(The “conjugation rewrite lemma”.) Let σ and τ be
permutations.
(a) Show that if σ maps x to y then στ maps τ(x) to
τ(y).
(b) Suppose that σ is a product of disjoint cycles. Show that
στ has the same cycle structure as
σ; indeed, wherever (... x y ...) occurs in σ, (... τ(x) τ(y)
...) occurs in στ.
Let T∈ L(V), and let p ∈ P(F) be a polynomial. Show that if p(λ)
is an eigenvalue of p(T), then λ is an eigenvalue of T. Under the
additional assumption that V is a complex vector space, and
conclude that {μ | λ an eigenvalue of p(T)} = {p(λ) | λan
eigenvalue of T}.
Let R be the relation on Q defined by a/b R c/d iff ad=bc. Show
that R is an equivalence relation. Describe the elements of the
equivalence class of 2/3.
Let μ=E(X), σ=stanard deviation of X. Find the probability P(μ-σ
≤ X ≤ μ+σ) if X has... (Round all your answers to 4 decimal
places.)
a. ... a Binomial distribution with n=23 and p=1/10
b. ... a Geometric distribution with p = 0.19.
c. ... a Poisson distribution with λ = 6.8.
Let gcd(a, p) = 1 with p a prime. Show that if a has at least
one square root, then a has exactly 2 roots. [hint: look at
generators or use x^2 = y^2 (mod p) and use the fact that ab = 0
(mod p) the one of a or b must be 0(why?) ]
Let p be a prime and d a divisor of p-1. show that the d th
powers form a subgroup of U(Z/pZ) of order (p-1)/d. Calculate this
subgroup for p=11, d=5; p=17,d=4 ;p=19,d=6