Exercise 3.F.10

Answers

Proof. Suppose $ψ \in W^{'}$ . For the additive property, we have

\begin{aligned} {(S + T)}^{'} (ψ) & = ψ \circ (S + T) \\ = ψ \circ S + ψ \circ T \\ = S^{'} (ψ) + T^{'} (ψ) \end{aligned}

Where the second equation follows from the distributive property of linear maps. Hence ${(S + T)}^{'} = S^{'} + T^{'}$ .

For the scalar multiplication, let $v \in V$ . We have

({(λT)}^{'} (ψ)) (v) = (ψ \circ (λT)) (v) = ψ \circ T (λv) = (T^{'} (ψ)) (λv) = λ (T^{'} (ψ)) (v)

Thus ${(λT)}^{'} (ψ) = λ T^{'} (ψ)$ which implies ${(λT)}^{'} = λ T^{'}$ , as desired. □

celiopassos

2017-10-06 00:00