HW6_Sol - Math 104 Fall 2009 Homework 6 solution set We use...

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Math 104 Fall 2009 Homework 6: 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 (a) Suppose that A = 1 0 0 1 1 0 = [ a 1 , a 2 ]. We will use the Classical Gram-Schmidt method to calculate the reduced and full QR decomposition of A . We begin with r 11 = k a 1 k = 2, q 1 = a 1 2 = 2 2 0 2 2 , r 12 = q * 1 a 2 = 0, r 22 = k a 2 - r 12 q 1 k = 1, q 2 = ( a 2 - r 12 q 1 ) r 22 = 0 1 0 . Therefore, we have the reduced QR decomposition A = ˆ Q ˆ R = 2 2 0 0 1 2 2 0 2 0 0 1 It is easy to extend ˆ Q into an orthogonal matrix Q = 2 2 0 2 2 0 1 0 2 2 0 - 2 2 , which gives the full QR decompostion A = QR = 2 2 0 2 2 0 1 0 2 2 0 - 2 2 2 0 0 1 0 0 . (b) Suppose that A = 1 2 0 1 1 0 = [ a 1 , a 2 ]. We will use the same method to calculate the reduced and full QR decomposition of A . Then r 11 = k a 1 k = 2, q 1 = a 1 k a 1 k = 2 2 0 2 2 , r 12 = q * 1 a 2 = 2, r 22 = k a 2 - r 12 q 1 k = 3, q 2 = ( a 2 - r 12 q 1 ) r 22 = 3 3 3 3 - 3 3 . This gives the reduced QR decomposition
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