Final Exam 2

# Final Exam 2 - Answer Key to Final Examination Version 2...

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

Answer Key to Final Examination, Version 2, Math. 33A, Spring Quarter, 2003 1. Which of the following four sets is a field? Subset yes or no reason The rational numbers . Yes One can add, substract, multiply and divide in . a : a in with a 0 No 1 is not in the set. a b b a : a and b are in Yes By computation, the set is closed under addition, substraction and multiplication, and one can do division if ab 0 a b 0 a : a and b are in No 0 1 0 0 does not have inverse. 2 . Answer the following questions: (a) Find all values of k when 1 1 1 1 2 k 1 4 k 2 is not invertible. We need to find when the rank of the matrix is less than 3. By row operation we get 1 1 1 1 2 k 1 4 k 2 2nd 1st, 3rd 2nd 1 1 1 0 1 k 1 0 2 k k 1 1st 2nd, 3rd 2 2nd 1 0 2 k 0 1 k 1 0 0 k 2 k 1 From this, the matrix is invertible if either k 1 or k 2. (b) Compute the determinant of 1 1 1 1 2 k 1 4 k 2 . By the above row operation, we find that the determinant is k 1 k 2 . (c) When 1 1 1 1 2 k 1 4 k 2 is not invertible, find all linearly independent vectors among column vectors of 1 1 1 1 2 k 1 4 k 2 ( Each redundant vector given costs 10 points ).

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