# Intersection of lines from lineair systems defines a conic section

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.

• What is a "lineair system of hyperplanes"? Do you mean a pencil of hyperplanes? – user10354138 Jun 9 at 2:46
• – amd Jun 9 at 5:14
• Yeah I mean a pencil of hyperplanes! This is a question that was originally in Dutch and I didn’t find a translation but now I know! – Mee98 Jun 9 at 6:52