Orthogonal complement/K/Test on generating system/Exercise

Let ${\displaystyle {}V}$ be a vector space over ${\displaystyle {}{\mathbb {K} }}$, endowed with an inner product ${\displaystyle {}\left\langle -,-\right\rangle }$, and let ${\displaystyle {}U\subseteq V}$ be a linear subspace. Let ${\displaystyle {}u_{i}}$, ${\displaystyle {}i\in I}$, be a generating system of ${\displaystyle {}U}$. Show that a vector ${\displaystyle {}v\in V}$ belongs to the orthogonal complement ${\displaystyle {}U^{\perp }}$ if and only if

${\displaystyle {}\left\langle v,u_{i}\right\rangle =0\,}$

holds for all ${\displaystyle {}i\in I}$.