20105ee205A_1_2010ee205A_1_HW6_sol

20105ee205A_1_2010ee205A_1_HW6_sol - Chapter 8 Linear Least...

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Chapter 8 Linear Least Squares Problems 1. For A IR m × n , b m , and any y n , check directly that ( I - A + A ) y and A + b are orthogonal vectors. Answer 8.1 (( I - A + A ) y ) T A + b = y T ( I - A + A ) T A + b = y T ( I - A + A ) A + b = y T ( A + - A + AA + ) b =0 . 2. Consider the following set of measurements ( x i ,y i ): (1 , 2) , (2 , 1) , (3 , 3) . (a) Find the best (in the 2-norm sense) line of the form y = αx + β that ±ts this data. Answer 8.2 ± α β ² =( A T A ) - 1 A T y where A = 11 21 31 and y = 2 1 3 29
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30 CHAPTER 8. LINEAR LEAST SQUARES PROBLEMS = 1 m i x 2 i - ( i x i ) 2 ± m i x i y i - i x i i y i i x 2 i i y i - i x i i x i y i ² = 1 42 - 36 ± 39 - 36 84 - 78 ² = ± 1 / 2 1 ² . Therefore, the best Ftting line is y = αx + β = x/ 2+1 . (b) Find the best (in the 2-norm sense) line of the form x = αy + β that ±ts this data. Answer 8.3 This time take A = 21 11 31 and x = 1 2 3 . Then ± α β ² =( A T A ) - 1 A T x = ± 1 / 2 1 ² (also obvious from symmetry of data).
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20105ee205A_1_2010ee205A_1_HW6_sol - Chapter 8 Linear Least...

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