Gaussian random variable in $\mathbb{R}^n$ question Let $X=(X_1,...,X_n)$ is a Gaussian random variable in $\mathbb{R}^n$ with mean $\mu$ and covariance matrix $V$. 
I want to show that we can write $X_2$ in the form $X_2 = aX_1 + Z$, where $Z$ is independent of $X_1$, and I want to find the distribution of $Z$.
Any help with this would be really appreciated.
Thanks!

Let $\bar{X}$ be the normalization of $X$ (so that its components have zero mean and unit variance). Then note that it suffices to show $$\bar{X}_2=a\bar{X}_1+Z$$ for some $Z$ independent of $\bar{X}_1$. So we can assume that the components of $X$ have unit variance and zero mean. Now write 
$$
\mathbb{E}[X_2X_1]=a \implies a=\sigma_{12}.
$$
So you want to show that $\sigma_Z=\mathrm{var}(Z)$ can be determined such that the  equality 
$$
X_2=\sigma_{12}X_1+Z
$$
holds (a.e.). This is equivalent to showing that
$$
\mathbb{E}[(X_2-\sigma_{12}X_1-Z)^2]=0.
$$ 
Expand both sides and solve for $\sigma_Z$.
