Student Solutions Manual for Gallian's Contemporary Abstract Algebra, 9th
9th Edition
ISBN: 9781305657977
Author: Gallian, Joseph
Publisher: Brooks Cole
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Chapter 2, Problem 8E
List the elements of U(20).
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Students have asked these similar questions
R denotes the field of real numbers, Q denotes the field of rationals, and
Fp denotes the field of p elements given by integers modulo p. You may refer to general
results from lectures.
Question 1
For each non-negative integer m, let R[x]m denote the
vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m.
x²+2, V3 = 5. Prove that (V1, V2, V3) is a linearly independent
(a) Let vi = x, V2 =
list in R[x] 3.
(b) Let V1, V2, V3 be as defined in (a). Find a vector v € R[×]3 such that (V1, V2, V3, V4)
is a basis of R[x] 3.
[8]
[6]
(c) Prove that the map ƒ from R[x] 2 to R[x]3 given by f(p(x)) = xp(x) — xp(0) is a
linear map.
[6]
(d) Write down the matrix for the map ƒ defined in (c) with respect to the basis
(2,2x + 1, x²) of R[x] 2 and the basis (1, x, x², x³) of R[x] 3.
[5]
Question 4
(a) The following matrices represent linear maps on R² with respect to an
orthonormal basis:
=
[1/√5 2/√5
[2/√5 -1/√5]
"
[1/√5 2/√5]
A =
B =
[2/√5 1/√5] 1
C =
D =
=
=
[ 1/3/5 2/35]
1/√5 2/√5
-2/√5 1/√5'
For each of the matrices A, B, C, D, state whether it represents a self-adjoint
linear map, an orthogonal linear map, both, or neither.
(b) For the quadratic form
q(x, y, z) = y² + 2xy +2yz
over R, write down a linear change of variables to u, v, w such that q in these
terms is in canonical form for Sylvester's Law of Inertia.
[6]
[4]
part b please
Chapter 2 Solutions
Student Solutions Manual for Gallian's Contemporary Abstract Algebra, 9th
Ch. 2 - Which of the following binary operations are...Ch. 2 - Which of the following binary operations are...Ch. 2 - Which of the following binary operations are...Ch. 2 - Which of the following sets are closed under the...Ch. 2 - In each case, find the inverse of the element...Ch. 2 - In each case, perform the indicated operation. a....Ch. 2 - Prob. 7ECh. 2 - List the elements of U(20).Ch. 2 - Show that {1, 2, 3} under multiplication modulo 4...Ch. 2 - Show that the group GL(2,R) of Example 9 is...
Ch. 2 - Let a belong to a group and a12=e . Express the...Ch. 2 - In U(9)find the inverse of 2, 7, and 8.Ch. 2 - Translate each of the following multiplicative...Ch. 2 - For group elements a, b, and c, express...Ch. 2 - Suppose that a and b belong to a group and...Ch. 2 - Show that the set {5, 15, 25, 35} is a group under...Ch. 2 - Let G be a group and let H=x1xG . Show that G=H as...Ch. 2 - List the members of K=x2xD4andL=xD4x2=e .Ch. 2 - Prove that the set of all 22 matrices with entries...Ch. 2 - For any integer n2 , show that there are at least...Ch. 2 - An abstract algebra teacher intended to give a...Ch. 2 - Let G be a group with the property that for any x,...Ch. 2 - (Law of Exponents for Abelian Groups) Let a and b...Ch. 2 - (SocksShoes Property) Draw an analogy between the...Ch. 2 - Prove that a group G is Abelian if and only if...Ch. 2 - Prove that in a group, (a1)1=a for all a.Ch. 2 - For any elements a and b from a group and any...Ch. 2 - If a1,a2,...,an belong to a group, what is the...Ch. 2 - The integers 5 and 15 are among a collection of 12...Ch. 2 - Prob. 30ECh. 2 - Prob. 31ECh. 2 - Construct a Cayley table for U(12).Ch. 2 - Suppose the table below is a group table. Fill in...Ch. 2 - Prove that in a group, (ab)2=a2b2 if and only if...Ch. 2 - Let a, b, and c be elements of a group. Solve the...Ch. 2 - Let a and b belong to a group G. Find an x in G...Ch. 2 - Let G be a finite group. Show that the number of...Ch. 2 - Give an example of a group with elements a, b, c,...Ch. 2 - Suppose that G is a group with the property that...Ch. 2 - Find an element X in D4 such that R90VXH=D .Ch. 2 - Suppose F1andF2 are distinct reflections in a...Ch. 2 - Suppose F1andF2 are distinct reflections in a...Ch. 2 - Let R be any fixed rotation and F any fixed...Ch. 2 - Let R be any fixed rotation and F any fixed...Ch. 2 - In the dihedral group Dn , let R=R360/n and let F...Ch. 2 - Prove that the set of all 33 matrices with real...Ch. 2 - Prove that if G is a group with the property that...Ch. 2 - In a finite group, show that the number of...Ch. 2 - List the six elements of GL(2,Z2) . Show that this...Ch. 2 - Prove the assertion made in Example 19 that the...Ch. 2 - Suppose that in the definition of a group G, the...Ch. 2 - Suppose that in the definition of a group G, the...
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- Question 5 (a) Let a, b, c, d, e, ƒ Є K where K is a field. Suppose that the determinant of the matrix a cl |df equals 3 and the determinant of determinant of the matrix a+3b cl d+3e f ГЪ e [ c ] equals 2. Compute the [5] (b) Calculate the adjugate Adj (A) of the 2 × 2 matrix [1 2 A = over R. (c) Working over the field F3 with 3 elements, use row and column operations to put the matrix [6] 0123] A = 3210 into canonical form for equivalence and write down the canonical form. What is the rank of A as a matrix over F3? 4arrow_forwardQuestion 2 In this question, V = Q4 and - U = {(x, y, z, w) EV | x+y2w+ z = 0}, W = {(x, y, z, w) € V | x − 2y + w − z = 0}, Z = {(x, y, z, w) € V | xyzw = 0}. (a) Determine which of U, W, Z are subspaces of V. Justify your answers. (b) Show that UW is a subspace of V and determine its dimension. (c) Is VU+W? Is V = UW? Justify your answers. [10] [7] '00'arrow_forwardTools Sign in Different masses and Indicated velocities Rotational inert > C C Chegg 39. The balls shown have different masses and speeds. Rank the following from greatest to least: 2.0 m/s 8.5 m/s 9.0 m/s 12.0 m/s 1.0 kg A 1.2 kg B 0.8 kg C 5.0 kg D C a. The momenta b. The impulses needed to stop the balls Solved 39. The balls shown have different masses and speeds. | Chegg.com Images may be subject to copyright. Learn More Share H Save Visit > quizlet.com%2FBoyE3qwOAUqXvw95Fgh5Rw.jpg&imgrefurl=https%3A%2F%2Fquizlet.com%2F529359992%2Fc. Xarrow_forward
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