In the proof of the fact that after a projective transformation, the equation for any nonsingular cubic curve can be put in the form , where , my notes in a middle-step arrives at the cubic
and then the projective transformation
is applied to transform it to
I can't really see how this works - shouldn't we use the inverse of this projective transformation? To me it seems that replacing with should give us the cubic , which is not really what we wanted.