MATH 231-01 (Fall 2010)
Dr. Kwong
Test 3 (Solutions and Remarks)
1. Since
given
1
3
5
we want a basis consisting of the original vectors, we need to form a matrix with the
vectors as columns, and red
MATH 231-01 (Fall 2010)
Dr. Kwong
Test 1 (9/20/2010)
Name
Instructions: Be sure to show all your work. No credits will be given if the answers are presented
without explanations. Use the back of the p
MATH 231-01 (Fall 2010)
Dr. Kwong
Test 2 (10/18/2010)
Name
Instructions: Be sure to show all your work. No credits will be given if the answers are presented
without explanations. Use the back of the
MATH 231-01 (Fall 2010)
Dr. Kwong
Final Exam (Solutions and Remarks)
1. Given any polynomial a + bx + cx2 P2 , we nd
T (a + bx + cx2 ) = x(a + bx + cx2 ) = ax + bx2 + cx3 .
Hence, for T (a + bx + cx2
MATH 231-01 (Fall 2010)
Dr. Kwong
Test 1 (Solutions and Remarks)
[
1. We nd AT B =
1 3
1 4
0
2
]
[
]
2
4
1
3 1 = 11
.
8 18
1
5
2. False, because
(A + B)2 = A(A + B) + B(A + B) = A2 + AB + BA + B 2 .
MATH 231-01 (Fall 2010)
Dr. Kwong
Final Exam (12/13/2010)
Name
Instructions: Be sure to show all your work. For examples, show the row/column reductions in
Gaussian elimination, and how you evaluate a
MATH 231-01 (Fall 2010)
Dr. Kwong
Test 3 (11/15/2010)
Name
Instructions: Be sure to show all your work. No credits will be given if the answers are presented
without explanations. Use the back of the
MATH 332 (Spring 2013)
Dr. Kwong
Homework 5 (40 Points)
Due Wednesday, 3/6/2013. No late homework will be accepted.
Remarks: The maximum score for this assignment is 48 points, but it will be counted
MATH 332 (Spring 2013)
Dr. Kwong
Homework 12: Solutions and Remarks
1. Complete the multiplication table for the nonzero elements in Z3 [x]/x2 + 1:
1
2
x
x+1
x+2
2x
2x + 1
2x + 2
1
1
2
x
x+1
x+2
2x
2x
MATH 332 (Spring 2013)
Dr. Kwong
Homework 4 (50 Points)
Due Wednesday, 2/27/2013. No late homework will be accepted.
Instructions: Be sure to explain how you obtain your answers, and write up your sol
MATH 332 (Spring 2013)
Dr. Kwong
Homework 6: Solutions and Remarks
1. For any integers n 1 and m 2, the ring Mn (Zm ) is non-commutative. The ring 2Z is
innite and does not have a unity.
2. It is easy
MATH 332 (Spring 2013)
Dr. Kwong
Homework 7 (40 Points)
Due Wednesday, 3/20/2013. No late homework will be accepted.
Remark: Although the maximum score for this assignment is 48 points, it will be cou
MATH 332 (Spring 2013)
Dr. Kwong
Homework 2: Solutions and Remarks
1. Obviously,(| ) 3 5i| = 34. Since 3 5i is located in the third quadrant, = arg(3 5i) =
+ tan1 35 . The sixth roots of 3 5i are
(
)
MATH 332 (Spring 2013)
Dr. Kwong
Homework 2 (40 Points)
Due Wednesday, 2/13/2013. No late homework will be accepted.
Instructions: Be sure to explain how you obtain your answers, and write up your sol
MATH 332 (Spring 2013)
Dr. Kwong
Homework 4: Solutions and Remarks
(
)(
)
1. Solution 1. From Aut(Z48 )
= U (48), we know |Aut(Z48 )| = (48) = 48 1 12 1 13 = 16.
Since 6 - 16, we determine that Aut(Z
MATH 332 (Spring 2013)
Dr. Kwong
Homework 6 (40 Points)
Due Wednesday, 3/13/2013. No late homework will be accepted.
1. Give an example of a finite non-commutative ring. Give an example of an infinite
MATH 332 (Spring 2013)
Dr. Kwong
Homework 1: Solutions and Remarks
1. (a) Solution 1: Suppose S is not closed under . Then there exist a and b such that a b
/ S,
that is, a b = 3. This means
0 = ab 3
MATH 332 (Spring 2013)
Dr. Kwong
Homework 11: Solutions and Remarks
1. () If a and b are associates, then a = bu for some unit u. This immediately tells us that
a b. Since u exists, we nd au = b. This
MATH 332 (Spring 2013)
Dr. Kwong
Homework 1 (40 Points)
Due Wednesday, 2/6/2013. No late homework will be accepted.
Instructions: Be sure to explain how you obtain your answers, and write up your solu
MATH 332 (Spring 2013)
Dr. Kwong
Homework 10 (40 Points)
Due Wednesday, 4/17/2013. No late homework will be accepted.
Remark: Although the maximum score for this assignment is 48 points, it will be co
MATH 332 (Spring 2013)
Dr. Kwong
Homework 11 (40 Points)
Due Wednesday, 4/24/2013. No late homework will be accepted.
Remark: Although the maximum score for this assignment is 48 points, it will be co
MATH 332 (Spring 2013)
Dr. Kwong
Homework 10: Solutions and Remarks
1. Solution 1. Suppose R has zero divisors a and b such that a and b are nonzero with ab = 0.
Since a = 0, it has a left multiplicat
MATH 332 (Spring 2013)
Dr. Kwong
Homework 9 (40 Points)
Due Wednesday, 4/10/2013. No late homework will be accepted.
Remark: Although the maximum score for this assignment is 48 points, it will be cou
MATH 332 (Spring 2013)
Dr. Kwong
Homework 12 (40 Points)
Due Wednesday, 5/1/2013. No late homework will be accepted.
Remark: Although the maximum score for this assignment is 48 points, it will be cou
MATH 332 (Spring 2013)
Dr. Kwong
Homework 3 (40 Points)
Due Wednesday, 2/20/2013. No late homework will be accepted.
Instructions: Be sure to explain how you obtain your answers, and write up your sol
MATH 332 (Spring 2013)
Dr. Kwong
Homework 7: Solutions and Remarks
1. First we need to show that both f and g are continuous. It is clear that our only concern is
at x = 2. Since
lim f (x) = lim+ f (x
MATH 332 (Spring 2013)
Dr. Kwong
Homework 8 (40 Points)
Due Wednesday, 4/3/2013. No late homework will be accepted.
Remark: Although the maximum score for this assignment is 48 points, it will be coun