Also, the case of the projective line is special, and hence generally treated differently.
A collineation from PG(V) to PG(W) is a map α : D(V) → D(W), such that: Given a projective space defined axiomatically in terms of an incidence structure (a set of points P, lines L, and an incidence relation I specifying which points lie on which lines, satisfying certain axioms), a collineation between projective spaces thus defined then being a bijective function f between the sets of points and a bijective function g between the set of lines, preserving the incidence relation.
[3] Every projective space of dimension greater than or equal to three is isomorphic to the projectivization of a linear space over a division ring, so in these dimensions this definition is no more general than the linear-algebraic one above, but in dimension two there are other projective planes, namely the non-Desarguesian planes, and this definition allows one to define collineations in such projective planes.
For dimension one, the set of points lying on a single projective line defines a projective space, and the resulting notion of collineation is just any bijection of the set.
This is different from the behavior in higher dimensions, and thus one gives a more restrictive definition, specified so that the fundamental theorem of projective geometry holds.
Projective linear transformations (homographies) are collineations (planes in a vector space correspond to lines in the associated projective space, and linear transformations map planes to planes, so projective linear transformations map lines to lines), but in general not all collineations are projective linear transformations.
The fundamental theorem of projective geometry states the converse: Suppose V is a vector space over a field K with dimension at least three, W is a vector space over a field L, and α is a collineation from PG(V) to PG(W).
Correspondingly, the quotient group PΓL / PGL ≅ Gal(K/k) corresponds to "choices of linear structure", with the identity (base point) being the existing linear structure.
Given a projective space without an identification as the projectivization of a linear space, there is no natural isomorphism between the collineation group and PΓL, and the choice of a linear structure (realization as projectivization of a linear space) corresponds to a choice of subgroup PGL < PΓL, these choices forming a torsor over Gal(K/k).
According to Wilhelm Blaschke[4] it was August Möbius that first abstracted this essence of geometrical transformation: Contemporary mathematicians view geometry as an incidence structure with an automorphism group consisting of mappings of the underlying space that preserve incidence.
Such a mapping permutes the lines of the incidence structure, and the notion of collineation persists.
As mentioned by Blaschke and Klein, Michel Chasles preferred the term homography to collineation.