Projective transformation

Definition

Let $V$ be a finite-dimensional vector space over a field $k$ . An invertible linear map $A:V\to V$ induces a projective transformation

\mathbb P(A):\mathbb P(V)\longrightarrow\mathbb P(V), \qquad [v]\longmapsto[Av].

More generally, a linear isomorphism $V\to W$ induces a projective isomorphism $\mathbb P(V)\to\mathbb P(W)$ .

Scalar ambiguity

Two invertible linear maps $A,B$ induce the same transformation exactly when $B=\lambda A$ for some $\lambda\in k^\times$ , provided $V\ne0$ . Thus the group of projective transformations is

\operatorname{GL}(V)/(k^\times I) =\operatorname{PGL}(V),

the projective general linear group.

Incidence preservation

Projective transformations send projective subspaces to projective subspaces and preserve incidence. A bijection with this line-preserving property is called a collineation. Projective transformations are collineations, but over a field with nontrivial automorphisms there can also be collineations induced by non-linear semilinear maps. The fundamental theorem of projective geometry accounts for all of them in projective dimension at least two.

References

Emil Artin, Geometric Algebra, Interscience, 1957. Relevant: Chapter II, §§3–4.
Joe Harris, Algebraic Geometry: A First Course, Springer, 1992. Publisher record. Relevant: Lecture 1, projective linear transformations.