I always thought a gaussian vector and a multivariate gaussian distribution were more or less the same thing but I've remembered that a gaussian vector have a more complex definition than that.
A gaussian vector is a vector such that every linear combination of its coefficients follows a gaussian distribution.
If the coefficients of a vector follow a multivariate distribution, the vector should be gaussian.
For the converse, each coefficient of a gaussian vector (as a trivial linear combination) should follow a gaussian distribution. The only difference I can see is about an assumption on correlation structure. A gaussian vector does not imply directly a linear correlation structure. But I feel that the condition "EVERY linear combination..." is strong enough so that we can't have the correlation structure we want.
Is the converse true? If so what is the difference between the two?