I have two pencils of hyperplanes $\Sigma_1$ and $\Sigma_2$ in $P^2(\mathbb{R})$ and a projective map $\phi \colon \Sigma_1 \to \Sigma_2$. Now I need to show that the the points that are defined by $l_1 \cap \phi(l_1)$, with $l_1 \in \Sigma_1$ forms a conic section in $P^2(\mathbb{R})$.
I first thought that I could show it by using the eigenvectors of the matrix associated with the projective map but any point on the line $l_1$ could map to itself so now I have no idea how I can prove this.