Find a sequence of linear transformations of $\mathbb{P} ^{2}$ whose composition transforms the ellipse $\frac{1}{4} x ^{2} + y ^{2} = z ^{2}$ to the parabola $zy=x^2$.
$P^2$ is projective plane.
I think if I let $x=a_1x+b_1y+c_1z, y=a_2x+b_2y+c_2z, z=a_3x+b_3y+c_3z$ and then plug in the ellipse equation to solve for $a_i,b_i,c_i (i=1,2,3)$, I will get the answer. It's a lot of work and I will not get a sequence of transformation unless I decompose the matrix.
I tried to map $[1:0:1] {\mapsto} [1:0:0] $ and $[0:1:1]{\mapsto}[0:1:0]$. All of these points are in the projective plane. Both$[1:0:0]$ and$[0:1:0]$ are points in the parabpla at infinity. But I just got $z^2+2xy+2xz+2yz=0$. I didn't know how to do the next step because it seemed that it's impossible to transform $z^2+2xy+2xz+2yz=0$ to $zy=x^2$. It still needed a bunch of work.
Does there exists any simple method to get the sequence of transformation?
Start by putting $z$ and $y$ on the same side of the original equation to get $\frac{1}{4}x^2=z^2-y^2$. Note that the RHS factors as $(z-y)(z+y)$. Set your new $z$ to be $z-y$ and your new $y$ to be $z+y$, while letting your new $x$ be $2x$. Now you have $zy=x^2$ and you're done.