So i p 2 i p i p i 2 p i ip p 2 i 2p p

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Unformatted text preview: n 2 S ⊆ R , prove that (I − P ) = I − P , where I is the n × n identity matrix. (b) (8 points) Give an example of a 3 × 3 matrix A whose columns are orthonormal and A has exactly one entry equal to 0. Solutions: (a) Since P is a projection matrix we know P 2 = P . So (I − P )2 = (I − P )(I − P ) = I 2 − P I − IP − P 2 = I − 2P + P 2 = I − 2P + P = I − P . (b) We’ll deal with the length 1 stuff at the end. First we get the ‘ortho’ stuff down. Let the first vector be in the xy -plane. It’ll have third component equal to zero. So (1, 1, 0) will do. The next vector has to be (a, b, c) where none of the entires are 0 and...
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