College Algebra in Context with Applications for the Managerial, Life, and Social Sciences (5th Edition)
5th Edition
ISBN: 9780134179025
Author: Ronald J. Harshbarger, Lisa S. Yocco
Publisher: PEARSON
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Chapter 3, Problem 21RE
To determine
To find: The exact solution to the equation
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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 3 Solutions
College Algebra in Context with Applications for the Managerial, Life, and Social Sciences (5th Edition)
Ch. 3.1 - In Exercises 16, (a) determine whether the...Ch. 3.1 - In Exercises 16, (a) determine whether the...Ch. 3.1 - In Exercises 16, (a) determine whether the...Ch. 3.1 - In Exercises 16, (a) determine whether the...Ch. 3.1 - In Exercises 16, (a) determine whether the...Ch. 3.1 - In Exercises 16, (a) determine whether the...Ch. 3.1 - Prob. 7ECh. 3.1 - Prob. 8ECh. 3.1 - Prob. 9ECh. 3.1 - In Exercises 714, (a) graph each quadratic...
Ch. 3.1 - Prob. 11ECh. 3.1 - Prob. 12ECh. 3.1 - In Exercises 714, (a) graph each quadratic...Ch. 3.1 - In Exercises 714, (a) graph each quadratic...Ch. 3.1 - Prob. 15ECh. 3.1 - Write the equation of the quadratic function whose...Ch. 3.1 - The two graphs shown have equations of the forni y...Ch. 3.1 - Prob. 18ECh. 3.1 - Prob. 19ECh. 3.1 - Prob. 20ECh. 3.1 - Prob. 21ECh. 3.1 - Prob. 22ECh. 3.1 - Prob. 23ECh. 3.1 - In Exercises 2130, (a) give the coordinates of the...Ch. 3.1 - Prob. 25ECh. 3.1 - Prob. 26ECh. 3.1 - In Exercises 2130, (a) give the coordinates of the...Ch. 3.1 - In Exercises 2130, (a) give the coordinates of the...Ch. 3.1 - Prob. 29ECh. 3.1 - Prob. 30ECh. 3.1 - Prob. 31ECh. 3.1 - Prob. 32ECh. 3.1 - Prob. 33ECh. 3.1 - Prob. 34ECh. 3.1 - Prob. 35ECh. 3.1 - Prob. 36ECh. 3.1 - Prob. 37ECh. 3.1 - Prob. 38ECh. 3.1 - Prob. 39ECh. 3.1 - Prob. 40ECh. 3.1 - Prob. 41ECh. 3.1 - Prob. 42ECh. 3.1 - Use the graph of each function in Exercises 4146...Ch. 3.1 - Prob. 44ECh. 3.1 - Use the graph of each function in Exercises 4146...Ch. 3.1 - Use the graph of each function in Exercises 4146...Ch. 3.1 - Profit The daily profit for a product is given by...Ch. 3.1 - Profit The daily profit for a product is given by...Ch. 3.1 - Worldwide Mobile Phone Shipments For selected...Ch. 3.1 - Prob. 50ECh. 3.1 - Tourism Spending The equation y = 1.69x2 0.92x +...Ch. 3.1 - Prob. 52ECh. 3.1 - Flight of a Ball If a ball is thrown upward at...Ch. 3.1 - Prob. 54ECh. 3.1 - Crude Oil The U.S. Crude Oil production, in...Ch. 3.1 - Prob. 56ECh. 3.1 - Profit The profit for a certain brand of MP3...Ch. 3.1 - Prob. 58ECh. 3.1 - Prob. 59ECh. 3.1 - Prob. 60ECh. 3.1 - Prob. 61ECh. 3.1 - Prob. 63ECh. 3.1 - Prob. 64ECh. 3.1 - Prob. 65ECh. 3.1 - Prob. 66ECh. 3.1 - Wind and Pollution The amount of particulate...Ch. 3.1 - Drug Sensitivity The sensitivity S to a drug is...Ch. 3.1 - Falling Object A tennis ball is thrown downward...Ch. 3.1 - Break-Even The profit for a product is given by P...Ch. 3.1 - Flight of a Ball A softball is hit with upward...Ch. 3.1 - Flight of a Ball A baseball is hit with upward...Ch. 3.1 - Apartment Rental The owner of an apartment...Ch. 3.1 - Rink Rental The owner of a skating rink rents the...Ch. 3.1 - World Population A