Journal of Statistical Software
October 2015, Volume 67, Book Review 1.
Reviewer: James P. Howard, II
University of Maryland University College
Data-Driven Modeling & Scientific Computation: Methods for Complex
Systems & Big
Math 225 Homework #1: due 9/9/2016
Let (F, +, .) be a field as given in the lectures. As mentioned in class, certain
conclusions can be inferred from these properties.
Here are some examples with proofs, followed by some exercises for you.
(i) For all a F
Math 225 Homework #4: due 9/30/2016
We proved in class that every ideal of Z is principal , that is, it has the form
nZ for some integer n.
(a) If a, b Z define (a, b) to be cfw_ma + nb | m, n Z. Prove that (a, b) is an
(b) Now suppose that at l
Math 225 Homework #2: due 9/16/2016
1. Let R be a ring with 1. We say that an element in R is a unit if v R such
that u.v = v.u = 1.
(a) Use Q1(c) to prove that if u is a unit then so is u.
(b) Let R, S be rings with 1 and let : R S be a homomorphism of r
1. Let B be an n n matrix with real entries satisfying BB t = I = B t B. Then
in particular AB is a unitary matrix, and we can apply the spectral theorem to it:
there is a unitary matrix U such that U BU = D, and the elements (= eigenv
1. Quotient rings and Subrings.
In class we discussed the following theorems.
First Isomorphism Theorem. Let : R S be a homomorphism of rings with
kernel K. Then there is a homomorphism : R/K S defined by (r) = (r).
This homomorphism is one to one ont
Math 225: On the spectral theorem
Suppose N is an n n matrix acting on elements of Cn . The spectral theorem
says that N is normal if and only if there is a unitary matrix U such that U N U = D
where D is a diagonal matrix.
(1) Lets think about what the s
Math 225 Homework #5: due 10/7/2016
1. Let I be the ideal (X 2 + 3, 5) = cfw_a(X) (X 2 + 3) + b(X) 5 | a(X), b(X) Z[X]
in the ring Z[X]. Let F5 = Z/5Z and define a map
p : Z[X] F5 [X]
by a0 + a1 X + + an X n 7 a0 + a1 X + + an X n where a0 , a1 , . . . ,
1. The rings Z/nZ.
1.1 Let n > 1 be a positive integer. If a Z we write a for the set cfw_a + qn
Z | q Z.
Lemma. We have b a if and only if a, b have the same integer remainder when
divided by n. In particular a = r if a has remainder r when divided by n
1. Homomorphisms between rings.
1.1 This is an expansion of what was discussed in class. First, let (R, +, .) and
S, , ) be two rings. We do not assume that they are commutative, nor do we
assume that they have multiplicative identities.
We say that a fun
Math 225 Homework #3: due 9/23/2016
1. Let a, b be two positive integers at least one of which is non zero.
(a) Let k be a third positive integer. Prove from the definition that gcd(a, b) =
gcd(a, b + ka):
(i) Let d = gcd(a, b): show that d|a, d|b + ka.