Solution_6 - THE CHINESE UNIVERSITY OF HONG KONG Department...

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Unformatted text preview: THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MAT2310 Linear Algebra and Applications (Fall 2007) Suggested Solutions For Homework 6 1. Solution: (i) Denote u 1 = 1- 1 1 and u 2 = 2- 2 1 , set v 1 = u 1 = 1- 1 1 , we calculate that v 2 = u 2- [ u 2 · v 1 / ( v 1 · v 1 )] v 1 = 1 / 3- 1 / 3- 2 / 3 . Hence, w 1 = 1 / √ 3- 1 / √ 3 1 / √ 3 and w 2 = 1 / √ 6 ,- 1 / √ 6- 2 / √ 6 , Hence the matrix Q should be as follows: 1 / √ 3 1 / √ 6- 1 / √ 3- 1 / √ 6 1 / √ 3- 2 / √ 6 . The ( i,j ) element of R is equal to u j · w i , so the matrix R should be as follows: • √ 3 5 √ 3 0 2 √ 6 ‚ . (ii) Denote u 1 = (1 ,- 1 ,- 1) T , u 2 = (0 , 2 ,- 2) T , u 3 = (2 , , 2) T , we calculate that v 2 = u 2- [ u 2 · v 1 / ( v 1 · v 1 )] v 1 = (0 , 2 ,- 2) T , v 3 = u 3- [ u 3 · v 1 / ( v 1 · v 1 )] v 1- [ u 3 · v 2 / ( v 2 · v 2 )] v 2 = (2 , 1 , 1) T . Hence w 1 = (1 / √ 3 ,- 1 / √ 3 ,- 1 / √ 3) T , w 2 = (0 , 1 / √ 2 ,- 1 / √ 2) T , w 3 = (2 / √ 6 , 1 / √ 6 , 1 / √ 6) T . Hence the matrix Q should be as follows: 1 / √ 3 0 2 / √ 6- 1 / √ 3 1 / √ 2 1 / √ 6- 1 / √ 3- 1 / √ 2 1 / √ 6 . The ( i,j ) element of R is equal to u j · w i , so the matrix R should be as follows: √ 3 4 / √ 3 2 √ 2- √ 2 0 4 / √ 6 . 2 2. Solution: (a)We assume that the least squares line y = b 1 x + b that ”best fits the data”. By assumptions, we obtain the linear system A x = b , where the matrix A is 3 1 4 1 5 1 6 1 7 1 , and the vectors are x = • b 1 b ‚ and b = 2 3 2 4 3 . The corresponding normal matrix system is A T A x = A T b . By calculations, we obtain A T A = • 3 4 5 6 7 1 1 1 1 1 ‚ 3 1 4 1 5 1 6 1 7 1 = • 135 25 25 5 ‚ A T b = • 3 4 5 6 7 1 1 1 1 1 ‚ 2 3 2 4 3 = • 73 14 ‚ . Using the Gauss-Jordan reduction, we obtain • . 3 1 . 3 ‚ . Hence the least squares line is y = 0 . 3 x + 1 . 3. (b)We assume that the quadratic least squares polynomial y = b 2 x 2 + b 1 x + b that ”best fits the data”. By assumptions, we obtain the linear system A x = b , where the matrix A is 9 3 1 16 4 1 25 5 1 36 6 1 49 7 1 , and the vectors are x = b 2 b 1 b and b =...
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Solution_6 - THE CHINESE UNIVERSITY OF HONG KONG Department...

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