Are uncorrelated linear combinations of the elements of a multivariate normal distribution always independent of each other?

Question

To make a simple example, we could have $X \sim N_2 \left( \begin{pmatrix} 0 \\ 0\end{pmatrix}, \begin{pmatrix} 4 & -4 \\ -4 & 2\end{pmatrix} \right)$ which gives $X_1 \sim N(0, 4)$ and $X_2 \sim N (0,2)$. Does this give us that $X_1$ is Independent of $X_1+X_2$?

NB: $\text{Cov}(X_1, X_1+X_2) = \text{Var}(X_1) + \text{Cov}(X_1, X_2) = 4 - 4 = 0$, so these two linear combinations are uncorrelated.

Answer 1

Yes: linear transforms of multivariate normal random variables are again multivariate normal, and uncorrelated implies independent for multivariate normal random variables.