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.
- 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 V6, V7,
and V8 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 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
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
- Problem 27.22.