Suppose that $V$ is a finite-dimensional vector space over $\mathbb{C}$, and that $\alpha: V \rightarrow V$ is a $\mathbb{C}$-linear map such that $\alpha^{n}=1$ for some $n>1$. Show that if $V_{1}$ is a subspace of $V$ such that $\alpha\left(V_{1}\right) \subset V_{1}$, then there is a subspace $V_{2}$ of $V$ such that $V=V_{1} \oplus V_{2}$ and $\alpha\left(V_{2}\right) \subset V_{2}$.

[Hint: Show, for example by picking bases, that there is a linear map $\pi: V \rightarrow V_{1}$ with $\pi(x)=x$ for all $x \in V_{1}$. Then consider $\rho: V \rightarrow V_{1}$ with $\left.\rho(y)=\frac{1}{n} \sum_{i=0}^{n-1} \alpha^{i} \pi \alpha^{-i}(y) .\right]$