MAT 150B
University of California
Winter 2018
Homework 4
due February 9, 2018
1. Let Q U2 (C) and = det Q. Show that if is a square root of , then
= 1, and det(Q) = 1.
2. Suppose that A, A0 SU2 (C) a
MAT 150B
University of California
Winter 2018
Homework 3
due February 2, 2018
1. Find a subgroup of GL2 (R) which is isomorphic to C .
2. (Artin 9.1.2) A matrix P is orthogonal if and only if its colu
MAT 150B
University of California
Winter 2018
Homework 2
due Friday January 26, 2018 in class
1. Let G = GLn (R) and let S = Mn (R) be the set of all n n matrices over
R. Show that the map G S S given
MAT 150B
University of California
Winter 2018
Homework 1
due Friday January 19, 2018 in class
1. (cf. Artin 8.1.1)
(a) Prove that every real square matrix is the sum of a symmetric matrix
and a skew-s
Math 150b: Modern Algebra
Homework 9 Solutions
Exercise 12.1.3.(a)
Solution. Let R be a Bezout domain and let m, n R be relatively prime. There will exist
u, v R such that um + vn = 1. Suppose we are
Math 150b: Modern Algebra
Homework 10 Solutions
Exercise 12.4.1(b)
Solution. The irreducible polynomials of degree less than five are listed in formula 12.4.4
of Artin. One immediately sees x and x +
Math 150b: Modern Algebra
Homework 4 Solutions
Exercise 3.7.1.
Solution. We have R = he1 , e2 , . . .i is the set of all sequences with only finitely many
nonzero terms. If we are allowed to add scala
Math 150b: Modern Algebra
Homework 7 Solutions
Exercise 11.3.5.(a).
Solution. We prove the Leibniz rule ( f g)0 = f 0 g + f g0 for polynomial multplication. Letting
m1
n1
n
k
0
k
0
k
f = m
k=0 ak x an
Math 150b: Modern Algebra
Homework 2 Solutions
Exercise 3.2.1
Solution. This exercise involves checking that the subset Q( 2) = cfw_a+b 2 | a, b Q C
is a field under the operations of addition and mul
Math 150b: Modern Algebra
Homework 5 Solutions
Exercise 8.2.1.
Solution. Let A be a real, positive definite, symmetric bilinear form on Rn and ai j its associated matrix in the standard basis. We will
Math 150b: Modern Algebra
Homework 8 Solutions
Exercise 11.7.2.
Solution. Since R is a domain, by definition it has no zero divisors. Assume we have
p(x), q(x) R[x] such that p(x)q(x) = 0. If p(x) = a
Math 150b: Modern Algebra
Homework 3 Solutions
Exercise 3.5.4.
Solution. For part (a), we need to create a bijection between bases for F2p and elements
of GL2 (F p ). Any basis cfw_v1 , v2 may be obt
Math 150b: Modern Algebra
Homework 1 Solutions
Exercise 3.2.2.
Solution. We first note that since 5 is not a multiple of any of the given primes, we expect an
inverse to exist. As the primes p = 7, 11