Dual space/Orthogonal space/Generators/Exercise

Let ${\displaystyle {}U=\langle u_{1},\ldots ,u_{r}\rangle }$ be a linear subspace of a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$. Show that

${\displaystyle {}U^{\perp }={\left\{f\in {V}^{*}\mid f(u_{i})=0{\text{ for all }}i=1,\ldots ,r\right\}}\,}$

holds.