Isomorphism/Dual mapping/Linear subspaces/Exercise

Let ${}\varphi \colon V\rightarrow W$ be an isomorphism between the ${}K$-vector spaces ${}V$ and ${}W$, and let ${}\varphi ^{*}\colon {W}^{*}\rightarrow {V}^{*}$ denote its dual mapping. Let ${}U\subseteq V$ be a linear subspace and ${}T:=\varphi (U)\subseteq W$ the corresponding linear subspace in ${}W$. Show that, in the dual spaces, the orthogonal spaces ${}U^{\perp }$ and ${}T^{\perp }$ correspond to each other.

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