We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. The objective is to show that $\mathbb{P}\left(Y - \mathbb{E}(Y) \in \operatorname{Im}\Sigma_Y\right) = 1$. We now assume $r < n$ where $r$ is the rank of $\Sigma_Y$.
Prove that the kernel and image of $\Sigma_Y$ are supplementary orthogonal subspaces in $\mathcal{M}_{n,1}(\mathbb{R})$.