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Math416_LeastSquares - Math 416 Abstract Linear Algebra...

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Math 416 - Abstract Linear Algebra Fall 2011, section E1 Least squares solution 1. Curve fitting The least squares solution can be used to fit certain functions through data points. Example: Find the best fit line through the points (1 , 0) , (2 , 1) , (3 , 1). Solution: We are looking for a line with equation y = a + bx that would ideally go through all the data points, i.e. satisfy all the equations a + b (1) = 0 a + b (2) = 1 a + b (3) = 1 . In matrix form, we want the unknown coefficients a b to satisfy the system 1 1 1 2 1 3 a b = 0 1 1 but the system has no solution. Instead, we find the least squares fit, i.e. minimize the sum of the squares of the errors 3 X i =1 | ( a + bx i ) - y i | 2 which is precisely finding the least squares solution of the system above. Writing the system as A~ c = ~ y , the normal equation is A T A~ c = A T ~ y and we compute A T A = 1 1 1 1 2 3 1 1 1 2 1 3 = 3 6 6 14 A T ~ y = 1 1 1 1 2 3 0 1 1 = 2 5 .
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