Bidual/Finite-dimensional/Orthogonal space/Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space, with its dual space ${}{V}^{*}$ and the bidual ${}{{V}^{*}}^{*}$. Let ${}F\subseteq {V}^{*}$ be a linear subspace. Show that the two orthogonal spaces ${}F^{\perp }\subseteq V$ (in the sense of definition) and ${}F^{\perp }\subseteq {{V}^{*}}^{*}$ (in the sense of definition) are equal via the natural identification of the space and its bidual.

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