| Instructor: Gábor Hetyei | Last update: Tuesday, April 22, 2008 |
| Disclaimer: The information below comes with no warranty. If, due to typographical error, there is a discrepancy between the exercises announced in class and the ones below, or this page is not completely up to date, the required homework consists of those exercises which were announced in class. Check for the time of last update above. If, by my mistake, a wrong exercise shows up below, I will allow you extra time to hand in the exercise that was announced in class. If, however, exercises are missing because this page is not up to date, it is your responsibility to contact me before the due date. (No extra time will be allowed in that case.) This page is up to date if the last update happened after the last class before the next due date. |
The deadlines do not apply to the Bonus questions,
which expire only once we solve them in class, or on April 24 at
latest.
Notation: 1.2/4a means exercise 4, part a, in chapter 1,
section 2.
| No. | Date due: | Problems: |
| 14 | Tue April 29 | 7.1/2bd, 7abcd. |
| 13 | Tue April 22 |
5.2/23; 5.4/7,17. Board problem: Consider the vector space of all polynomials with real coefficients. Define the inner product of two polynomials as the definite integral of their product from -1 to 1. Assume that, p0, p1, p2, ... is an orthonormal basis (such that the degree of pn is n). Introduce now a new inner product of two polynomials as the definite integral of their product from 0 to 1. Using a linear substitution, find an orthonormal basis q0, q1, q2, ... associated to this new inner product, such that for each n, qn is expressed in terms of pn. |
| 12 | Tue April 22 |
6.4/2 bde (you do not have diagonalize only state which properties the
operators have), 6, 13, 16; 6.5/2be, 10, 11. Bonus: complete the proof shown in class of the theorem that every self-adjoint operator on a real vector space has a diagonal matrix with respect to an orthonormal basis. |
| 11 | Tue April 8 | 6.2/2ah,6,8,16a; 6.3/2b,4,18 |
| 10 | Tue April 1 |
6.1/8,10,11. Bonus: Adapt the proof of the Cauchy-Schwartz inequality for real scalars presented in class to complex scalars. |
| 9 | Tue March 25 |
5.1/5, 7ae, 11, 17a, 17b (for n=2 only), 17c; 5.2/8. Our second test is on Tuesday March 25. You may download the Study Guide for Test 2 I distributed in class. |
| 8 | Tue March 19 |
4.3/2, 10, 11, 12, 15, 21 Bonus: Prove the Lemma in 4.2 using elementary product expansion. |
| 7 | Tue March 11 |
3.4/6; 4.1/5,8,9; 4.2/4,23,30. Bonus: 3.4/15. |
| 6 | Tue Feb 26 |
3.1/7: you are allowed to give a "proof by example". Work out
only the 3x3 matrices of the following elementary operations:
3.2/2df, 11,14,17 3.3/5 (provide an example for each n); 3.3/8,10. |
| 5 | Tue Feb 19 |
2.4/13; 2.5/6bd, 9, 10; 2.6/4, 10 ab Bonus: 2.6/8 |
| 4 | Tue Feb 12 |
2.3/9, 10, 12, 13; 2.4/5,6. Our first test is on Tuesday February 12. You may download the Study Guide for Test 1 I distributed in class. |
| 3 | Tue Feb 5 | 2.1/3, 12, 15, 26, 28; 2.2/6, 7, 14, 16. |
| 2 | Tue Jan 29 | Explain 1.4/Example 4; 1.4/2df, 4bd, 11; 1.5/3,9,14; 1.6/10b; 1.7/2,3. |
| 1 | Tue Jan 22 |
1.1/4,6 1.2/4bd, 12, 15 (explain!), 16 (explain!), 18
1.3/8bdf, 10. Bonus questions:
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