How To Prove that $v=r\cdot z\ $ is Distributed Uniformly at Random

Question

I consider a finite field $\mathbb{F}_q$, where $q=2p+1$, and $p$ , $q$ are prime numbers.

Let $z$ be a fixed element of the field. Also let $r$ be a value picked uniformly random from the field where $r>\frac{q}{2}$

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Question: How to prove that $v=r\cdot z\ $ is distributed uniformly random in subset of the field where the subset's size is $p$?

Answer 1

The map $x \mapsto xz$ is a bijection unless $z = 0$. This immediately yields the assertion.