Let $X \sim \mathcal{N}(\mu, \Sigma)$ in $\mathbb{R}^k$. (a) For a $d \times k$ matrix $A$ and vector $b \in \mathbb{R}^d$: $AX + b \sim \mathcal{N}(A\mu + b,\; A\Sigma A^\top)$. (b) If $A_n \xrightarrow{\mathbb{P}} A$ entrywise (with $A$ deterministic) and $X_n \xrightarrow{d} \mathcal{N}(\mu, \Sigma)$, then $A_n X_n \xrightarrow{d} \mathcal{N}(A\mu,\; A\Sigma A^\top)$. (c) If $\Sigma$ is diagonal, the components $X^{(1)}, \ldots, X^{(k)}$ are independent.