MAT
246
University of California, Davis
Winter 2011
Homework 4
due Feb 25, 2011 , in class
I'm going to list several to start on and think about.
I'll pick out some subset to write up nicely and hand in.
Without a reader, I'm not sure how many problems will get
graded. But you should do many hw problems to learn the material,
regardless of whether they are collected!
And if you've taken the time to write up more than
what's required to hand in, feel free to turn that in
too--especially if I may be lazy and use that as the
posted solution (I'll certainly factor those extra efforts
into your grade).
Make sure your homework paper is legible, stapled,
and has your
name
clearly showing on each page.
Also, for alphabetizing purposes, in the upper LeftHand corner of your paper
please put the
first letter of your last name,
large and circled.
No late homeworks will be accepted! (but good arguments to shift the due date might be)
Collaboration with classmates is encouraged, so long as you write up
your solutions in your own words. Please note on the front page of
your HW who you have collaborated with.
° Read EC II, Sections 7.15, 7.16, 7.17, and A1 Appendix
(maybe 3.4 and 3.5 of Sagan too)
(that list may well change as I fine tune the HW)
1.
7.30ab (p 459)
2.
7.35ab (optional c) (p 460)
Please
WRITE
this one up to hand in.
3.
7.36 (p 460)
4.
7.41 (p 461)
(you might want to read 7.42 for interest)
5.
Please
WRITE
this one up to hand in.
Let A be an n × n matrix. Let cA(t) be the characteristic
polynomial of A. Let ak = tr(Ak). Write the coefficients
of cA(t) in terms of the ak.
I'll start you out-- the leading coeff is 1 which we can think of as 1/n*a0;
the coeff of tn-1 is +/- a1. And so on...
6.
Show Δ is co-associative. That is, as maps
(1 ⊗ Δ) ° Δ = (Δ ⊗ 1) ° Δ
where Δ : Λ -- > Λ ⊗ Λ = Λ(x) ⊗ Λ(y)
is given by Δ(f) = f(x,y).
(1 is the identity map above.)
(Think about how co-associativity looks/feels like the associativity
of multiplication/usual product if you were to write that out formally.)
7.
Recall Young's lattice of partitions where we'll write μ ⊂ λ
if μ is contained in λ and the skew shape λ/μ contains 1 box.
Define operators D and U by
D(λ) = ∑μ ⊂ λ μ and
U(λ) = ∑λ ⊂ ν ν .
Show
(a)
DU - UD = I
(look familiar?!!)
and
(b)
DUk = k Uk-1 + UkD.
(c)
Derive anew that ∑|λ|=n ( fλ)2 = n!