assig3 - MATH 245 Linear Algebra 2, Assignment 3 Due Fri...

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Unformatted text preview: MATH 245 Linear Algebra 2, Assignment 3 Due Fri Oct 15 1: (a) Find the least-squares best fit quadratic f ∈ P 2 for the following data points. x i- 1 0 1 2 3 y i 2 3 2- 2 (b) Let { p 1 ,p 2 , ··· ,p l } be a linearly independent set of polynomials in P m and let ( x 1 ,y 1 ) , ( x 2 ,y 2 ) , ··· , ( x n ,y n ) be points in R 2 such that at least m +1 of the x-coordinates x i are distinct. Show that there exists a unique polynomial f ∈ Span { p 1 ,p 2 , ··· ,p l } which minimizes the sum n ∑ i =1 ( y i- f ( x i ) ) 2 . 2: (a) Show that for all u,v,w ∈ R 3 we have ( u × v ) × w = ( u . w ) v- ( v . w ) u . (b) Show that for all u 1 ,u 2 , ··· ,u n- 1 ,v 1 ,v 2 , ··· ,v n- 1 ∈ R n we have X ( u 1 ,u 2 , ··· ,u n- 1 ) . X ( v 1 ,v 2 , ··· ,v n- 1 ) = det( A t B ) , where A = ( u 1 ,u 2 , ··· ,u n- 1 ) ∈ M n × ( n- 1) and B = ( v 1 ,v 2 , ··· ,v n- 1 ) ∈ M n × ( n- 1) ....
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This note was uploaded on 12/08/2011 for the course MATH 245 taught by Professor New during the Fall '09 term at Waterloo.

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