A Hungerfords Algebra
Solutions Manual
Volume I: Introduction through Chapter IV
James Wilson
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a2 , b
b
a2 , ab
a2 b
I
a
a2
ab
a3 b
0
0 = C0 (G)
Cn1 (G) Cn (G) = G
0 = Gn
Gn1
G1
G0 = G
0 = n+1 G
II
C1 (G)
n G
2 G
1 G = G
Local
Ring
Commutative
Ri
Math 266.3 Midterm Solutions
1. 20% Determine an equation of the quadratic polynomial which passes through the
points
(1, 1), (1, 5), and (2, 2).
Solution:
have:
y = ax 2 + bx + c must be satised at the three points, so we must
1 = a(1)1 + b(1) + c
5 = a(
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Department of Mathematics and Statistics
Course Outline Math 363 (Q2) Abstract Algebra
2014 Summer Term, July 22nd August 12th 2014
Instructor: Dr. Sara Madariaga
Oce: 215 McLean Hall
Phone: (306) 966-6104
e-mail: madariaga@math.usask.ca
Oce hours by appo
Chapter 1
Rings
We have spent the term studying groups. A group is a set with a binary
operation that satises certain properties. But many algebraic structuressuch
as R, Z, and Zn come with two binary operations, usually called addition and
multiplication
1
Abel
1
1.1
R R
;
(a, b) a b ( a + b
).
R R R
a b R, a, b R.
Z M (R)
(ring) R
+ ,
R1. (R, +) Abel
0,
R2. (R, ) ( R );
R3.
a(b+c) = ab+ac; (b+c)a = ba+ca, a, b, c R.
(
)
1.1
(bridge).
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Math 266 - Solution to Assignment 7- Winter 2013.
1. In each of the following cases, nd all
0 4 1
1
B = 1 2 2 , C = 0
2 1 1
2
matrices A satisfying AB = C where
52
1 3 .
11
x1 y1 z1
Solution: We are looking for matrix A = x2 y2 z2 such that AB =
x3 y3 z3
Math 266 - Solution to Assignment 8- Winter 2013.
1. Denition. An n n matrix E is called an elementary matrix if it can be
obtained from the identity In by a single elementary row operation.
100
100
001
Let A = 0 1 2 , B = 0 1 0 , C = 0 1 0 . Determine
0
Math 266 - Comments on Midterm 2 - Winter 2013.
1. (4 marks) Consider the vector space V = R3 with vector addition and scalar
multiplication. In each of the following cases, determine whether the set W is
a subspace of V or not. Justify your answer.
5x
Math 266 - Solutions to Assignment 6 - Winter 2013.
1. Let A be an m n matrix (i.e. a matrix with m rows and n columns). Let
X Rn and c R. Use the denition of matrix-vector multiplication to prove
that A(cX ) = cAX .
Solution: Let A be an m n matrix writt
Math 266 - Solution to Assignment 5 - Winter 2013.
1. Let P3 be the vector space of all the polynomials in x of degree less than or
equal to 3. Prove that cfw_x2 + x3 , x 3, 2x2 , x + 1 forms a basis for P3 .
Solution: We need to show that the set cfw_x2
Math 266 - Assignment 2 - Winter 2013.
1. Determine if each of the following matrices are in reduced row-echelon form,
row-echelon form, or neither. In each case, nd the rank of the matrix.
(i)
1 0 0 1
021 0
(ii)
1 2 3 1
0000
(iii)
001
001
1 6 8 1
(iv) 0
Math 266 - Solution of Assignment 3
1. In each of the following cases, determine whether the set V together with the
proposed vector addition and scalar multiplication forms a vector space or not.
Support your answer.
x
y
(i) V =
as
x1
y1
+
: x, y R , wit
Math 266 - Solution for Assignment 4 - Winter 2013.
1. Let v1 , v2 , and v3 be three vectors in a vector space V . Show that
Spancfw_v1 , v2 , v3 = Spancfw_v1 + v2 , v2 + v3 , v1 + v3
Solution: We rst show that Spancfw_v1 , v2 , v3 Spancfw_v1 + v2 , v2
Math 266 - Solutions for Assignment 1 - Winter 2013.
1. Find the unique solution of this system of two linear equations in two variables.
Draw a graph to represent the solution as the intersection of two lines in the
xy -plane:
2x + y = 5
3x + 2y = 6
Solu