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NOTE:
- The sequence Math 4377—Math 4378 satisfies the requirements of the B.S. or B.A. in Mathematics (https://uh.edu/nsm/math/undergraduate/major-minor-programs/): additional 12 advanced Math elective hours with a minimum of 9 hours at 4000-level, which includes a Math Senior Sequence.
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Do not skip steps that are important. E.g., in 5.2 #13(a) there are two inclusions to prove in order to show the equality of the two eigenspaces. Same for 5.1, #9(a, b).
[Actually, in these cases it would suffice to say that one can do the same proof for \(T^{-1}\) because \((T^{-1})^{-1}=T\) etc.]
Please write the solutions with all the relevant details (including computations when needed).
| ASSIGNMENTS (FUTURE ONES MIGHT CHANGE) | |||
|---|---|---|---|
| Section | TO DO | Optional | |
| 5.1 page 256 |
4(d), 5(d), 9(a,b), 16(a) | 1s, 3s, 4s, 5s, 8, 9(c), 12, 13, 15do, 20, 21, 22d, 23 | |
| 5.2 page 277 (direct sums) |
2(b, d, f), 3(b), 8, 13 | 1s, 2s, 3s, 4, 9, 10, 11, 12*o, 14, 21, 22, 23 | |
| 5.4 page 319 |
5, 6(b,d), 17* (use one Thm from this section), 18(a, b), 21 | 1s, 2s, 3, 4, 8, 11, 13 & 14, 20*, 25, 26 | |
| 6.1 page 334 |
5 [Hint: for checking prop (d), can use that \(Re(\imath z_1
\bar{z_2})\le |z_1| |z_2|\), which leads to an inequality for real
numbers], 9, 10do, 11, 12 |
1s, 15, 17, 20, 21, 22 | |
| 6.2 A page 350 |
2(a, h), 3 | 1s (d, e, f, g), 17 | |
| 6.2 B page 350 |
12, 16, 19(c) and 20(c) | 1s, 6, 11o [Note that this is an "if and only if" statement, so both implications should be addressed], 13 (see 23* too), 14, 15 | |
| 6.3
page 362 |
2(b, c), 3(b), 8, 9, 11, 20s, 22(c) | 1s, 2(a,c)s, 3(a,c)s, 6, 10, 12, 13, 14s, 15, 18do, 19, 21s, 22 | |
| 6.4
page 371 |
2(b,d,f) [one method: use an o.n. basis in each space], 4, 9, 10, 11, 12 |
1s, 2s, 3, 6, 7 (see 8 too), 8*, 14, 15, 16, 17-19, 20 & 22 | |
| 6.5
page 389 |
2(b,e), 8, 10 [Hint: can use that the change of coordinate matrix between two o.n. bases is unitary/orthogonal] | 1, 5s, 7o (the scalars must be the complex numbers), 9 | |
| 6.5 cont'ed |
13, 17 [Hint: can use problem 19 in section 6.3 (no need to assume
the columns are different); see also the proof in the textbook of
Thm. 6.16],
19 |
21, 23 (can derive this from Thm. 6.22) | |
| 6.6
page 400 |
3 (b, e); 7 (b,c,d) [hint: use (a)] For #3: what you have to do is explicitly find the projections (that is, their matrices), check that these are indeed orthogonal projections, and check (d), (e) of Thm. 6.25. |
1s, 2s, 3s, 4, 5, 6, 8, 10 | |
| 7.1
page 487 Jordan normal form, due in 1870 to Camille Jordan (1838 - 1922) |
2 (a, d) [Hint for (d): eigenvalues are 2,2,3,3], 3 (b), 4do | 1s, 2s, 3s, 7 | |
| 7.2
page 502 |
2, 4 (a,b,c), 5 (b, c, f) | 1s, 5, 6 and 7 | |
| 7.3
page 515 |
2 (a, b, d) | 1s, 8, 9, 10, 13 (minimal polynomial from Jordan normal form) | |