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 )| =
x2 + + x2 .
n
1
Example 1.1. The identity transfo
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
Please print the assignment (three pages), write your
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
Student Number
Please print the assignment (ve pages),
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_(x, y) R2 | x + y = 2
L0 = cfw_(x, y) R2 | x + y = 0
1
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) Find the formula for RH (v), the reection of v in H.
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 reversing?
c) Is f a reection, a rotation or a spin reect
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.
for v 2 R2 .
a
2 R2 for which f (u) f (v) = u v for ev
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
Stillwell [3] section 5.7 and 5.8.
Denition 8.1. An isometry
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 v for every u, v R2 .
c) (Bonus) Show that if g : Rn R
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 (v) = u v for every u, v 2 R2 .
2
1 0
cos + sin2
0
t
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) f (v) = u v for every u, v 2 R2 .
2. [Exercise 1.2.1] a
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 line in R2 . That is , show that
L = cfw_v 2 R2 | f (v
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 v and
let 0 be the line through 0 with direction v. Th
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 complex number. Let z = x + iy be a complex number with rea
2 it? Eggg Emma-Jae %” a
1. Let H1 and H2 be the two planes
H1:{(3:,y,z) f 13:20} and H2={(3:}y,z) I a:+y=:1}.
a) Find the formula for Rgz R31.
b) Find the vector parametric form for the line L 2: H1 0 H2, and give a unit direction vector for
K ‘ L.
c)
Assignment 1, MAT 2355
September 21, 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,
prove it. If not, show a counterxample (e.g. two points P, Q such th
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
(Bonus) 5
0
Total
PLEASE READ THESE INSTRUCTIONS CAREF
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 phones is not permitted.
3. The correct answer require
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 phones is not permitted.
3. The correct answer require
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 following spherical triangles (i.e. nd all vertex and arc angl
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 following spherical triangles (i.e. nd all vertex and arc angl
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,
prove it. If not, show a counterxample (e.g. two points
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 functions f : S 2 S 2 , determine if f is an isometry
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 functions f : S 2 S 2 , determine if f is an isometry o
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 functions f : Rn Rn , determine if f is an isometry. I
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 functions f : Rn Rn , determine if f is an isometry. If
Assignment 2, MAT 2355
October 10, 2015
Exercise 1.
1. (10 points) Consider the following isometries f, g : R3 R3 .
f (x, y, z) = (1/3)x 2(y + z)/3 + 2, (1/3)y 2(x + z)/3 + 2, (1/3)z 2(y + x)/3 + 2)
g(x, y, z) = (y + 1, z + 1, x + 2)
Determine if g, f and
Assignment 2, MAT 2355
October 6, 2015
Exercise 1.
1. (10 points) Consider the following isometries f, g : R3 R3 .
f (x, y, z) = (1/3x 2(y + z)/3 + 2, 1/3y 2(x + z)/3 + 2, 1/3z 2(y + x)/3 + 2)
g(x, y, z) = (y + 1, z + 1, x + 2)
Determine if g, f and f g p