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# sol55 - Partial Solution Set Leon 5.5 5.5.2(a Straight...

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Partial Solution Set, Leon § 5.5 5.5.2 (a) Straight forward computations (b) We have u 1 = 1 3 2 , 1 3 2 - 4 3 2 T , u 2 = ( 2 3 , 2 3 , 1 3 ) T , and u 3 = 1 2 , 1 2 , 0 T . Let x = (1 , 1 , 1) T . Write x as a linear combination of u 1 , u 2 , and u 3 , and use Parseval’s formula to compute || x || . Solution : We know from part (a) that [ u 1 , u 2 , u 3 ] is an orthonormal basis for R 3 . By Theorem 5.5.2, we know that x = ( x T u 1 ) u 1 + ( x T u 2 ) u 2 + ( x T u 3 ) u 3 = - 2 3 2 u 1 + 5 3 u 2 + 0 u 3 = - 2 3 2 u 1 + 5 3 u 2 By Parseval’s formula, || x || = ( 4 18 + 25 9 ) 1 / 2 = 3. 5.5.3 We are given S , the subspace spanned by u 2 and u 3 of the preceding exercise, and x = (1 , 2 , 2) T . We are to find the projection p of x onto S , and to verify that p - x S . Solution : The projection is p = ( x T u 2 ) u 2 + ( x T u 3 ) u 3 = 8 3 u 2 - 1 2 u 3 = 23 18 , 41 18 , 8 9 T So p - x = ( 5 18 , 5 18 , - 10 9 ) T . It is easy to show that p - x S , by showing that it is orthogonal to each of u 2 , u 3 . Note: A close look at the computation by which the projection was obtained is consistent with the observation (Corollary 5.5.9) that the projection operator is

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