Linear Algebra. Math-4100, Fall 2007
Assignment 3
Due Thursday, September 27, by 4pm. (Either in class, or my mailbox in AE 301, or
under my door AE 405).
Reading
Sep. 17 and 20: Strang Sections 3.4–3.6, 4.1; Gelfand Section 2.
Sep. 24: Strang Sections 4.1-4.2. There will be a pre-test review on Sep. 27
Problems
You are welcome to consult the text and notes and discuss the problems with other people.
However, the solutions should be yours. Please indicated on your papers, who you discussed
the problems with. Please submit extra credit problems on a separate sheet of paper.
R1. Read (and understand!) worked examples 3.4 A and 3.4 B (the latter is
very tempting as a source of examination questions). Nothing to hand in for this
problem.
1. Problem 3.4 #1.
2. Problem 3.4 #5.
3. Problem 3.4 #10.
4. Problem 3.4 #21
5. Problem 3.4 #24. Demonstrate that your examples have the desired properties.
6. Problems 3.4 #33 and 34. The latter is similar to Problem 3.2 #31.
7. Problem 3.5 #5. Comment: You have to understand that the question of whether the
given column vectors v1, v2, v3are dependent or independent is the same as the question
whether the system V x = 0 has a nontrivial solution. (of course, V= [v1, v2, v3] is the
matrix formed of vs as columns).
8. Problem 3.5 #10.
R2. Read (and understand!) worked example 3.6 A. It is rather compressed, so you’ll
need to work a little.
9. Problem 3.6 #5.
10. Problem 3.6 #14.
11. Problem 3.6 #23.
R3. Read worked examples 4.1 A and 4.1B.
12. Problem 4.1 #11.
E4∗. (Extra credit) Let Abe a square matrix. Prove that if AX =XA for any
matrix Xthen A=cI for some constant c. [This is a serious problem, please
take it seriously. If you don’t get it now, you are welcome to revisit it and submit
a solution in the coming weeks.]