proof of the invariance theorem

Invariance Theorem
Suppose that C is a projective conic, and t a projective transformation, then
(1) t(C) is a projective conic, and
(2) if L is the polar of P with respect to C, then
      t(L) is the polar of t(P) with respect to t(C).

proof
Suppose that C has equation xTMx = 0, with M symmetric,
and that t is the transformation t([x]) = [Ax], with A non-singular.

(1)
[xt(C)  if and only if  x = Ay, with yTMy = 0
i.e.  xT(A)-1)TMA-1x = 0, as y = A-1x.
Thus t(C) is the p-conic with matrix N = (A)-1)TMA-1.

(2)
Suppose that P = [p]. By Joachimsthal's formulae, L is pTMx = 0.

Now t(P) = [Ap], so its polar with respect to t(C) is (Ap)TNx = 0,
where N is as in (1). Using the formula in (1) and simplifying, the polar
has equation pTMA-1x = 0.

We leave it as an exercise (along the lines of (1)) to show that, if K is
the p-line bTx = 0, then t(K) is bTA-1x = 0.

Thus t(L) is pTMA-1x = 0, so is the required polar.

the invariance theorem