333 Presentation and Problem Set List
Chapters 1-2
Presentation
1A 8,
and prove (ab)^m =
(a^m)b^m by induction for positive integers m
1B 2, 4
1C 7-8 as one presentation, 2
2A Prove 2.21 if v_1 = 0 and j = 1.
How does 2.23 need to be adapted
to account for all possible cases?
2B 1, 5
2C 1, one of 4-8
Problem Set
1A 1, 16
1B 5, 6
1C 12-13 as one problem, 24
2A 10, 16
2B 7
2C 16
Chapter 3
Presentation
3A Prove that the maps in 3.6 are indeed linear.
Separately, finish the work for 3.7.
3B 2, 4
3C 7, 8
3D 2, 3
3E 1, 3
3F 10
Problem Set
3A 9, 11
3B 24-25 as one, 26
3C 14, prove or find a proof that function composition is
associative when it is defined (it is not difficult to prove) - use this to
rejustify 14.
3D 10, 19
3E 13, 20
3F What can you say about L(F, V)? How does it
compare to L(V, F)?
Chapter 5 (4 will be done in a one-day review without presentations or
problems)
Presentation
5A 9, 13
5B 4, 8
5C 1, 2
Problem Set
5A 18, 34
5B 16 or 17, 19
5C 12 is required, 13 or 14
Chapter 6
Presentation
6A 4, 19-20 as one
6B 4, Prove 6.25 by induction
6C 1, 5
Problem Set
6A 21, 26
6B 15,16
6C 9, 12
Chapter 7
Presentation
7A 9, Justify or fix 7.14
7B 1, 7 or 8
7C prove P_U is positive. prove R in 7.35 b=>c is
positive.
7D in the polar decomposition argument "verify that
S_1 is a linear map", 6
Problem Set
7A 10, 18
7B 6, 10, 13 for 2 extra credit points
7C 10, 14
7D 8, create your own example of finding a singular values
decomposition of a nonsquare nonzero matrix (not already in \Sigma
form), following the more material in ยง7.4.2
here
Chapter 8 + Differential Equations Supplement
Presentation
8A Give an example of an operator T on a vector space
V such that V is not the direct sum of null T and im T, 6.
8B 2, prove that R^2 = T in 8.33
8C 9, one of 4-6 (not 3, in fact)
8D 2, 3
2.8.1 1-2, 3-4 both as one
2.8.2 9-10 as one, 12
Problem Set
8A 21, 16-17 as one problem.
8B 7, 10
8C 11, 18
8D 6, 8
2.8.1 5-6 as one, 8
2.8.2 13, 14-15 as one