Relationship between univariate normal distribution and multivariate normal distribution

Question

Let $a_1, a_2, a_3$ is column vector and $H = [a_1 a_2 a_3]$. If $a_i$ have standard normal distribution, is this following statement true ? $$ vec(H) = [(a_1)^T (a_2)^T (a_3)^T]^T$$ have multivariate normal distribution, where $T$ denote transpose. I think the statement is true but I just guessing.

Answer 1

First of all, $\mathrm{vec}(H)$ will have the joint distribution of $9$ i.i.d. standard normal distribtuions. This is by definition the same as a standard multivariate distribution of dimension $9$ (with covariance matrix $I_9$ and expected value $0\in\mathbb R^9$). So the answer is yes.