Which distributions preserve its family after linear transformation? For example, if $X$ obeys a multivariate gaussian distribution and let $Y = AX + B$ where $A$ is the matrix for the transformation and $B$ a constant vector. Then $Y$ still obeys a gaussian distribution. Are there any other distributions that share this property?
 A: Yes them exists. Elliptical distributions share this property. Gaussian is a special case. 
More formally, if a random vector $X$ have an elliptical distribution $El (\mu, \Sigma, \phi)$  then any linear combination of that, $Y = AX + B$, maintain an elliptical distribution with the same characteristic generator. More precisely:
$Y$ follow the distribution $El (A\mu +B, A\Sigma A’, \phi)$ 
Where: $\mu$ stand for location vector; $\Sigma$ for scale matrix and $\phi$ for characteristic generator. 
Elliptical is a family of distributions. In the notable example of multivariate t-distribution, $\phi$ stand for degree of freedom (shared among all marginals). If $\phi$ go to infinity we come back to the Normal distribution case invoked in the question. 
As reference I suggest: Quantitative Risk Management: Concepts, Techniques and Tools – McNeil, Frey and  Embrechts; Princeton University Press 2005 (pag 95).
A: For now, I'm studying about the theory of extreme value and there is the Generalized extreme value distribution (GEV) which preserve its family after linear transformation (which includes Gumbel, Weibull and Frechet). But the constant $a_{n}$ and $b_{n}$ depend on $n$
