I have the following problem: Let $X,Y,Z$ be iid Gaussian rv and $U = 2X-Y-Z , V=3X+Y-4Z$
So, I know that (U,V) is a Gaussian vector. The goal is to calculate $E(V|U)$ The hint says to write it as $aU +b$ for a,b real... But this would mean that $E(V|U)$ is Gaussian and that we can describe it as a gauss variable.
But why should be this true?
I am not going to give the answer to you because I think you should be intelligent enough to get it yourself, but here are the results you need.
Lemma 1: Let $(X,Y)$ be a Gaussian vector, then $X$ and $Y$ are independent iff $Cov(X,Y)=0$
Lemma 2: Let $(U,V)$ be a Gaussian vector, then $V = aU + (V-aU)$ for some $a$ such that $(V-aU)$ is independent of $U$.
Let's find $a$, by lemma 1, it must be the case $Cov(V-aU,U) = 0\Rightarrow Cov(V,U) - a Var(U)=0$, this means $a$ is...
For your problem, you can use lemma 2, then
$E(V|U) = E(aU|U) + E(V-aU|U)= aU + ? $