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

• Notes
• 3

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern