ASSIGNMENT 1MATH 2100
Due Monday, September 19 at the beginning of tutorial.
1. Take
W =
/5
a b
b
a
: a, b R
together with the usual operations of matrix addition and matrix multiplication in M22 (R) (the set of all 22 real matrices). Prove that W is a el
ASSIGNMENT 14MATH 2100
Due Thursday, February 27th at the beginning of tutorial.
1. Suppose G is a group and , Aut(G). Let = . Recall that the
composition of two bijections is again a bijection. Prove that Aut(G)
by showing that satises the homomorphic pr
MATH 2100 Assignment 12 Solutions
Problem 1. Show that |ai | = |aj | if and only if (|a|, i) = (|a|, j).
Proof. This comes almost immediately from Theorem 4.2. In fact, a chain of if-and-only-ifs is safe to
use here because its so short:
|a|
|a|
=
(|a|,
MATH 2100 Assignment 11 Solutions
Problem 1. If n, a Z+ and d = (n, a), show that the equation ax 1 (mod n) has a solution if
and only if d = 1.
Proof. ( = ): Let x be a solution to the equation. Then
ax 1
(mod n)
ax = 1 + kn
for some k Z
ax kn = 1
Thus 1
ASSIGNMENT 16MATH 2100
Due Thursday, March 13th at the beginning of tutorial.
1. Suppose G and H are groups and e is the identity of G. Prove that /5
cfw_e H is a subgroup of G H and that it is isomorphic to H.
2. Prove that Z8 Z2 is not isomorphic to Z4
ASSIGNMENT 11MATH 2100
Due Thursday, January 23rd at the beginning of tutorial.
1. Exercise 11 in Chapter 0. You may use Theorem 0.2 without proof.
/4
2. Suppose a, x, n Z, and n > 1. Prove that if ax mod n = 1 then
ar mod n = 1 where r = x mod n.
/3
3. S
ASSIGNMENT 17MATH 2100
Due Thursday, March 20th at the beginning of tutorial.
1. Let H be the subgroup of upper-triangular matrices in GL(2, R). De- /2
termine whether H is a normal subgroup or not.
2. Suppose G is a cyclic group and H
G. Prove that G/H i
ASSIGNMENT 18MATH 2100
Due Thursday, March 27th at the beginning of tutorial.
1. Suppose G is an abelian group. Prove by induction on n 1 that
(ab)n = an bn for all a, b G.
/3
2. Suppose G is an abelian group and n is a positive integer. Dene
: G G by (a
ASSIGNMENT 19MATH 2100
Due Thursday, April 3rd at the beginning of tutorial.
1. Suppose R is a ring. Dene the centre of R to be the set Z = cfw_z R :
za = az for all a R. Prove that the center is a subring. Is it also
an ideal?
/4
2. The quaternion ring i
ASSIGNMENT 20MATH 2100
Due Tuesday, April 8th at the beginning of lecture.
1. Prove part 6 of Theorem 15.1.
/4
2. Suppose n and m are are positive integers and dene : Mnn (Z)
Mnn (Zm ) by
[(A)]jk = [A]jk mod m, 1 j, k n
for all matrices A Mnn (Z). You ma
February 13, 2014
1
TEST 2 SOLUTIONSMATH 2100
1. Suppose G is a cyclic group. Provide the denition of a generator
of G.
Solution: A generator of G is an element a G such that
G = a = cfw_ak : k Z.
2. Suppose G is a nite group and a G. Dene the order of a.
CARLETON UNIVERSITY
MID-YEAR
EXAMINATION
December 12, 2013
DURATION:
3
No.
HOURS
Department Name & Course Number:
Course Instructor(s) Paul Mezo
of Students
Mathematics and Statistics MATH 2100
AUTHORIZED MEMORANDA
NO AIDS ALLOWED
Students MUST count the
MATH 2100 Assignment 13 Solutions
Problem 1. Given a group G and element a with |a| = 48, nd a divisor k of 48 such that
a. a21 = ak
b. a14 = ak
c. a18 = ak
Proof. All we need is Theorem 4.2:
a21 = a(48,21) = a3
a14 = a(48,14) = a2
a18 = a(48,18) = a6
Th
ASSIGNMENT 10MATH 2100
Due Thursday, December 5 at the beginning of tutorial.
