HW7 - u U . (b) Prove the Pythagorean theorem : (1) k x-u k...

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Math 601 HOMEWORK 7 Solve the problems 1. , 2. below, and also from Strang, 3rd ed., problems 3.1.9(use any method), 3.1.12, 3.1.16, 3.1.19 on p.142, and 3.2.3, 3.2.10, 3.2.12 on p.151 (see the new .pdf file handed out). 1. Let ( V, , · , i ) be an inner product space, U a subspace of V , and denote by P the orthogonal projection onto U . (a) Explain why the triangle with vertices x , u , P x is a right triangle if
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Unformatted text preview: u U . (b) Prove the Pythagorean theorem : (1) k x-u k 2 = k u-P x k 2 + k x-P x k 2 , for any u U ( Hint: expand the inner products and use the fact that x-P x U , that P 2 = P , that P * = P etc.) 2. Let M an m n matrix. Prove that the null space of M * M is the same as the null space of M . 1...
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