Collineations of an affine geometry

1. Mar 3, 2010

samkolb

An affine geometry is a nonempty set of points A, together with a set of lines, where each line connects one or more of the points in A.

A collineation of A is a bijection f: A --> A that carries lines to lines. That is, if P,Q are points in A lying on the same line, then f(P), f(Q) are points in A lying on the same line.

My question:

Does this definition imply that each pair of points not lying on the same line is carried to a pair of points not on the same line? That is, if P, Q do not lie on the same line, then do f(P) and f(Q) also not lie on the same line?