Collineations in a finite projective geometry
Oswald Veblen · Transactions of the American Mathematical Society · 1907
In the paper by Mr. Bussey and myself on Finite Protective Geometries it was shown f that in a PG(k, p") any transformation of the form m *'* aiixr + ai2< + ■ ■ • + ai*+i°€li fi_, h, xk+i ak+nxi ~r an-i2a:2 + ■--+ a*+i*+ia;i:+i where wi is zero or an integer less than n, is a collineation.As a result of Dr. Levi's article J it is possible to prove the converse proposition, namely, that every collineation in PG(k,pn)is of type (1).The following argument connects this theorem directly with our former article.It will be sufficient to give the argument for the case, k = 2.A projective collineation, § or, in other words, a linear transformation, is determined by the quadrangle into which it transforms the quadrangle (0 0 1),