1. Let R and
/5
A=
cos()
sin()
sin()
cos()
M22 (C).
Prove that LA L(C2 ) has eigenvalues ei and ei . (You do not need
to compute the eigenvectors!) Use this to prove that LA h
ASSIGNMENT 6MATH 2100
Due Thursday, November 7 at the beginning of tutorial.
In all of the exercises below V is a vector space over F = R or C and , is
an inner product on V .
1. Suppose v, w V . Prove that v = w if and only if
/4
v, y = w, y
for all y V
ASSIGNMENT 5MATH 2100
Due Thursday, October 17 at the beginning of tutorial.
1. Suppose U1 and U2 are subspaces of a nite-dimensional vector space
V over a eld F , V = U1 U2 , and T L(V ). Suppose further that
cfw_u1 , . . . , um is an ordered basis for
ASSIGNMENT 2MATH 2100
Due Thursday, September 26 at the beginning of tutorial.
1. Suppose U1 , . . . Um are subspaces of a nite-dimensional vector space V
over a eld F . Prove by induction on m 2 that
m
dim(U1 + + Um ) =
/6
m
dim(Uj )
j=1
dim(U1 + + Uk1
ASSIGNMENT 3MATH 2100
Due Thursday, October 3 at the beginning of tutorial.
1. Suppose V and W are nite-dimensional vector spaces over a eld F . Suppose further that B = cfw_v1 , . . . , vn is a basis for V and B = cfw_w1 , . . . , wm
is a basis for W .
ASSIGNMENT 4MATH 2100
Due Thursday, October 10 at the beginning of tutorial.
/4
1. Suppose V and W are vector spaces over a eld F , T L(V, W ) and
T 1 : W V is any function such that T T 1 = I and T 1 T = I.
Prove that T 1 is a linear map. (Hint: See proo
ASSIGNMENT 7MATH 2100
Due Thursday, November 14 at the beginning of tutorial.
In all of the exercises below V is a nite-dimensional vector space over F =
R or C, and , is an inner product on V .
1. Suppose U is a subspace of V and PU is the orthogonal pro
ASSIGNMENT 8MATH 2100
Due Thursday, November 21 at the beginning of tutorial.
In all of the exercises below V is a nite-dimensional vector space over F =
R or C, and , is an inner product on V .
1. Suppose F = R and let V = L(V, R). This is called the dua
ASSIGNMENT 12MATH 2100
Due Thursday, January 30th at the beginning of tutorial.
1. Prove the second part of Corollary 2 to Theorem 4.2, namely |ai | = |aj | /4
if and only if gcd(|a|, i) = gcd(|a|, j).
2. Prove the rst part of Corollary 3 to Theorem 4.2,
ASSIGNMENT 9MATH 2100
Due Thursday, November 28 at the beginning of tutorial.
In all of the exercises below V is a nite-dimensional vector space over F =
R or C, and , is an inner product on V .
1. Suppose F = C and T L(V ) satises T 3 = T 2 .
(a) Using t
ASSIGNMENT 12MATH 2100
Due Thursday, February 6th at the beginning of tutorial.
1. Exercise 72 in chapter 4 begins with a group element a such that |a| = 48.
In each case below it asks you to nd a divisor k of 48 such that
/3
(a) a21 = ak
(b) a14 = ak
(c)
ASSIGNMENT 15MATH 2100
Due Thursday, March 6th at the beginning of tutorial.
1. Let n be a positive integer. Prove that |Z : nZ| = n. (Hint: You must /4
nd n distinct cosets whose union is Z. Be sure to prove that your
cosets are indeed distinct.)
2. Supp
October 24, 2013
1
TEST 1 SOLUTIONSMATH 2100
Name:
/6
Student number:
1. Suppose F is a eld and U = cfw_a + bX 2 P(F ) : a, b F . Prove that U
is a subspace of the vector space of polynomials P(F ).
Solution: First, 0 = 0+0X 2 U . Second, suppose a1 +b1 X