We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We assume $r < n$ where $r$ is the rank of $\Sigma_Y$. We denote by $d = \dim \ker \Sigma_Y$ and we consider an orthonormal basis $(V_1, \ldots, V_d)$ of $\ker \Sigma_Y$, and we have shown that $\mathbb{P}\left(V_j^\top(Y - \mathbb{E}(Y)) = 0\right) = 1$ for all $j \in \llbracket 1, d \rrbracket$. Conclude that $\mathbb{P}\left(Y - \mathbb{E}(Y) \in \operatorname{Im}\Sigma_Y\right) = 1$.
We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We assume $r < n$ where $r$ is the rank of $\Sigma_Y$. We denote by $d = \dim \ker \Sigma_Y$ and we consider an orthonormal basis $(V_1, \ldots, V_d)$ of $\ker \Sigma_Y$, and we have shown that $\mathbb{P}\left(V_j^\top(Y - \mathbb{E}(Y)) = 0\right) = 1$ for all $j \in \llbracket 1, d \rrbracket$.
Conclude that $\mathbb{P}\left(Y - \mathbb{E}(Y) \in \operatorname{Im}\Sigma_Y\right) = 1$.