Q: Consider $\displaystyle U_{1}=Y_{1}+Y_{2}$ and $\displaystyle U_{2}=Y_{1}-Y_{2}$. Assume $\displaystyle U_{1}$ and $\displaystyle U_{2}$ have a bivariate normal distribution. Show that $\displaystyle U_{1}$ and $\displaystyle U_{2}$ are independent.

My work: Suppose $\displaystyle U_{1}=Y_{1}+Y_{2}$ and $\displaystyle U_{2}=Y_{1}-Y_{2}$ have a bivariate normal distribution. To show $\displaystyle U_{1}$ and $\displaystyle U_{2}$ are independent we must show $\displaystyle Cov(U_{1},U_{2})=\sum_{i}\sum_{j}\\a_{i}b_{j}Cov(U _{1},U_{2})=0$.

I am having trouble with the latter computation:

$\displaystyle Cov(U_{1},U_{2})=\sum_{i}\sum_{j}\\a_{i}b_{j}Cov(U _{1},U_{2})

=a_{1}b_{1}Cov(Y_{1},Y_{1})+$

$\displaystyle a_{1}b_{2}Cov(Y_{1},Y_{2})+

a_{2}b_{2}Cov(Y_{2},Y_{2})$

How do I treat $\displaystyle Cov(Y_{1},Y_{1})$ given $\displaystyle U_{1}=Y_{1}+Y_{2}$ and $\displaystyle U_{2}=Y_{1}-Y_{2}$?

All in all, I'm pretty stuck. Any help would be greatly appreciated.

Thanks