Dual mapping/Linear subspaces/Exercise

Let

$\varphi \colon V\longrightarrow W$

be a linear mapping between the finite-dimensional ${}K$-vector spaces ${}V$ and ${}W$, and let

$\varphi ^{*}\colon {W}^{*}\longrightarrow {V}^{*}$

denote its dual mapping.

a) Show that, for a linear subspace ${}S\subseteq V$, the relation

${}\varphi ^{*-1}(S^{\perp })=(\varphi (S))^{\perp }\,$

holds.

b) Show that, for a linear subspace ${}T\subseteq W$, the relation

${}\varphi ^{*}(T^{\perp })=(\varphi ^{-1}(T))^{\perp }\,$

holds.

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