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Exercise 6.C.5

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Proof. Suppose $v \in V$ . By 6.47, we can write $v = u + w$ , where $u \in U$ and $w \in U^{⊥}$ . Note that, by 6.51, ${(U^{⊥})}^{⊥} = U$ . Thus $P_{U^{⊥}} v = w$ , because $u \in {(U^{⊥})}^{⊥}$ . Therefore

P_{U^{⊥}} v = w = v - u = Iv - P_{U} v = (I - P_{U}) v .

Hence $P_{U^{⊥}} = I - P_{U}$ . □

celiopassos

2017-10-06 00:00

Exercise 6.C.5

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