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sol54 - Partial Solution Set Leon Section 5.4 5.4.2 Given x...

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Partial Solution Set, Leon Section 5.4 5.4.2 Given x = (1 , 1 , 1 , 1) T and y = (8 , 2 , 2 , 0) T , 1. Determine the angle θ between x and y . 2. Find the vector projection p of x onto y . 3. Verify that x - p is orthogonal to p . 4. Compute k x - p k 2 , k p k 2 , and k x k 2 , and verify that the Pythagorean law holds. Solution : (a) θ = arccos 12 2 · 6 2 = arccos 1 2 = π 4 . (b) The vector projection in question is p = 1 6 y = ( 4 3 , 1 3 , 1 3 , 0) T . (d) We have k x - p k 2 = 2, k p k 2 = 2, and k x k 2 = 2. It follows that k x - p k 2 2 + k p k 2 2 = 2 + 2 = 4 = k x k 2 2 . 5.4.3 Let w = ( 1 4 , 1 2 , 1 4 ) T , and use the weighted inner product h x , y i = n X i =1 x i y i w i . Let x = (1 , 1 , 1) T and y = ( - 5 , 1 , 3) T . 1. Show that x and y are orthogonal with respect to this inner product. 2. Compute the values of k x k and k y k with respect to this inner product. Solution : 1. Since h x , y i = - 5 4 + 1 2 + 3 4 = 0, it follows that x and y are orthogonal. 2. Using the same weighted inner product, we find k x k = q 1 4 + 1 2 + 1 4 = 1, and k y k = q 25 4 + 1 2 + 9 4 = 3.
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