Let we have $X$, $N$ dimensional vector of independent random variables. If we multiply this vector by some matrix $V$ with size $r\times N$, with property $V*V'=I$, where $I$ is identity matrix, and ' is transpose operation, we get set of $r$ independent random variables collected in vector $Y$. If $X$ variables are with Gaussian distribution, how can we find PDF of derived elements of vector $Y$? Is it still Gaussian?
Recall that every affine function of a normal vector is normal, and that every vector with independent normal entries is normal.
In your case, the vector $X$ has independent normal entries hence $Y=VX$ is normal, with mean $E(Y)=VE(X)$ and covariance matrix $V\Sigma V'$, where $\Sigma$ is the diagonal matrix with diagonal entries the variances of the random variables $X_k$. There is no reason to expect that the random variables $Y_j$ are independent.