sol56 - Selected Solutions, Leon 5.6 5.6.2 (b) Factor A = 2...

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Unformatted text preview: Selected Solutions, Leon 5.6 5.6.2 (b) Factor A = 2 5 1 10 into a product QR , where Q is an orthogonal matrix and R is upper triangular. Solution : We use the modified Gram-Schmidt algorithm. Let a 1 , a 2 denote the columns of A . We begin by finding r 11 = k a 1 k = 5, and normalizing a 1 to obtain q 1 = ( 2 5 , 1 5 ) T . Next, we find r 12 = q T 1 a 2 = 4 5. We then update a 2 by subtracting from a 2 its projection onto q 1 , i.e., a 2 := a 2- r 12 q 1 = (- 3 , 6) T . This completes one iteration of the algorithm. We now calculate r 22 = k a 2 k = 3 5, and normalize a 2 to obtain q 2 = (- 1 5 , 2 5 ) T . Assembling these pieces, we have A = 2 5 1 10 = 2 / 5- 1 / 5 1 / 5 2 / 5 5 4 5 3 5 = QR. 5.6.3 Given the basis { (1 , 2 ,- 2) T , (4 , 3 , 2) T , (1 , 2 , 1) T } for R 3 , use the Gram-Schmidt process to obtain an orthonormal basis....
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sol56 - Selected Solutions, Leon 5.6 5.6.2 (b) Factor A = 2...

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