Dual space/Orthogonal space/Sum and intersection/Exercise

Let ${}V$ be a ${}K$-vector space, and let ${}U_{1},U_{2}\subseteq V$ denote linear subspaces. Show that in the dual space ${}{V}^{*}$, the equality

${}{\left(U_{1}+U_{2}\right)}^{\perp }=U_{1}^{\perp }\cap U_{2}^{\perp }\,$

holds.

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