T HE U NIVERSITY OF S YDNEY
S CHOOL OF M ATHEMATICS AND S TATISTICS
Solutions to MATH2968 Assignment 2
MATH2968: Algebra (Advanced)
Semester 2, 2013
1. Let p = 2 be a prime number and consider
G=
a b
0 1
a, b F p , a = 0 .
You may assume that G is a subgr
Linear Algebra
Lecture Notes for MATH2961
Daniel Daners
School of Mathematics and Statistics
The University of Sydney
Semester 1, 2012
2
Lecture Notes for MATH2961
Linear Algebra (Advanced)
Daniel Daners
School of Mathematics and Statistics
The University
2
Cosets
Ill start this section with a brief review of equivalence relations and partitions. They will carry us through this
section, which is devoted to Lagranges theorem and a consequence in number theory. The last subsection gives
a nice geometric appl
The University of Sydney
School of Mathematics and Statistics
Solutions to Quiz 2 (Week 11)
MATH2968: Algebra (Advanced)
Semester 2, 2012
Name:
SID:
This quiz lasts 45 minutes.
There are a total of 36 marks.
No calculators are allowed. Answers must be wri
The University of Sydney
School of Mathematics and Statistics
Solutions to Quiz 2 (Week 12)
MATH2968: Algebra (Advanced)
Semester 2, 2013
Name:
SID:
This quiz lasts 45 minutes.
There are a total of 36 marks.
No calculators are allowed. Answers must be wri
The University of Sydney
School of Mathematics and Statistics
Solutions to MATH2968 Assignment
MATH2968: Algebra (Advanced)
Semester 2, 2013
a b
| a, b R, a > 0 and let H =
0 1
(a) Show that G is a subgroup of GL(2, R).
c 0
0 1
1. Let G =
| c R, c > 0 .
S
The University of Sydney
School of Mathematics and Statistics
MATH2968 Assignment
MATH2968: Algebra (Advanced)
Semester 2, 2012
This assignment is worth 5% of your assessment for MATH2968. It is due before 4pm
on Wednesday 14 August (Week 3). It must be h
Abstract Algebra:
Supplementary
Lecture Notes
JOHN A. BEACHY
Northern Illinois University
1995
Revised, 1999, 2006
ii
To accompany
Abstract Algebra, Third Edition
by John A. Beachy and William D. Blair
ISBN 1577664344, Copyright 2006
Waveland Press, Inc.
Jordan Normal Form
Alastair Fletcher
January 5th 2003
1
Introduction
Any matrix over C (or any algebraically closed eld, if that means anything
to you!) is similar to an upper triangular matrix, but not necessarily similar
to a diagonal matrix. Despite th
8038
Semester 2, 2011
The University of Sydney
Faculties of Arts, Economics, Education,
Engineering and Science
MATH2968: Algebra (Advanced)
Lecturer: Anthony Henderson
Time allowed: 2 hours plus 10 minutes reading time
This booklet contains 4 pages.
This
5
Functions and
Graphs
TERMINOLOGY
Arc of a curve: Part or a section of a curve between two
points
Even function: An even function has line symmetry
(reflection) about the y-axis, and f ] - x g = - f ] x g
Asymptote: A line towards which a curve approache
T HE U NIVERSITY OF S YDNEY
S CHOOL OF M ATHEMATICS AND S TATISTICS
Practice Class 3
MATH2968: Algebra (Advanced)
Main topics: Actions of the symmetic group
1. Action on the set of subsets
If S cfw_1, . . . , n and g 2 Sym(n), then Ill write g(S) = cfw_g(
The University of Sydney
School of Mathematics and Statistics
Aut(Q8 )
MATH2968: Algebra (Advanced)
The automorphism group of a group is the set of all structure preserving bijections from
the group to itself, and hence can be thought of as the symmetry g
10
The Quadratic
Function
TERMINOLOGY
Axis of symmetry: A line about which two parts of
a graph are symmetrical. One half of the graph is a
reflection of the other
Coefficient: A constant multiplied by a pronumeral in an
algebraic term e.g. in ax3 the a i
8
Introduction to
Calculus
TERMINOLOGY
Composite function: A function of a function. One
function, f (x), is a composite of one function to another
function, for example g(x)
Continuity: Describing a line or curve that is unbroken
over its domain
Continuo
3
Equations
TERMINOLOGY
Absolute value: the distance of a number from zero on a
number line.
pronumeral that is solved to find values that make the
statement true e.g. 2x - 3 2 4
Equation: A mathematical statement that has a
pronumeral or unknown number a
7
Linear Functions
TERMINOLOGY
Collinear points: Two or more points that lie on the same
straight line
Interval: A section of a straight line including the end
points
Concurrent lines: Two or more lines that intersect at a
single point
Midpoint: A point l
4
Geometry 1
TERMINOLOGY
Altitude: Height. Any line segment from a vertex to the
opposite side of a polygon that is perpendicular to that side
Polygon: General term for a many sided plane figure. A
closed plane (two dimensional) figure with straight sides
11
Locus and the
Parabola
TERMINOLOGY
Axis: A line around which a curve is reflected e.g. the axis
of symmetry of a parabola
Latus rectum: A focal chord that is perpendicular to the
axis of the parabola
Cartesian equation: An equation involving two variab
2
Algebra and
Surds
TERMINOLOGY
Binomial: A mathematical expression consisting of
two terms such as x + 3 or 3x - 1
Binomial product: The product of two binomial
expressions such as (x + 3) (2x - 4)
Expression: A mathematical statement involving numbers,
9
Properties of
the Circle
TERMINOLOGY
Arc: Part of a curve, most commonly a portion of the
distance around the circumference of a circle
Chord: A straight line joining two points on the
circumference of a circle
Concentric circles: Circles that have the
The University of Sydney
School of Mathematics and Statistics
Practice Class 2
MATH2968: Algebra (Advanced)
Main topics: Groups actions, the orbit-stabiliser relation
1. Let G = GL2 (R) act on X = R2 in the usual way: A x = Ax for A GL2 (R) and
x R2 a col