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Exercise 7.A.12

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Proof. Let $u$ and $w$ denote normalized eigenvectors corresponding to $3$ and $4$ . By 7.22, $u$ and $w$ are orthogonal. From the Pythagorean Theorem, we have

| | u + w | |^{2} = | | u | |^{2} + | | w | |^{2} = 2

and

| | T (u + w) | |^{2} = | | 3 u + 4 w | |^{2} = | | 3 u | |^{2} + | | 4 w | |^{2} = 9 + 16 = 25 .

Letting $v = u + w$ and taking the square root of each equation gives the desired result. □

celiopassos

2017-10-06 00:00

Exercise 7.A.12

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