Exercise 7.A.14

Answers

Proof. Since $v$ and $w$ are eigenvectors corresponding to distinct eigenvalues, by 7.22 they are orthogonal. Thus

\begin{aligned} | | T (v + w) | |^{2} & = | | Tv + Tw | |^{2} \\ = | | 3 v + 4 w | |^{2} \\ = | | 3 v | |^{2} + | | 4 w | |^{2} \\ = 9 | | v | |^{2} + 16 | | w | |^{2} \\ = 100, \end{aligned}

where the third line follows from the Pythagorean Theorem. □

celiopassos

2017-10-06 00:00