A Transition to Advanced Mathematics
8th Edition
ISBN: 9781285463261
Author: Douglas Smith, Maurice Eggen, Richard St. Andre
Publisher: Cengage Learning
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Chapter 4.1, Problem 19E
Prove that every irrational number is "missing" from
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= 1. Show
(a) Let G = Z/nZ be a cyclic group, so G = {1, 9, 92,...,g" } with g":
that the group algebra KG has a presentation KG = K(X)/(X” — 1).
(b) Let A = K[X] be the algebra of polynomials in X. Let V be the A-module
with vector space K2 and where the action of X is given by the matrix
Compute End(V) in the cases
(i) x = p,
(ii) xμl.
(67) ·
(c) If M and N are submodules of a module L, prove that there is an isomorphism
M/MON (M+N)/N.
(The Second Isomorphism Theorem for modules.)
You may assume that MON is a submodule of M, M + N is a submodule of L
and the First Isomorphism Theorem for modules.
(a) Define the notion of an ideal I in an algebra A. Define the product on the quotient
algebra A/I, and show that it is well-defined.
(b) If I is an ideal in A and S is a subalgebra of A, show that S + I is a subalgebra
of A and that SnI is an ideal in S.
(c) Let A be the subset of M3 (K) given by matrices of the form
a b
0 a 0
00 d
Show that A is a subalgebra of M3(K).
Ꮖ
Compute the ideal I of A generated by the element and show that A/I K as
algebras, where
0 1 0
x =
0 0 0
001
(a) Let HI be the algebra of quaternions. Write out the multiplication table for 1, i, j,
k. Define the notion of a pure quaternion, and the absolute value of a quaternion.
Show that if p is a pure quaternion, then p² = -|p|².
(b) Define the notion of an (associative) algebra.
(c) Let A be a vector space with basis 1, a, b. Which (if any) of the following rules
turn A into an algebra? (You may assume that 1 is a unit.)
(i) a² = a, b²=ab = ba 0.
(ii) a²
(iii) a²
=
b, b² = abba = 0.
=
b, b²
=
b, ab = ba = 0.
(d) Let u1, 2 and 3 be in the Temperley-Lieb algebra TL4(8).
ገ
12
13
Compute (u3+ Augu2)² where A EK and hence find a non-zero x € TL4 (8) such
that ² = 0.
Chapter 4 Solutions
A Transition to Advanced Mathematics
Ch. 4.1 - Find two upper bounds (if any exits) for each of...Ch. 4.1 - Assign a grade of A (correct), C (partially...Ch. 4.1 - Prob. 3ECh. 4.1 - Prob. 4ECh. 4.1 - Let A and B be subsets of . Prove that if A is...Ch. 4.1 - Let x be an upper bound for A. Prove that if xy,...Ch. 4.1 - Let A. Prove that if A is bounded above, then Ac...Ch. 4.1 - Give an example of a set A for which both A and Ac...Ch. 4.1 - Let A. Prove that if sup(A) exists, then it is...Ch. 4.1 - Formulate and prove a characterization of greatest...
