Econ 519: David Reiley

Due Wednesday, 2 August 2006

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.

- Problem 11.6.
- Problem 11.7.
- Problem 11.9.
- Problem 11.10.
- Problem 11.12.
- Problem 11.14.
- Problem 27.1.
- Problem 27.5.
- Problem 27.6. Restrict your attention to the sets V
_{6}, V_{7}, and V_{8}in the chapter's examples. - Problem 27.8.
- 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**. - Problem 27.12.
- Problem 27.13.
- 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 c_{1}, c_{2}, 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 (c_{1}, c_{2}, 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**x**to the equation_{0}**Ax**=**b**, then you can generate the whole set of solutions to**Ax**=**b**by shifting each point in the nullspace by the amount**x**That is, take each vector_{0}**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 (c_{1}, c_{2}, etc.) to specify each**b**and its corresponding solution set**x**.

- Problem 27.22.