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.

  • $\begingroup$ What is a "lineair system of hyperplanes"? Do you mean a pencil of hyperplanes? $\endgroup$ – user10354138 Jun 9 at 2:46
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    $\begingroup$ See en.m.wikipedia.org/wiki/Steiner_conic. $\endgroup$ – amd Jun 9 at 5:14
  • $\begingroup$ 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! $\endgroup$ – Mee98 Jun 9 at 6:52

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