Exercise 9.A.5

Answers

Proof. Suppose $u, v \in V$ . Then

\begin{aligned} {(S + T)}_{ℂ} (u + iv) & = (S + T) u + i (S + T) v \\ = (Su + iSv) + (Tu + iTv) \\ = S_{ℂ} (u + iv) + T_{ℂ} (u + iv) . \end{aligned}

Thus ${(S + T)}_{ℂ} = S_{ℂ} + T_{ℂ}$ . Now suppose $λ \in ℝ$ . Then

{(λT)}_{ℂ} (u + iv) = λTu + λiTv = λ (Tu + iTv) = λ T_{ℂ} (u + iv) .

Therefore ${(λT)}_{ℂ} = λ T_{ℂ}$ . □

celiopassos

2017-10-06 00:00