M AT2355 I NTRODUCTION TO G EOMETRY F IRST A SSIGNMENT
P ROFESSOR : H ADI S ALMASIAN
Deadline: September 19 at 2:00p.m.
The drop box will be available on September 19 from 8:30a.m. until 2:00p.m.
Name
Solutions to Assignment 1, MAT 2355 Due: 4:00 p.m, September 20.
1.
Let 2 R, set R =
sin
cos
cos
sin
and x v0 =
a
2 R2 , and dene
b
f : R2 ! R2
by
f (v) = R v + v0 ,
a) Show that f is an isometry.
MAT 2355 Fall 2013 Mid-Term
30th October, 2013.
Duration: 80 minutes
Instructor: Barry Jessup
0
Family Name:
First Name:
6
sin
0
3
1
2
p
2
2
p
3
2
2
1
4
Student number:
cos
1
p
3
2
p
2
2
1
2
3
4
1
2
Midterm, MAT 2355
November 8, 2015
1. You have 80 minutes to complete this exam.
2. This is a closed book exam, and no notes of any kind are permitted. Calculators are not
allowed, and the use of cell
ISOMETRIES OF Rn
KEITH CONRAD
1. Introduction
An isometry of Rn is a function h : Rn Rn that preserves the distance between vectors:
|h(v) h(w)| = |v w|
for all v and w in Rn , where |(x1 , . . . , xn
MAT 2355 Practice Problems.
cos
sin
x0
1. Let 2 R, set R =
and x v0 =
sin cos
y0
2
2
f : R ! R by f (v) = R v + v0
2 R2 , and dene
a) Show that f is an isometry.
b) Show that if v0 = 0, then f (u)
M AT2355 I NTRODUCTION TO G EOMETRY F IFTH A SSIGNMENT
P ROFESSOR : H ADI S ALMASIAN
Deadline: December 3 at 3:30p.m.
The drop box will be available as of 8:30a.m. on the due date.
Name
Student Number
MAT 2355 Fall 2013 Final Exam
11-December, 2013.
Duration: 3 hours
Instructor: Barry Jessup
1
2
Family Name:
3
First Name:
4
5
Student number:
6
7 (Bonus)
Total
PLEASE READ THESE INSTRUCTIONS CAREFULL
Midterm, MAT 2355
November 8, 2015
1. You have 80 minutes to complete this exam.
2. This is a closed book exam, and no notes of any kind are permitted. Calculators are not
allowed, and the use of cell
Solutions to nal exam, MAT 2355
December 21, 2015, Fall session
Question 1. Consider the following four lines in the plane:
L1 = cfw_(x, y) R2 | x 3y = 0cfw_
L2 = cfw_(x, y) R2 | 3x y = 0
L3 = cfw_
MATH 130
SPRING, 2012
ISOMETRIES OF THE EUCLIDEAN PLANE
MARCH 4TH, 2012
M. J. HOPKINS
8. Isometries of the plane
Much of this is in Ryan [1], in chapters 1 and 2. I also like the presentation in
Still
Problem Solutions -1
cos sin
sin
cos
2
2
f : R R by f (v) = R v + v0
1. Let R, set R =
and x v0 = (x0 , y0 ) R2 , and dene
a) Show that f is an isometry.
b) Show that if v0 = 0, then f (u) f(v) = u
MAT 2355 Practice Problems.
cos
sin
x0
1. Let 2 R, set R =
and x v0 =
sin cos
y0
f : R2 ! R2 by f (v) = R v + v0
2 R2 , and dene
a) Show that f is an isometry.
b) Show that if v0 = 0, then f (u) f
MAT 2355 Assignment 2 - Solutions
1. Let 2 R and f : R ! R be dened by f (v) =
2
2
cos
sin
sin
v.
cos
a) Show that f is an isometry of R2 .
b) Show that the set L of points which are xed by f is a
MAT2355-WEEK5
MOHAMMAD BARDESTANI
Contents
1. Rotation in R2
2. Translation in R2
1
3
1. Rotation in R2
Let us rst recall briey the previous lecture. Let = P + [v] be a line with unit direction vector
MAT2355-WEEK5
MOHAMMAD BARDESTANI
Contents
1.
2.
3.
Complex numbers
Isometries in C
Orthogonal matrices and isometries of Rn
1
2
3
1. Complex numbers
In this section we recall some basic facts of comp
MAT2355-WEEK5
MOHAMMAD BARDESTANI
Contents
1.
Structure of the isometry group
1
1. Structure of the isometry group
We saw in the previous note that every rotation and every translation is a compositio
Memes; a?) mam $96»
1. 3.) Express the translation of R2 deﬁned by f (say) = (:c -'3 ,3; +4) as a product of two reﬂections,»
g and sketch your solution (with each line labelled) on a set of labelled
MAT 2355 Assignment 5 Solutions
2
0
41
1. Let A =
0
3
0 1
0 0 5 and consider f : S2 ! S2 dened by f (x) = Ax.
1 0
a) Show that f is an isometry of S2 .
b) Is f orientation preserving or orientation re
2
2. Let P = Q be distinct points in Rn . Let H be the hyperplane
H = cfw_v Rn | |v P | = |v Q|
a) Find 0 = a Rn and b R such that
H = cfw_v Rn | a v = b.
(Hint: see solutions to assignment 1, Q3a.)
b
Assignment 5, MAT 2355
December 10, 2015
To get a perfect score in this assignment you only need to obtain 100 points. Extra points will
be counted up to 120/100.
Exercise 1. Solve each of the followi
Assignment 5, MAT 2355
November 23, 2015
To get a perfect score in this assignment you only need to obtain 100 points. Extra points will
be counted up to 120/100.
Exercise 1. Solve each of the followi
Assignment 1, MAT 2355 with solutions
October 6, 2015
This assignment is due Monday, September 28.
Exercise 1. For each of the following functions f , determine if f is an isometry or not. If it is,
p
Assignment 4, MAT 2355
November 26, 2015
To get a perfect score in this assignment you only need to obtain 100 points. Extra points will
be counted up to 120/100.
Exercise 1. For each of the following
Assignment 4, MAT 2355
November 8, 2015
To get a perfect score in this assignment you only need to obtain 100 points. Extra points will
be counted up to 120/100.
Exercise 1. For each of the following
Assignment 3, MAT 2355
November 16, 2015
To get a perfect score in this assignment you only need to obtain 100 points. Extra points will
be counted up to 120/100.
Exercise 1. For each of the following
Assignment 3, MAT 2355
October 31, 2015
To get a perfect score in this assignment you only need to obtain 100 points. Extra points will
be counted up to 120/100.
Exercise 1. For each of the following
MAT2362: Foundations of Mathematics
Solutions to Homework Assignment #1.
1. Construct the truth-table for the following propositional formulas. In each case, explain
whether the formula is a tautology