Proof on distribution of $d$-dimensional Gaussian vectors I'm reading the following proof on this notes (Proposition 3.5)
Proposition If $X$ and $Y$ are $d$-dimensional Gaussian random vectors with $E(X) = E(Y)$ and $\mathrm{Cov}(X) = \mathrm{Cov}(Y)$, then $X$ and $Y$ have the same distribution.

I've marked in yellow my doubts.
The first one is that I don't really understand why $B=AL^T$.
I get that $B_i=A_i L^T$ for every $i=1, \cdots, l$ but, how do I know that the rest of the rows on $A$ follow the same linear dependence as the ones in $B$?
My second doubt is about why by Lemma 3.3 we have that $X$ and $Y$ have the same distribution.
My guess is that, if two random variables $X$ and $Y$ have the same distribution, then for every matrix $AX$ and $AY$ have also the same distribution. Is that correct? And, it it is correct, where can I find a proof?
 A: *

*Since $\ L\ $ is an isomorphism and $\ A_1, A_2, \dots, A_l\ $ are linearly independent, then so are $\ B_1, B_2, \dots, B_l\ $, and since $\ \dim\mathcal{A}=$$\dim\mathcal{B}=$$l\ $, then $\ B_1, B_2, \dots, B_l\ $ must be a basis for $\ \mathcal{B}\ $.  Now if $\ B_k=\sum_\limits{i=1}^l v_iB_i\ $ and we let $\ v\ $ be the row vector whose $\ i^\text{th}\ $ entry is $\ v_i\ $ for $\ i=1,2,\dots,l\ $, $\ {-1}\ $ for $\ i=k\ $, and $\ 0\ $ for all other $\ i\ $, then
$$
   0=vBB^Tv^T=vAA^Tv^T\ ,
$$
and so $\ vA=0\ $, or, in other words, $\ A_k=\sum_\limits{i=1}^l v_iA_i\ $. Therefore,
\begin{align}
L\big(A_k\big)&=\sum_\limits{i=1}^l v_iL\big(A_i\big)\\
&=\sum_\limits{i=1}^l v_iB_i\\
&=B_k\ .
\end{align}

*You're correct that if $\ X\ $ and $\ Y\ $ have the same distribution, then so do $\ AX\ $ and $\ AY\ $.  In fact, $\ f(X)\ $ and $\ f(Y)\ $ have the same distribution for any measurable $\ f:\mathbb{R}^k\mapsto M\ $, with $\ \big(M,\mathcal{M}\big)\ $ being any measurable space. For any $\ W\in \mathcal{M}\ $ we have
\begin{align}
P\big(f(X)\in W\big)&=P\big(X\in f^{-1}(W)\big)\\
&=P\big(Y\in f^{-1}(W)\big)\ \ \ (\text{because }\ X\ \text{ and }\ Y\ \text{ are }\\
&\hspace{8.5em}\text{identically distributed})\\
&=P\big(f(Y)\in W\big)\ .
\end{align}
