Exercise 7.A.20

Answers

Proof. Let $v \in V$ and $w \in W$ . Then

\begin{aligned} ((Φ_{V} \circ T^{∗}) (w)) (v) & = (Φ_{V} (T^{∗} w)) (v) \\ = ⟨ v, T^{∗} v ⟩ \\ = ⟨ Tv, w ⟩ . \end{aligned}

On the other hand, we have

\begin{aligned} ((T^{'} \circ Φ_{W}) (w)) (v) & = (T^{'} \circ Φ_{W} (w)) (v) \\ = (Φ_{W} (w) \circ T) (v) \\ = (Φ_{W} (w)) (Tv) \\ = ⟨ Tv, w ⟩ . \end{aligned}

Therefore $Φ_{V} \circ T^{∗} = T^{'} \circ Φ_{W}$ . □

celiopassos

2017-10-06 00:00