Sol8 - SOLUTION TO ASSIGNMENT 8 - MATH 251, WINTER 2007 In...

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SOLUTION TO ASSIGNMENT 8 - MATH 251, WINTER 2007 In this assignment F = R or C . 1. Consider results of an experiment given by a series of points: ( x 1 ,y 1 ) , ( x 2 ,y 2 ) ,..., ( x n ,y n ) , where x 1 < x 2 < ··· < x n and the x i ,y i are real numbers. We assume that the actual law governing this data is linear. Namely, that there is an equation of the form f A,B ( x ) = Ax + B that fits the data up to experimental errors. Therefore, we look for such an equation Ax + B that fits the data best. Our measure for that is “the method of list squares”. Namely, given a line Ax + B , let d i = | y i - ( Ax i + B ) | (the distance between the theoretical y and the observed y ). Then we seek to minimize d 2 1 + d 2 2 + ··· + d 2 n . Let T : R 2 R n , be the map T ( A,B ) = ( f A,B ( x 1 ) ,...,f A,B ( x n )) . Prove that T is a linear map: We can represent T by a matrix ( A,B ) 7→ x 1 1 x 2 1 . . . . . . x n 1 ± A B . So clearly it is linear. and that the problem we seek to solve is to minimize k T ( A,B ) - ( y 1 ,...,y n ) k 2 . Let us calculate: k T ( A,B ) - ( y 1 ,...,y n ) k 2 = k ( Ax 1 + B,. ..,Ax n + B ) - ( y 1 ,...,y n ) k 2 = n i =1 ( Ax i + B - y i ) 2 = n i =1 d 2 i . Let W be the subspace of R n which is the image of T . Prove that W is two dimensional and that { s 1 ,s 2 } is a basis for W , where s 1 = (1 , 1 ,..., 1) ,s 2 = ( x 1 ,x 2 ,...,x n ). W is spanned by T (0 , 1) and T (1 , 0) , which are just s 1 and s 2 . W is two dimensional, unless x 1 = x 2 = ··· = x n , which is not the case. Assume for simplicity that n i =1 x i = 0 (this can always be achieved by shifting the data). Find an orthonormal basis for W and use it to find the vector in W closest to ( y 1 ,...,y n ). 1
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2 SOLUTION TO ASSIGNMENT 8 - MATH 251, WINTER 2007 The vector closest to ( y 1 ,...,y n ) is its orthogonal projection on W . Let v 1 = n - 1 / 2 (1 , 1 ,..., 1) , v 2 = ( n X i =1 x 2 i ) - 1 / 2 ( x 1 ,...,x n ) . Then { v 1 ,v 2 } are an orthonormal basis for W . The projection is the function v 7→ h v,v 1 i v 1 + h v,v 2 i v 2 . Taking
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This note was uploaded on 08/26/2008 for the course MATH 251 taught by Professor Goren during the Spring '08 term at McGill.

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Sol8 - SOLUTION TO ASSIGNMENT 8 - MATH 251, WINTER 2007 In...

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