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- Apr 14, 2013

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Let $\phi_1:\mathbb{R}^3\rightarrow \mathbb{R}^3$, $v\mapsto \begin{pmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}\cdot v$, $\phi_2:\mathbb{R}^3\rightarrow \mathbb{R}^3$, $v\mapsto \frac{1}{3}\begin{pmatrix}2 & - 1& - 1\\ - 1& 2 & -1\\ - 1&-1 & 2\end{pmatrix}\cdot v$ which describes the shadow of the sun on a plane.

Give the direction of the sun by giving a vector. How do we find this vector?

To give the equation of the plane do we have to find the image of the map?

How can we explain why at a projection $\psi$ it holds that $\psi\circ\psi=\psi$ and $\text{im} \psi=\text{fix} \psi$?