 Solution

Put $D_j=X_j-\overline{X}$. The supplied linear-transformation result shows that $(\overline{X},D_2,\dots ,D_n)$ is jointly multivariate normal. Its marginal distributions areIndeed, $\mathrm{Cov}(X_j,\overline{X})={\sigma }^2/n$, soFor a jointly Gaussian vector, zero cross-covariance between blocks implies their independence. Thus $\overline{X}$ is independent of the entire vector $(D_2,\dots ,D_n)$, not merely of each coordinate separately. The residual coordinates themselves need not be independent: for $j\ne k$, their covariance is $-{\sigma }^2/n$.