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 reduce it to row echelon form:
1 R2
6
1 2 5 3R1+R2 1 2 5

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 page if you need more space.
1 1
2
4
4 and B = 3 1 .
1.

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 page if you need more space.
1. [16%] Determine whether

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 ) = 0, we need a = b = c = 0. Therefore ker(T ) = cfw_0

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 .
Hence, (A + B)2 = A2 + 2AB + B 2 only when AB = BA.
3.

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 determinant. No credits will be given if the answers
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 page if you need more space.
1. [12%] Find a basis for

MATH 231-01 (Fall 2010)
Dr. Kwong
Test 2 (Solutions and Remarks)
1. The linear combination
[
]
[
]
[
] [
1 1
4 3
1 8
0
+ k2
+ k3
=
k1
4
5
2 3
22 23
0
0
0
]
leads to the homogeneous linear system
k1
k1
4k1
5k1
+ 4k2
+ 3k2
2k2
+ 3k2
We solve it with an aug

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 as a 40-point
assignment. Hence, there is a built-in 8-

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 + 1
2x + 2
2
2
1
2x
2x + 2
2x + 1
x
x+2
x+1
x
x
2x
2
x

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 solutions neatly
and clearly, in a manner that could be un

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 to verify that 6 is the unity, as illustrated below:
6

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 counted as a
40-point assignment. Consider the 8-point die

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
(
)
+ 2(k 1)
12
34 cis
,
1 k 6.
zk =
6
Numerically, in 4

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 solutions neatly
and clearly, in a manner that could be un

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(Z48 ) does not have any element with order 6.
U (16) U

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 noncommutative ring that does not have a unity.
2. The

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 3a 3b + 9 = (a 3)(b 3),
which would require either a = 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 implies that b a. Therefore, a = b.
() If a = b, then

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 solutions neatly
and clearly, in a manner that could be und

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 counted as a
40-point assignment. Consider the 8-point di

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 counted as a
40-point assignment. Consider the 8-point di

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 multiplicative inverse a such that a a = 1. Then
0 = a 0 = a ab =

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 counted as a
40-point assignment. Consider the 8-point die

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 counted as a
40-point assignment. Consider the 8-point die

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 solutions neatly
and clearly, in a manner that could be un

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) = 0,
x2
x2
it is clear that limx2 f (x) exists and eq

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 counted as a
40-point assignment. Consider the 8-point dier