Are the entries in matrix/vector product independent This question has been confusing me.  Choose a random square matrix $M$ with $M_{i,j} \in \{1,-1\}$ so that $M_{i,j} = 1$ with probability $1/2$ and $M_{i,j} = -1$ with probability $1/2$. Say all the $M_{i,j}$ are i.i.d. Now select a random vector $v$ so that $v_i = 1$ with probability $1/2$ and $v_i = -1$ with probability $1/2$  and all the $v_i$ are i.i.d. We also have that $v$ and $M$ are independent. Let $y = Mv$.

Are the entries $y_i$ independent of each other?


Since the matrix rows of $M$ are i.i.d. and the column vector $v$ is i.i.d and $y_i$ is the $i$th row of $M$ times $v$, you won't get any dependence among $y_i$, since there is no dependence among the rows of $M$.
So the answer is yes!
