Econ 519: David Reiley
Due Wednesday, 2 August 2006

## Problem Set #2

These problems come from the text by Simon & Blume. As usual, though you can check some answers in the back of the book, I expect you to attempt each problem independently, and check your answer only after you have either solved the problem or struggled with the problem unsuccessfully for 15 minutes. Similarly, I expect you to struggle for 15 minutes on a problem before getting help from a classmate, the TA, or the instructor. Because the course is so fast-paced, I suggest you limit yourself to 15 minutes if you're totally stuck; make a note to ask for a hint, and then move on to the next problem.

1. Problem 11.6.
2. Problem 11.7.
3. Problem 11.9.
4. Problem 11.10.
5. Problem 11.12.
6. Problem 11.14.
7. Problem 27.1.
8. Problem 27.5.
9. Problem 27.6. Restrict your attention to the sets V6, V7, and V8 in the chapter's examples.
10. Problem 27.8.
11. Problem 27.11. To clarify, I want you to find a basis for the set of vectors b that yield a solution to the equation Ax=b.
12. Problem 27.12.
13. Problem 27.13.
14. Problem 27.14. To clarify, I want you to find the set of b for which there exist solutions to the equation Ax=b. Then parametrize this set using constants like c1, c2, etc. Finally, for each b in this set, find the value(s) of x that satisfy the equation. You can express your values of x in terms of the parametrizing constants (c1, c2, etc.), so that each x corresponds to a different b.

Hint: use Theorem 27.9, which states that the solutions to Ax=b are related to the solutions to Ax=0 (the nullspace). If you can find just one particular solution x0 to the equation Ax=b, then you can generate the whole set of solutions to Ax=b by shifting each point in the nullspace by the amount x0 That is, take each vector x in the set of solutions to Ax=0 and add to it the vector x0; that gives you the whole set of solutions to Ax=b.

Hint 2: If you have trouble solving these problems in full generality, then start by simply choosing a single b (specify numbers for concreteness) that you know yields a solution x. Make sure you find the whole set of solutions for that b. Finally, consider how you can generalize your answer to give the set of solutions x for each possible b. This involves some algebraic work: where initially you had numbers, now you will use parameters (c1, c2, etc.) to specify each b and its corresponding solution set x.
15. Problem 27.22.