Affine Transformations of Gaussian Random Vectors are Gaussian

Let $(X_1,\dots,X_d)$ be a Gaussian random vector on a probability space $(\Omega,\mathcal{F},P)$, let $p$ be a natural number, and let $c_i$ and $M_{ik}$ ($1\le i\le p$, $1\le k\le d$) be real numbers. Define

Then $(Y_1,\dots,Y_p)$ is a Gaussian random vector, and its mean vector and covariances are given by

with the covariance of square-integrable random variables.

In particular: every linear combination $c+\sum_{k}c_kX_k$ of jointly Gaussian random variables is a Gaussian random variable; every subfamily $(X_{i_1},\dots,X_{i_p})$ with $1\le i_1<\dots<i_p\le d$ of a jointly Gaussian family is jointly Gaussian; and sums and differences of jointly Gaussian random variables are Gaussian.