MAT 444 Test 4
Instructor: Helene Barcelo
April 19, 2004
Name:
You can take up to 2 hours for completing this exam.
Close book, notes and calculator.
Do not use your own scratch paper.
Write each solution in the space provided, not on scratch paper.
MAT 444 H. Barcelo
Spring 2004
Homework 3 Part 2 Solutions
Section 5.2
Question:
6.
Prove that a linear operator on 2 is a reflection if and only if its eigenvalues are 1 and 1,
and its eigenvectors are orthogonal.
Answer:
If T : 2 2 is a reflection along
MAT 444 H. Barcelo
Spring 2004
Homework 3 Solutions
Section 2.10
Question:
4.
a) Consider the presentation (1.17) of the symmetric group S 3 . Let H be the subgroup
cfw_1, y . Compute the product sets (1H )( xH ) and (1H )( x 2 H ) , and verify that they
MAT 444 H. Barcelo
Spring 2004
Homework 2 Solutions
Section 2.5
Question:
6.
a) Prove that the relation x conjugate to y in a group G is an equivalence relation on G.
Answer:
a) Let x ~ y if and only if y = g x g 1 for some g G .
It is straightforward to
MAT 444 H. Barcelo
Spring 2004
Homework 1 Solutions
Section 2.1
Questions:
5.
Assume that the equation xyz = 1 holds in a group G. Does it follow that yzx = 1 ? That
yxz = 1 ?
Answers:
a) If xyz = 1 then yz = x 1 and yzx = x 1 x = 1
b)
If x y z = 1 but G
MAT 444
Final Exam
Instructor: Helene Barcelo
May 6, 2004
Question:
1. (10 points)
Prove that if G is a group of order 385, then G contains a normal cyclic subgroup of order 77.
Solution:
385 = 5 7 11 . Let n p be the number of Sylow p-subgroups of G for
MAT240 Assignment 10 Partial Solutions
Arthur Fischer April 1, 2004
Exercise (#17, p.229). Let A, B Mnn (F ) be such that AB = BA. Prove that if n is odd and F is not a eld of characteristic two, then A or B is not invertible. Solution. Assume as in the s
115A PROBLEMS FROM CLASS, NOV 29
Sec 6.2.
15. (a) Parsevals identity. Let = cfw_v1 , . . . , vn be an orthonormal basis for an inner product space V . For any x, y V prove that
n
x, y =
i=1
x, vi y , vi .
By Theorem 6.5, since is an orthonormal basis, we
115A HW 5
Sec 2.3.
11. Let V be a vector space and let T : V V be a linear operator on V . Show that T 2 = T0 (that is, T 2 is the zero transformation) if and only if R(T ) N (T ). Suppose T 2 = T0 . Let u R(T ); that is, there exists v V such that u = T
MAT 444 H. Barcelo
Spring 2004
Homework 4 Solutions
Section 5.3
Question:
2.
List all subgroups of the group D4 , and determine which are normal.
Answer:
Recall that Dn = cfw_1, x, x 2 , , x n 1 , y, xy, x 2 y, , x n 1 y with relations:
x n = 1 , y 2 = 1
MAT 444 H. Barcelo
Spring 2004
Homework 5 Solutions
Section 5.6
Question:
2.
Let G be a group, and let H be the cyclic subgroup generated by an element x of G. Show
that if left multiplication by x fixes every coset of H in G, then H is a normal subgroup.
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MAT 444 Test 2
Instructor: Helene Barcelo
March 1, 2004
Name:
You can take up to 2 hours for completing this exam.
Close book, notes and calculator.
Do not use your own scratch paper.
Write each solution in the space provided, not on scratch paper.
I
MAT 444 Test 1
Instructor: Hl`ne Barcelo
ee
February 14, 2004
Name:
You can take up to
2 hours for completing this exam.
Close book, notes and calculator.
Do not use your own scratch paper.
Write each solution in the space provided, not on scratch pap
MAT 444 H. Barcelo
Spring 2004
Homework 13 Solutions
Chapter 13
Section 13.1
Question:
4.
Let F be a field containing exactly eight elements. Prove or disprove: The characteristic of
F is 2.
Answer:
Let F be a field with | F | = 8 since F is a finite fiel
MAT 444 H. Barcelo
Spring 2004
Homework 9 Solutions
Chapter 10
Section 10.3
Question:
11. a) Prove that the kernel of the homomorphism : [ x, y ] [t ] defined by
x t 2 , y t 3 is the principal ideal generated by the polynomial y 2 x 3 .
