Assume $A$ is an $m\times n$ random matrix with i.i.d entries, and $x\in\mathbb{R}^n$ be a fixed vector with $\Vert x\Vert_2=1$. Then can we say something about $y:=Ax$?
Does $y$ still have i.i.d components? If not, what is a possible counter-example, and what additional assumptions maybe necessary to make $y$ has i.i.d components?
If you see $A$ as a stack of m vectors in $\mathbb{R}^n$, then the $i$'th component of $y$ is just the projection of the $i$'th vector in $A$ on the vector $X$. Since all the vectors in $A$ are i.i.d, these projections will also be i.i.d, whatever the vector, given $X$.
Even if $X$ were a random unit vector, since all the vectors in $A$ had no correlations between them, their projections on $X$ - whatever $X$ is - cannont be corelated to each other - so in the end, $A X$ is a vector of i.i.d. variables, whether $X$ is given or not.