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!

Assuming that $X_2 = aX_1 + Z$ holds (with $Z$ and $X_1$ independent), you can find $a$ in terms of particular entries of the covariance matrix. Once you have $a$, you know that $Z=X_2 - aX_1$, so you can find the distribution of $Z$, and check that it is independent of $X_1$.