Answer:
a) Let I =
MAT 444 H. Barcelo
Spring 2004
Homework 8 Solutions
Chapter 3
Section 3.2
Question:
14. a) Let p be a prime. Use the fact that Fp is a group to prove that a p 1 1 (modulo p) for
every integer a not congruent to zero.
Answer:
a) Fp = cfw_1, 2, , p 1 for a
MAT 444 H. Barcelo
Spring 2004
Homework 7 Solutions
Section 6.5
Question:
4.
Let G be a group of order 55.
a) Prove that G is generated by two elements x, y, with the relations
x11 = 1, y 5 = 1, yxy 1 = x r , for some r, 1 r < 11 .
Answer:
a) Assume | G |
MAT 444 H. Barcelo
Spring 2004
Homework 6 Solutions
Section 6.1
Question:
2.
Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe
the orbits for this operation.
Answer:
The orbits of G are the right cosets of H.
Question
115A PROBLEM FROM CLASS, OCT 26
Sec 2.2.
16. Let V and W be vector spaces such that dim V = dim W , and let T : V W be linear. Show there exist ordered bases for V and for W such that [T ] is diagonal. Let dim V = dim W = n. If T is onto, we can simply ta
115A HW 4 SELECTED SOLUTIONS
Sec 2.1. 2. Consider T : R3 R2 given by T (a1 , a2 , a3 ) = (a1 a2 , 2a3 ). We see that the vector (1, 1, 0) is a basis for N (T ). By the dimension theorem, rank(T ) = 3 1 = 2, which means that R(T ) = R2 (and thus T is onto)
115A HW 3 SELECTED SOLUTIONS
Sec 1.5. 9. Let u and v be distinct vectors in a vector space V . Show that cfw_u, v is linearly dependent if and only if u or v is a multiple of the other. If u and v are linearly dependent, then by denition there exist scal
MAT 442: Test 1
Instructor: J. Jones
Name:
1
2
3
4
5
6
7
Total
10 pts.
15 pts.
25 pts.
10 pts.
20 pts.
30 pts.
20 pts.
130 pts.
Write your answers on separate paper. Make sure your name is on every sheet.
If a problem asks you to prove a result already co
Solution 7
Sec. 5.2
2.
For the matrix A M nxn (R ) , test A for diagonal and ability and if A is diagonalizable,
find an invertible matrix Q and a diagonal matrix D such that Q-1AQ=D
7 4 0
(d) 8 5 0
6 6 3
Ans.:
(d) (3 )[(5 ) + 32] = 0 = 3,3,1
4 4 0
(
Qualifying Exam in Linear Algebra and Field Theory August, 2006 Answer as many questions as you can, but be sure to answer suciently many questions to comprise 120 points, with at least 30 points from each section. Detailed, completely correct answers to
Field Axioms
Definition 1. A binary operation, , on a set A, is a function :AAA Note: has to be defined for every pair a, b A, and give an element of A input is an ordered pair the value on an ordered pair (a, b) is written a b Example. On Z, +, -, and ,
Math 442 Final Examination and Qualifying Examination in Linear Algebra and Field Theory December, 2008 Write your answer to each problem on a separate sheet, placing the problem number in the upper right corner. Be sure to justify all your answers. Do no
MAT 442 Final Exam
December 2007
Linear Algebra Qualifying Exam
Write your answers on separate paper. Make sure your name is on every sheet. You may use any results from the course or text, except when asked to prove those results (which are marked as res
Alex Gittens
February 9, 2007
ACM 104: HW 4
Exercise 1. Find the Jordan canonical form and the corresponding Jordan canonical basis for the following
linear operators.
(a) T : R3 R3 dened by T x = Ax for x R3 where
11 4
A = 21 8
3
1
5
11 .
0
(b) T : P2 (R
Homework 2
Problem. (5-2. # 20) Prove that
k
V=
Wk
i=1
is the direct sum of W1 , . . . , Wk if and only if
k
dim(Wk ).
dim(V ) =
i=1
Solution. () This is trivial.
() Now suppose that
k
dim(Wk ).
dim(V ) =
i=1
Let i be a basis for Wi for i = 1, . . . , k .