We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. Let $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$ and $q_Y(U) = \mathbb{V}(U^\top Y)$.
In the general case, prove that the function $q_Y$ admits a maximum on $C$. Specify the value of this maximum as well as a vector $U_0 \in C$ such that
$$\max_{U \in C} \mathbb{V}\left(U^\top Y\right) = \mathbb{V}\left(U_0^\top Y\right).$$