If $X \in \mathbb{R}^n$ and $Y \in \mathbb{R}^m$ are jointly Gaussian random vectors, then $E[X|Y] = AY+b$ and $E[(X-E[X|Y])(X-E[X|Y])^t|Y] = C$ where $A \in \mathbb{R}^{n\times m}$, $b \in \mathbb{R}^n$, and $C \in \mathbb{R}^{n \times n}$. Moreover $C$ is positive definite. If $X$ and $Y$ are zero-mean, then $b = \vec{0}$ and the conditional expectation is a linear function of $Y$.
I stumbled across this statement in my reading course, and am unsure how to begin to prove this. Any advice or hints would be much appreciated.