quiz2_S07_solutions

# quiz2_S07_solutions - Princeton University Department of...

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

Princeton University Department of Mathematics MAT 202 – Linear Algebra with Applications QUIZ 2 – Spring 2007 SOLUTIONS (1) Let A = 0 1 1 1 1 1 and b = 2 3 5 . (a) Find the least-squares solution of the system Ax = b . (b) What is the vector in the image of A which is closest to the vector b ? Solution: (a) Since A T A = » 0 1 1 1 1 1 2 4 0 1 1 1 1 1 3 5 = » 2 2 2 3 is invertible and A T b = » 0 1 1 1 1 1 2 4 2 3 5 3 5 = » 8 10 the least-squares solution of the system Ax = b is x * = ( A T A ) - 1 A T b = 1 2 » 3 - 2 - 2 2 – » 8 10 = » 2 2 . (b) The vector in the image of A which is closest to the vector b is Ax * = 2 4 0 1 1 1 1 1 3 5 » 2 2 = 2 4 2 4 4 3 5 . (2) Let A = 1 1 0 1 0 1 0 1 0 0 0 1 . (a) Find the 3-volume of the parallelepiped defined by the columns of A . (b) Let v be a unit vector in R 4 . The augmented matrix [ A | v ] is 4 × 4 . What is the maximum value det[ A | v ] may take? Solution: (a) We have that det( A T A ) = det 0 B B @ 2 4 1 1 0 0 1 0 1 0 0 1 0 1 3 5 2 6 6 4 1 1 0 1 0 1 0 1 0 0 0 1 3 7 7 5 1 C C A = det

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

View Full Document
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