A well known fact exists which is that if a multivariate normal distribution undergoes a linear transformation it's also multivariate normal.
There are two proofs I have seen, If the transformation is invertible an application of the change of variables theorem is used.
If it's not an argument using either characteristic function or moment generating function is used.
My question is, can it proved in the general case without using characteristic function?
IE how could the change of variables approach be modified to work for singular transformations?