Dual space/Orthogonal space/Generators/Exercise

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

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

holds.

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