Let $W_{\ell} = \bigoplus_{0 \leq i < \ell} \mathbf{C} v_i$ and $a \in \mathbf{C}^*$. We define a linear map $P_a : V \rightarrow V$ by $P_a(v_i) = a^p v_r$ where for $i \in \mathbf{Z}$, we define $r$ and $p$ respectively as the remainder and quotient of the Euclidean division of $i$ by $\ell$; in other words, $i = p\ell + r$ where $0 \leq r < \ell$ and $p \in \mathbf{Z}$. Show that $P_a$ is a projector with image $W_{\ell}$.