Given a random vector X with the multivariate normal distribution F(X), we know that, for two vectors a and b, the projections $A=\sum_j a_j X_j $ and $B=\sum_i b_i X_i $ are univariate normal.
I'm interested in the joint distribution of A and B. Is their joint distribution normal? Is the dependence between A and B described only by their correlation? (do they have only linear dependence?) Thank you for any insight. References are highly appreciated as well.
The vector (A,B) consisting of two scalars is a "two-dimensional projection" of the multivairate normal vector X. More generally, if you multiply X with any matrix such that you get a new vector with dimension <= than the original dimension of X, you again get a multivariate normal vector. That is, all the components of the new vector will be normal, and their dependence is determined only by their correlations.