mathrm{PGL}(n + 1, k)$. They are the symmetries of projective geometry, as Euclidean motions are those of Euclidean geometry.