HW5_Sol - Math 104 Fall 2009 Homework 5 solution set We use...

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Math 104 Fall 2009 Homework 5: solution set We use the notations below to ease readability. Matrices are bold capital, vectors are bold lowercase and scalars or entries are not bold. For instance, A is a matrix and a ij its ( i, j )th entry. Likewise x is a vector and x j its j th component. The linear span generated by a group of vecotors v 1 , v 2 , . . . , v n is denoted by span( v 1 , v 2 , ..., v n ). Problem 1 The equation of a line is y = a + bx . We want to minimize (1 - a ) 2 + ( - 1 - a - b ) 2 + ( - 2 - a - 2 b ) 2 . In matrix form, we need to minimize k c - Az k 2 with A = 1 0 1 1 1 2 , c = 1 - 1 - 2 , z = a b Solving the equation in the sense of least squares sense gives z = ( A * A ) - 1 A * c = 5 6 - 3 2 Therefore, the linear equation is y = 5 6 - 3 2 x . Problem 2 Put A = 1 x 1 . . . . . . 1 x n , c = y 1 . . . y n , z = a b c - Az = y 1 - a - bx 1 . . . y n - a - bx n . Thus the pair ( a, b ) minimizing the sum n i =1 ( y i - a - bx i ) 2 is the vector c minimizing k c - Az k 2 . Therefore, z = ( A * A ) - 1 A * c We introduce some notations and let ¯
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