Ch. 4.1 - If possible, give an example of a set A such that...Ch. 4.1 - Let A. Prove that if sup(A) exists, then...Ch. 4.1 - Let A and B be subsets of . Prove that if sup(A)...Ch. 4.1 - (a)Give an example of sets A and B of real numbers...Ch. 4.1 - (a)Give an example of sets A and B of real numbers...Ch. 4.1 - An alternate version of the Archimedean Principle...Ch. 4.1 - Prob. 17ECh. 4.1 - Prove that an ordered field F is complete iff...Ch. 4.1 - Prove that every irrational number is "missing"...Ch. 4.2 - Let A and B be compact subsets of . Use the...Ch. 4.2 - Prob. 2ECh. 4.2 - Prob. 3ECh. 4.2 - Prob. 4ECh. 4.2 - Assign a grade of A (correct), C (partially...Ch. 4.2 - For real numbers x,1,2,...n, describe i=1nN(x,i)....Ch. 4.2 - State the definition of continuity of the function...Ch. 4.2 - Find the set of interior point for each of these...Ch. 4.2 - Suppose that x is an interior point of a set A....Ch. 4.2 - Let AB. Prove that if sup(A) and sup(B) both...Ch. 4.2 - Let Abe a nonempty collection of closed subsets of...Ch. 4.2 - Prob. 12ECh. 4.2 - Prob. 13ECh. 4.2 - Prob. 14ECh. 4.2 - Prob. 15ECh. 4.2 - Prob. 16ECh. 4.2 - Prove Lemma 7.2.4.Ch. 4.2 - Which of the following subsets of are compact? ...Ch. 4.2 - Give an example of a bounded subset of and a...Ch. 4.3 - Let A and F be sets of real numbers, and let F be...Ch. 4.3 - In the proof of Theorem 7.3.1 that =, it is...Ch. 4.3 - Assign a grade of A (correct), C (partially...Ch. 4.3 - Prove that 7 is an accumulation point for [3,7). 5...Ch. 4.3 - Find an example of an infinite subset of that has...Ch. 4.3 - Find the derived set of each of the following...Ch. 4.3 - Let S=(0,1]. Find S(Sc).Ch. 4.3 - Prob. 8ECh. 4.3 - (a)Prove that if AB, then AB. (b)Is the converse...Ch. 4.3 - Show by example that the intersection of...Ch. 4.3 - Prob. 11ECh. 4.3 - Prob. 12ECh. 4.3 - Let a, b. Prove that every closed interval [a,b]...Ch. 4.3 - Prob. 14ECh. 4.3 - Prob. 15ECh. 4.4 - Prob. 1ECh. 4.4 - Prove that if x is an interior point of the set A,...Ch. 4.4 - Recall from Exercise 11 of Section 4.6 that the...Ch. 4.4 - A sequence x of real numbers is a Cauchy* sequence...Ch. 4.4 - Prob. 5ECh. 4.4 - Assign a grade of A (correct), C (partially...Ch. 4.4 - Prob. 7ECh. 4.4 - Give an example of a bounded sequence that is not...Ch. 4.4 - Prob. 9ECh. 4.4 - Let A and B be subsets of . Prove that (AB)=AB....Ch. 4.5 - For the sequence y defined in the proof of Theorem...Ch. 4.5 - Prob. 2ECh. 4.5 - Prob. 3ECh. 4.5 - Let I be a sequence of intervals. Then for each...Ch. 4.5 - Prob. 5ECh. 4.5 - Prob. 6ECh. 4.5 - Find all divisors of zero in 14. 15. 10. 101.Ch. 4.5 - Prob. 8ECh. 4.5 - Suppose m and m2. Prove that 1 and m1 are distinct...Ch. 4.5 - Prob. 10ECh. 4.5 - Prob. 11ECh. 4.5 - Determine whether each sequence is monotone. For...Ch. 4.5 - Prob. 13ECh. 4.5 - Complete the proof that xn=(1+1n)n is increasing...Ch. 4.5 - Prob. 15ECh. 4.5 - Prob. 16ECh. 4.5 - Prob. 17ECh. 4.6 - Prob. 1ECh. 4.6 - Repeat Exercise 2 with the operation * given by...Ch. 4.6 - Prob. 3ECh. 4.6 - Let m,n and M=A:A is an mn matrix with real number...Ch. 4.6 - Let be an associative operation on nonempty set A...Ch. 4.6 - Let be an associative operation on nonempty set A...Ch. 4.6 - Suppose that (A,*) is an algebraic system and * is...Ch. 4.6 - Let (A,o) be an algebra structure. An element lA...Ch. 4.6 - Let G be a group. Prove that if a2=e for all aG,...Ch. 4.6 - Prob. 10ECh. 4.6 - Complete the proof of Theorem 6.1.4. First, show...Ch. 4.6 - Prob. 12ECh. 4.6 - Prob. 13ECh. 4.7 - Give an example of an algebraic structure of order...Ch. 4.7 - Let G be a group. Prove that G is abelian if and...Ch. 4.7 - Prob. 3ECh. 4.7 - (a)In the group G of Exercise 2, find x such that...Ch. 4.7 - Show that (,), with operation # defined by...Ch. 4.7 - Let m be a prime natural number and a(Um,). Prove...Ch. 4.7 - Prob. 7ECh. 4.7 - Prob. 8ECh. 4.7 - Prob. 9E
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