Orthonormal Linear Combinations of Independent Standard Normal Random Variables

Let $m$ and $p$ be natural numbers, let $Z_1,\dots,Z_m$ be independent standard normal random variables on a probability space $(\Omega,\mathcal{F},P)$, and let $w_1,\dots,w_p$, with coordinates $w_i=(w_{i1},\dots,w_{im})$, be an orthonormal family in the Euclidean space $\mathbb{R}^{m}$:

with the dot product of $\mathbb{R}^{m}$. Define

Then $W_1,\dots,W_p$ are independent standard normal random variables on $(\Omega,\mathcal{F},P)$.