Difference between a gaussian vector and a multivariate gaussian distribution 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?
 A: The two conditions (or definitions if you prefer)  are equivalent

A random vector $x = (X_1, \ldots, X_k)'$ is said to have the multivariate normal distribution if it satisfies the following equivalent conditions.

*

*Every linear combination of its components $Y = a_1 X_1 + \ldots + a_k X_k$ is normally distributed. That is, for any constant vector $a \in R^k$, the random variable $Y = a′x$ has a univariate normal distribution, where a univariate normal distribution with zero variance is a point mass on its mean.

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Wikipeida is the source.
A: Just to add Matthew's answer and give you some intuition.  The covariance matrix of a multivariate normal distribution is symmetric positive definite.  That means you can take the singular-value-decomposition of this matrix, and change coordinates so that the coordinates are independent normally distributed random variables.  A linear combination of the original normal coordinates can be rewritten as a linear combination of these independent normal coordinates and will therefore be normal.
