Solutions to Abstract Algebra (Dummit and Foote 3e)
Chapter 2 : Subgroups
Jason Rosendale
jason.rosendale@gmail.com
February 11, 2012
This work was done as an undergraduate student: if you really dont understand something in one of these
proofs, it is ver
Fall 2004 AMS526: Homework 3
Due on Oct. 13, 2004
1. Let A be an m by n (m > n) matrix of full rank and cfw_1 , . . . , n be the singular values of A in the
non-increasing order. The pseudoinverse of A is written as A+ = (A A)1 A . What are the singular
MIT 18.335, Fall 2007: Homework 2, Solutions
1. (Trefethen/Bau 6.3)
Consider the SVD factorization A = U V , which gives that
A A = V U U V = V 2 V
Here, we see that the singular values of A A are the squares of the singular values of A. The
n n matrix A
Solutions to Abstract Algebra (Dummit and Foote 3e)
Chapter 1 : Group Theory
Jason Rosendale
jason.rosendale@gmail.com
February 11, 2012
This work was done as an undergraduate student: if you really dont understand something in one of these
proofs, it is
Chapter 1.6, Page 37
Problem 2:
(a) Prove that x is in the Cantor set i x has a ternary expansion that
uses only 0s and 2s.
(b) The Cantor-Lebesgue function is dened on the Cantor set by writing xs ternary expansion in 0s and 2s, switching 2s to 1s, and
r
Math 121 Homework, Week 8
Michael Von Kor
December 12, 2004
5.2: 2(d,f),7; 5.4: 13,14,15,17,18,21,36,38.
Problem 1. For each of the following matrices A Mnxn (R), test A for diagonalizability, and if A is diagonalizable, nd an invertible matrix Q and a
di
MATH2040 HW1 Solution
Tsang Chi Shing, Sidney Due Date: 20-1-2011
Sec. 5.1: 3(b), 3(c), 4(e), 6, 7 3(b) (i) The characteristic polynomial of A is: -t -2 -3 det(A - tI) = -1 1 - t -1 = -(t - 1)(t - 2)(t - 3) 2 2 5-t Therefore the eigenvalues of A are 1, 2
MATH 2040A ssignment 6 Solution
Tsang Chi Shing, Sidney Due Date: 26-2-2009
Sec. 6.1: 9, 11, 13, 15, 17 Sec. 6.1 9) Let = v1 , ., vn be a basis for a finite-dimensional inner product space V (dim V = n). (a) Let x = x
2 n j=1
aj vj V . If x, z = 0 for all
MAT2040 Assignment 4 Solution
Tsang Chi Shing, Sidney February 12, 2011
Sec. 5.2: 17(a), 18(a), 20, 21, 22 17(a) Suppose that T and U are simultaneously diagonalizable linear operators on a finite-dimensional vector space V , then there exists an ordered
MAT2040 Assignment 3 Solution
Tsang Chi Shing, Sidney February 12, 2011
Sec. 5.2: 2(b), 2(e), 2(f), 3(a), 3(c), 7, 8, 10, 12, 13 2(b) The characteristic polynomial of A is: det(A - tI) = 1-t 3 3 1-t = (t + 2)(t - 4)
Therefore the eigenvalues of A are -2 a
Math 121 Homework, Week 2
Michael Von Kor
October 14, 2004
1.4: 14,15; 1.5: 8,13,18; 1.6: 14,15,17,21; 1.7: 3
Problem 1. 1.4 14. Show that if S1 and S2 are arbitrary subsets of a vector
space V, then span(S1 S2 ) = span(S1 ) + span(S2 ).
Proof. Suppose s
Math 121 Homework, Week 5
Michael Von Kor
November 2, 2004
2.6: 4,7,10,14,15,20 (nite-dimensional case only) ; 3.2: 6(d,e),14,17
Problem 1. 2.6 4. Let V = R3 , and dene f1 , f2 , f3 V as follows:
f1 (x, y, z) = x 2y, f2 (x, y, z) = x + y + z, f3 (x, y, z)
Math 121 Homework, Week 4
Michael Von Kor
October 22, 2004
2.3: 9,11,12,16 ; 2.4: 14,16,23; 2.5: 6(b,c),7,8
Problem 1. 9. Find linear transformations U, T : F 2 F 2 such that U T = T0
but T U = T0 . Use your answer to nd matrices A and B such that AB = 0
Math 121 Homework, Week 3
Michael Von Kor October 21, 2004
1.6: 31,33; 2.1: 9,14,15,19,35; 2.2: 3,5,13 Problem 1. 1.6 31. Let W1 and W2 be subspaces of a vector space V having dimensions m and n, respectively, where m n. (a) Prove that dim(W1 W2 ) n. (b)
Math 121 Homework, Week 1
Michael Von Kor October 1, 2004
Problem 1. A real-valued function f dened on the real line is called an even function if f (x) = f (x) for each real number x. Prove that the set of even functions dened on the real line with the o
Suggested Solutions to HW 10
December 1, 2010
1. (6) If B can be obtained from A by an element row(column) operation. by Th3.1, we have B =
t
t
t
t
E1 A(B = AE2 ), where E1 , E2 are elementary matrices, so B t = At E1 (B t = E2 At ). Since E1 , E2
are sti
MAT2040B Assignment 2 Solution
Tsang Chi Shing, Sidney January 30, 2011
Sec. 5.1: 8, 9, 11, 12, 13, 17, 21, 22 8 (a) Let T be a linear operator on a finite-dimensional vector space V . T is invertible N (T ) = cfw_0 (N (T ) is the null space of T in V ) T
MAT2040B Assignment 5 Solution
Tsang Chi Shing, Sidney February 15, 2011
Sec. 5.4: 2(a), 2(b), 6(a), 6(b), 11, 13, 16, 17, 19, 20, 23, 24, 33, 37, 38 2(a) V = P3 (R), T (f (x) = f (x). For all f (x) W = P2 (R), f (x) = a0 + a1 x + a2 x2 for some ai R, 0 i
Math 235 Summer 2010 Solutions to Homework 5
Section 4.1.
2. The determinants are 30, 17, and 8, respectively.
3. The determinants are 10 + 15i, 36 + 41i, and 24, respectively.
Section 4.2.
1. See back of book. Be sure you can justify each answer; if not,
MAT2040 HW 13 Solution
Tsang Chi Shing, Sidney April 25, 2011
Sec. 7.1: 2, 4, 5, 8, 10 2) (a) The characteristic polynomial of A is: det(A - tI) = det 1-t -1 1 3-t = (t - 2)2
Hence = 2 is the only eigenvalue of A with multiplicity 2 (so dim(K ) = 2). We h