Lectures and Homework 642:550, Summer 2002
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 24 |
3.6 |
205 - 207 |
14 (both parts).(see note) |
| 3.Review |
208 - 210 |
28. |
| S1 |
3 |
A. |
| Jun 25 |
4.2 |
218 - 221 |
6, 11 (see note), 16. |
| 4.3 |
228 - 230 |
4. |
| 4.Review |
241 - 242 |
3, 13. |
| Jun 27 |
4.4 |
238 - 240 |
1. |
| 4.Review |
241 - 242 |
20, 23. |
| 5.1 |
251 - 253 |
5, 16. |
| Jul 01 |
S4 |
3 |
1, 2. |
5.2 |
260 - 261 |
2, 3 (see note), 7. |
5.3 |
272 - 274 |
8 |
| Jul 02 |
5.4 |
286 - 289 |
1, 6, 12. |
| 5.Review |
319 - 321 |
1, 6, 10. |
| Jul 08 |
5.5 |
301 - 303 |
6, 7. |
| 5.Review |
319 - 321 |
15. |
| S6 |
7 - 8 |
1, 2, 3, 4. |
| Jul 09 |
5.6 |
315 - 318 |
16. |
| S7 |
4 |
1, 2, 3. |
| Jul 11 |
6.1 |
328 - 329 |
12. |
| 6.2 |
337 - 338 |
2, 7. |
| Jul 15 |
6.3 |
345 - 346 |
1, 2, 5 (see note). |
| Jul 16 |
6.4 |
352 - 354 |
6, 11 (see note). |
| Appendix A |
451 - 452 |
2, 4. |
| Jul 18 |
S8 |
6 |
1, 2, 3, 4. |
| Jul 22 |
7.2 |
369 |
4, 10 (see note). |
| Jul 23 |
7.3 |
378 - 379 |
1, 5. |
| 7.4 |
386 - 387 |
5 (see note). |
Notes
- 3.6.14
- The desired factorization is the reduced factorization
described in item 2 on page 197.
- 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.
- 5.2.3
- The two diagonalizing matrices should be
essentially different. The rescaling mentioned in
Remark 2, and permutation of columns of S, which should have
been mentioned there, are always available. Since the matrix A
of this problem is diagonalizable in spite of having a repeated
eigenvalue, additional diagonalizing matrices are available. At least
one column of your second S should not be a multiple of any
column of your first choice of S.
- 6.3.5
- This problem mentions a matrix A that is never identified.
As part of your solution, find A and
show that it is positive definite. In
addition, find the pivots of A-4I. What does this tell you
about the eigenvalues of A?
- 6.4.11
- This problem uses a simple form of Remark 2 on page 352.
- 7.2.10
- Note that the Euclidean norm that was featured in most of the
discussion is not used in this problem. What does it
mean for a vector to be of norm 1 in this sense? How does that bound
the same norm of the image under the matrix A?
(This norm was briefly mentioned in connection with the
Perron-Frobenius Theorem.)
- 7.4.5
- Matrix A was introduced in problem 4 and Gershgorin's
theorem is stated on page 386, just before that problem.
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