Let $w,v$ be two different vectors in the finite vector space $Z_p^m$ over $Z_p$ where $p$ is prime. Let $u$ be a vector chosen uniformly at random from $Z_p^m$. Are the random variables $u \cdot w$ and $u \cdot v$ independent (the dot product calculated over $Z_p$)? If not, can we estimate their covariance somehow?
Thank you for your help.