We have given a projective plane $ (P, G) = PG(2, \mathbb{R}) $ and a linear transformation: \begin{equation} f_1: \mathbb{R}^3 \to \mathbb{R}^3, x \mapsto \left( \begin{matrix} 1 &0&1\\ 0&1&1\\ 0&0&2 \end{matrix}\right) x \end{equation}
And are supposed to calculate the center (by definition the point where all lines trough that point are fixed point lines, $f_1(l) =l$ for all $l$ on such a line) and the axis of $f_1$. We got the solution from our professor: the axis is the eigenspace of the eigenvalue 1, that means \begin{equation} \mathbb{R}(1, 1,1)^T + \mathbb{R} (1,-1,0)^T \end{equation}
And as a center \begin{equation} \mathbb{R}(0,1,-1)^T \end{equation}
But we don’t get to that, can someone verify that solution? And help us out on that?