Assume that the set of random variables $V, W, Z$ are independent with Gaussian distributions where $V \sim \mathcal{N}(0,\sigma_V)$, $W \sim \mathcal{N}(0,\sigma_W)$, and $Z \sim \mathcal{N}(0,\sigma_Z)$. From this we can say that $V, W, Z$ are jointly Gaussian. Now consider two following affine transformation of $V, W, Z$: \begin{align} X = a_1V + a_2W + a_3 Z, \quad Y = b_1V + b_2W + b_3 Z. \end{align}
I want to show that $X,Y, V,W,Z$ are jointly Gaussian. Any idea?
My approach: By definition the collection of random variables $F_1, \ldots, F_n$ are jointly Gaussian iff $\sum_{i=1}^n c_i F_i$ is Gaussian for any vector $c = [c_1, c_2, \ldots, c_n]$. If we use this definition, by setting $F_1 = X, F_2 = Y, F_3 = V, F_4 = W, F_5 = Z$ \begin{align} \sum_{i=1}^5 c_i F_i = c_1 (a_1V + a_2W + a_3 Z) + c_2 (b_1V + b_2W + b_3 Z) + c_3 V + c_4W + c_5Z = (c_1a_1 + c_2b_1 + c_3)V + (c_1a_2 + c_2b_2 +c_4)W + (c_1a_3 + c_2b_3 + c_5)Z \end{align} which is Gaussian since $V,W,Z$ are jointly Gaussian. Is this correct?