Lectures and Homework 642:550, Summer 2004
The exercises listed here should be prepared for the second class
following the one in which they are assigned. Problems will be
listed here shortly before they are announced in class. A table
of assigned problems will evolve in this space.
| Date |
Section |
Pages |
Problems |
| Jun 28 |
3.6 |
205 - 207 |
14 (both parts from text and extra part from S1,
see note). |
| S1 |
7 |
A
(see note). |
| Jun 29 |
4.2 |
218 - 221 |
6, 11 (see note). |
| 4.3 |
228 - 230 |
4. |
| 4.Review |
241 - 242 |
13. |
| S2 |
4 |
A. |
| Jul 01 |
4.4 |
238 - 240 |
1, 12. |
| S3 |
3 |
A, B (see note). |
| 5.1 |
251 - 253 |
12, 16
(see note). |
| Jul 05 |
5.2 |
260 - 261 |
7. |
| 5.3 |
272 - 274 |
8 |
| S4 |
2, 5 and 6 |
1, 2, and 3. |
| Jul 06 |
S5 |
9 |
1, 2, 3. |
| Jul 08 |
S6 |
6 |
1, 2, 3, 4. |
| Jul 12 |
5.5 |
301 - 303 |
6, 7, 11, 19. |
| 5.Review |
319 - 321 |
15. |
| Jul 13 |
S7 |
4 |
1, 2, 3. |
| Jul 15 |
S8 |
8 - 9 |
1, 2, 3, 4. |
| Jul 19 |
6.2 |
337 - 338 |
2, 7, 11. |
| Jul 20 |
6.3 |
345 - 346 |
1, 2, 11 (see note). |
| 6.4 |
352 - 354 |
11 (see note). |
Jul 22 |
Appendix A |
451 - 452 |
2, 4, 5. |
| Jul 26 |
3.3 |
162 - 165 |
13. |
| 3.4 |
180 - 182 |
15. |
| 3R |
208 - 210 |
33. |
| no more homework! |
Notes
- 3.6.14
- The desired factorization is the reduced factorization
described in item 2 on page 197. More details and an
extra part are in Supplement 1, section 3 (p.4). There is a misprint
in the textbook: the number of columns of L that must be dropped is
m-r (matching the rows of U that are discarded), not n-r (as written
in the text). [This was noticed by a student in Summer 2003].
- S1.A
- Part (a) should be an easy calculation, but part (b) may be
tedious. It is included to show that the intersection of the
subspaces does not depend on the order in which the spaces are
written, but the effort involved in a solutions depends on how the
problem is formulated.
- 4.2.11
- One part asks to use properties of the determinant to show that
the determinant of every 3 by 3 skew-symmetric matrix is
zero. Another part asks for one example of a 4 by 4
skew-symmetric matrix with nonzero determinant. This example
should be a numerical matrix, and you should find the
determinant to be sure that it isn't zero.
- S3. A and B
- There are some errors in the statements of the problems, but they
should have no effect on the solution. Specifically,
the determinants give the squares of the measures of
the figures, and the measure should be called "area" in A.
- 5.1.16
- The problem statement includes a reference to 4.3.10. That
problem suggests some direct ways to evaluate this determinant. None
of those methods need be submitted with the solution to 5.1.16, but
you are free to use them to check your answer. The point of the
exercise if that the use of eigenvalues often leads to a much simpler
evaluation of the determinant although it uses deeper theory. A later
supplement will say more.
- 6.3.11
- This is based on the generalized eigenvalue
problem discussed on pages 343 and 344. It is tempting to
solve this problem as described in the footnote on page 344, and the
ease with which that succeeds in this case reinforces that instinct.
However, a complete solution should use a factorization of M as a
product of a matrix and its transpose. Three such factorizations have
been described (see page 334), and you should consider all three
before using one to find the generalized eigenvalues and their
eigenvectors.
- 6.4.11
- This problem uses a simple form of Remark 2 on page 352.
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