I understand how to create random variables with a prespecified correlational structure using a Cholsesky decomposition. But I would like to be able to solve the inverse problem: Given random variables $X\_1, X\_3, \dots X\_n$ ,and two different linear sums of those variables $V\_1=a\_{11}X\_1+a\_{12}X\_2+\dots a\_{1n}X\_n$, and $V\_2=a\_{21}X\_1 + a\_{22}X\_2 +\dots+a\_{2n}X\_n$, I wish to calculate the correlation between the $V\_1$ and $V\_2$. 

I have searched for terms like "linear combination random variables correlation" and have found plenty of material discussing how the correlation affects the variance of the sum of random variables. Unfortunately I have found nothing that seems to relate to the problem described. I would appreciate any information at all, including either an appropriate book chapter or web page reference.