MAT 150A
Homework #8
Solutions
Shawn Witte
December 3, 2013
1
Let G be the set of integer sequences (ai ) for
i=1
which there is some N N such that ai = 0 for
all i > N .
(a) Show that G is an abelian group under the operation (ai ) +
i=1
(bi ) = (ai + bi
MAT 150A
Homework #7
Solutions
Shawn Witte
December 1, 2013
1
Show by explicitly stating the isomorphisms and
their inverses that
(a) C S 1 R
=
>0
Recall that all non-zero complex numbers can be uniquely written in
polar form with radius r R>0 and angle 0
MAT 150A
Homework #9
Solutions
Shawn Witte
December 6, 2013
1
Artin 6.3.6
(a) We can write this rotation as sx = t(1,1) /2 t(1,1) x which is translating
the point (1, 1) to the origin, rotating /2, then translating the origin
back to (1, 1). Using the for
MAT 150A
Homework #6
Solutions
Shawn Witte
December 3, 2013
1
(a) List all elements of (Z/Z21) . What is the order of this group?
This is the multiplicative group of all numbers between 1 and 21 which
are coprime with 21. These numbers are 1, 2, 4, 5, 8,
MAT 150A
Homework #5
Solutions
Shawn Witte
December 1, 2013
1
For which integers n does 2 have a multiplicative
inverse in Z/Zn?
Suppose x is the inverse of 2 in Z/Zn. Then 2x = 1 mod n which means
2x 1 = kn for some k. This means that kn is odd, which im
Math 150A Practice midterm solutions
(1) Question 1. Prove or disprove each statement.
(a) True, since S 1 is the kernel of the modulus map C R .
(b) False, since if g = e then lg (e) = g, and not the identity.
(c) True since Cg (ab) = g(ab)g 1 = (gag 1 )
Math 150A Sections A01-A02 Homework 3 Solutions
(1) Artin 2.5.1:
Suppose : G G is surjective.
If G is abelian, then ab = ba for all a, b G. Given c, d G , nd a, b G such that
c = (a), d = (b). Then cd = (a)(b) = (ab) = (ba) = (b)(a) = dc, so G is
abelian.
Math 150A Solutions to practice problems
(1) Artin 6.9.1:
Let G be the group of rotational symmetries of a cube. The cube has 8 vertices, all
of degree 3, so if v is a vertex then |Gv | = 3. Since G acts transitively on the set of
vertices, we have that |
Math 150A Sections A01-A02 Practice nal solutions
(1) Question 1.
(a) True, there is always the trivial homomorphism G G : g e.
(b) True, if C2 = g and : C2 C2 is an automorphism then (e) = e, and since
is a bijection we have (g) = g. So is the identity
Math 150A Sections A01-A02 Homework 4 Solutions
(1) Artin 2.7.3:
Let Q = R R , where R and R are equivalence relations on S. Since (a, a) R
and (a, a) R for all a S, we have (a, a) Q for all a S, and Q is reexive.
If (a, b) Q, then (a, b) R and (a, b) R ,
Math 150A Sections A01-A02 Homework 2 Solutions
(1) (a) Let x = a + bi and y = c + di. Then
|xy|2 = |ac bd + (ad + bc)i|2 = (ac bd)2 + (ad + bc)2
= a2 c2 + b2 d2 + a2 d2 + b2 c2 = (a2 + b2 )(c2 + d2 ) = |x|2 |y|2 .
(b) Let x be the conjugate of x. If x =