In the proof of the fact thatafter 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.