Moment Generating Function with Jointly Gaussian Vector The goal is to prove that for $X$ a jointly Gaussian random vector and $A \in \mathbb{R}^{M \times N}$ a rank $M$ matrix that $Y=AX$ is also jointly Gaussian.
I've found that a common way to prove this is using the moment generating function. Here to save space, we'll let $t = (t_1, \ldots, t_n)$. So $M_Y(t) = E[e^{t'Y}] = E[e^{t'AX}]$. What I'm struggling with is what to do next.
Any hints or suggestions would be much appreciated.

You could write $t'AX = (A't)'X$, so $M_Y(t) = M_X(A't)$
