HOMEWORK 1, MATH 150A, FALL, 2014
Due date: Friday, 10/10/2014, 10AM
Suggested readings: Artin, 2nd Edition, 1.1-1.5, 2.1-2.4
(1) Artin 1.7. Find a closed formula and prove it by induction on n:
n
1 1 1
0 1 1
0 0 1
Whats the order of the matrix as a
MAT 150A, Fall 2015
Solutions to Homework Assignment 5
Page 126. 2.3. Find all real 2 2 matrices that carry the line y = x to the line y = 3x.
Solution: Let
a b
c d
A=
then for y = x we get
a b
x
ax + bx
=
,
c d
x
cx + dx
so cx + dx = 3(ax + bx) for all x
HOMEWORK 1, MATH 150A, FALL, 2016
Due date: Monday, 9/26/2016, 10AM
Suggested readings: Artin, 2nd Edition, 1.1-1.5
(1) Artin 1.7. Find a closed formula and prove it by induction on n:
n
1 1 1
0 1 1
0 0 1
Whats the order of the matrix as an element of G
Math 150A, Lecture 3
Every element of order 2 is its own inverse.
Given a cycle (a1 a2 an ) we can write (not uniquely) it as a product of transpositions:
(a1 an )(a1 an1 ) (a1 a2 )
To every permutation, we can associate a matrix which is obtained by p
Math 150A, Lecture 2
GLn (R) is n n invertible matrices with real entries. SLn (R) GLn (R) is the set of
all matrices with determinant 1. It is also a group. It is an example of a subgroup.
Basically it is a subset of a group and is a group in its own ri
MAT 150A HW1 Solutions, Fall 2014
(1) Artin 1.7. Write
0
1
1 1 1
M = @0 1 1A .
0 0 1
To get some sense of what a closed formula for M n might look like, we probably ought to
compute some easy examples first:
0
1
1 2 3
M 2 = @0 1 2A ,
0 0 1
0
1
1 3 6
M 3 =
D-MATH
Prof. Brent Doran
Algebra I
HS 2013
Solution 1
Groups, Cyclic groups
1. Prove the following properties of inverses.
(a) If an element a has both a left inverse l and a right inverse r, then r = l, a
is invertible and r is its inverse.
Solution Sinc
Homework 3 M 373K solutions
Thanks to Mark Lindberg
We will use left and right inverses slightly more generally below than in the case of a
monoid. Namely, we will talk about left and right inverses of functions f : X Y and of
non-square matrices. A left
Math 150A, Lecture 1 Summary
Definition of a group: A group is a set G with an operation ? such that
(0) the set is closed under the operation.
(1) ? is associative.
(2) the identity e is in the set.
(3) every element has an inverse.
e is called an ident
HOMEWORK 4, DUE 10/17/2016, MONDAY 10AM, MATH 150A
Suggested Readings: Sections 2.5, 2.6.
(1) (a) Show (but not turn in) (R , ), where R is the set of non-zero real numbers is
a group. (b) Show that f : R R defined by f(x) = x3 is an isomorphism. (c)
Show
MAT 150A HW2 Solutions, Fall 2014
(1) (a) Let e1 =
and e2 = (0, 1)T be the standard basis vectors of R2 . Let A 2
GL2 (R) be a matrix which fixes the x-axis. In other words, A is an invertible 2 2 matrix
with entries in R with the special property that
(1
MAT 150A, Fall 2015
Solutions to Homework Assignment 4
6.2. (25 points) Describe all homomorphisms : Z Z. Determine which are injective,
which are surjective and which are isomorphisms.
Solution: If is an homomorphism then (0) = 0, and for all x, y (x + y
MAT 150A, Fall 2015
Solutions to Homework Assignment 3
8.5. (25 points) A nite group contains an element x of order 10 and an element y of
order 6. What can be said about the order of G?
Solution: By Lagrange theorem, the order of G is divisible both by 1
HOMEWORK 2, DUE 10/20/2014, MONDAY 10AM, MATH 150A
Suggested Readings: Sections 1.5, 2.1-2.6
(1) (a) Determine the subset of GL2 (R) that xes the x-axis. (Notice that each element
M GL2 (R) acts on x R2 as M.) (b) Show that the set forms a subgroup.
x
(2)
Syllabus
Math 150A Modern Algebra
Fall, 2014
Instructor: Kevin Luli; Oce: Math Science Building (MSB) 3136; email: kluli@math.ucdavis.edu
Lead Teaching Assistant: TBA
Lecture Time and Location: ART 204 MWF 1000-1050 AM
Monday
3:10 PM - 4:10 PM
Instructor
Math 150A, Lecture 4
Let An be all the even permutations of Sn . Ex. A3 = cfw_e, (123), (132). An is a group
and is called the alternating group. We will see that |An | = n!/2.
Subgroup: Let S be a subset of a group G. S is a subgroup (i) it is closed u
10-5-2016
Let G be a group. x; y 2 G are said to be conjugate if there exists g 2 G
such that y = gxg 1 .
Conjugation by an element g is a homomorphism.
Two cycles in Sn are conjugate if and only if they have the same cycle
type (length).
Proposition: Let
Math 150A, Lecture 5
Last Let a, b be positive integers. Then aZ + bZ = dZ for some integer d. Last time
we claimed that d = gcd(a, b), i.e. d divides both a and b and if k divides a and b
then k divides d.
Corollary: gcd(a, b) = ma + nb, for some integ
Math 150A, Lecture 8
Def. Two groups are isomorphic if there is an isomorphism between them. If G is
isomorphic to H, we write G ' H.
Examples: (R, +) ' (R+ , ).
Theorem: Any two cyclic groups of order n are isomorphic.
Properties of homomorphisms: (1
Math 150A, Lecture 6
Every cyclic group is abelian.
a
Given a subset S of a group G, < S >= cfw_sai 1 sj j : si , , sj S, a1 , , aj Z is
called the group generated by S. It is the smallest group containing S as a subset. If
S has only finitely finitely
Jazzmin Yuson
MAT 150 SSII
Homework 2
2.1: For any prime p, show that Cp2 and Cp Cp are not isomorphic. (Hint:
There are many bijections between them but one can show that any bijection
: Cp2 ! Cp Cp satisfying the property of a homomorphism that the iden
Jazzmin Yuson
MAT 150 SSII
Homework 2
2.1: For any prime p, show that Cp2 and Cp Cp are not isomorphic. (Hint:
There are many bijections between them but one can show that any bijection
: Cp2 ! Cp Cp satisfying the property of a homomorphism that the iden
MAT 150A, Fall 2015
Solutions to Homework Assignment 2
2.4. (25 points) Determine if the following subset H is a subgroup of G:
b) (8 points) G = R and H = cfw_1, 1.
Solution: This is a subgroup, in fact, H is a cyclic subgroup of R
generated by (1).
c) (
2
To determine the order of M, we need to find the smallest positive n such that M n
is the identity matrix I; if no such n exists, then the order of M is infinity. Writing
n is never 0. Thus M n can never
M n = (Minj )1i, j3 , notice, for example, that t