low-projection scenario for...Ch. 3.2 - In Exercises 110, use factoring to solve the...Ch. 3.2 - In Exercises 110, use factoring to solve the...Ch. 3.2 - In Exercises 110, use factoring to solve the...Ch. 3.2 - In Exercises 110, use factoring to solve the...Ch. 3.2 - In Exercises 110, use factoring to solve the...Ch. 3.2 - In Exercises 110, use factoring to solve the...Ch. 3.2 - In Exercises 110, use factoring to solve the...Ch. 3.2 - In Exercises 110, use factoring to solve the...Ch. 3.2 - In Exercises 110, use factoring to solve the...Ch. 3.2 - In Exercises 110, use factoring to solve the...Ch. 3.2 - In Exercises 116, find the x-intercepts...Ch. 3.2 - In Exercises 116, find the x-intercepts...Ch. 3.2 - In Exercises 116, find the x-intercepts...Ch. 3.2 - In Exercises 116, find the x-intercepts...Ch. 3.2 - In Exercises 116, find the x-intercepts...Ch. 3.2 - In Exercises 116, find the x-intercepts...Ch. 3.2 - Use a graphing utility to find or to approximate...Ch. 3.2 - Prob. 18ECh. 3.2 - Use a graphing utility to find or to approximate...Ch. 3.2 - Prob. 20ECh. 3.2 - Prob. 21ECh. 3.2 - Prob. 22ECh. 3.2 - Prob. 23ECh. 3.2 - Prob. 24ECh. 3.2 - Prob. 25ECh. 3.2 - Prob. 26ECh. 3.2 - Prob. 27ECh. 3.2 - Prob. 28ECh. 3.2 - In Exercises 2934, use the square root method to...Ch. 3.2 - Prob. 30ECh. 3.2 - Prob. 31ECh. 3.2 - Prob. 32ECh. 3.2 - Prob. 33ECh. 3.2 - Prob. 34ECh. 3.2 - Prob. 35ECh. 3.2 - Prob. 36ECh. 3.2 - Prob. 37ECh. 3.2 - Prob. 38ECh. 3.2 - Prob. 39ECh. 3.2 - Prob. 40ECh. 3.2 - Prob. 41ECh. 3.2 - Prob. 42ECh. 3.2 - Prob. 43ECh. 3.2 - Prob. 44ECh. 3.2 - Prob. 45ECh. 3.2 - Prob. 46ECh. 3.2 - Prob. 47ECh. 3.2 - Prob. 48ECh. 3.2 - Prob. 49ECh. 3.2 - Prob. 50ECh. 3.2 - Prob. 51ECh. 3.2 - In Exercises 4954, find the exact solutions to...Ch. 3.2 - Prob. 53ECh. 3.2 - Prob. 54ECh. 3.2 - Prob. 55ECh. 3.2 - Prob. 56ECh. 3.2 - In Exercises 5558, you are given the graphs of...Ch. 3.2 - Prob. 58ECh. 3.2 - Prob. 59ECh. 3.2 - Prob. 60ECh. 3.2 - Prob. 61ECh. 3.2 - Prob. 62ECh. 3.2 - In Exercises 6374, solve analytically and then...Ch. 3.2 - Prob. 64ECh. 3.2 - Prob. 65ECh. 3.2 - In Exercises 6374, solve analytically and then...Ch. 3.2 - In Exercises 6374, solve analytically and then...Ch. 3.2 - Prob. 68ECh. 3.2 - Prob. 69ECh. 3.2 - Prob. 70ECh. 3.2 - Prob. 71ECh. 3.2 - In Exercises 6374, solve analytically and then...Ch. 3.2 - Prob. 73ECh. 3.2 - In Exercises 6374, solve analytically and then...Ch. 3.2 - Prob. 75ECh. 3.2 - Prob. 76ECh. 3.2 - Prob. 77ECh. 3.2 - Prob. 78ECh. 3.2 - Prob. 79ECh. 3.2 - Prob. 80ECh. 3.2 - Prob. 81ECh. 3.2 - International Travel to the United States The...Ch. 3.2 - Gold Prices The price of an ounce of gold in U.S....Ch. 3.2 - Non-White Population The number of millions of...Ch. 3.2 - Retail Sales November and December retail sales,...Ch. 3.2 - Prob. 86ECh. 3.2 - World Population One projection of the world...Ch. 3.3 - In Exercises 110, sketch the graph of each...Ch. 3.3 - Prob. 2ECh. 3.3 - Prob. 3ECh. 3.3 - Prob. 4ECh. 3.3 - Prob. 5ECh. 3.3 - Prob. 6ECh. 3.3 - Prob. 7ECh. 3.3 - Prob. 8ECh. 3.3 - Prob. 9ECh. 3.3 - Prob. 10ECh. 3.3 - Prob. 11ECh. 3.3 - Prob. 12ECh. 3.3 - a.Graph f(x)={4x3ifx3x2ifx3 b.Find f(2) and f(4)....Ch. 3.3 - Prob. 14ECh. 3.3 - Prob. 15ECh. 3.3 - Prob. 17ECh. 3.3 - Prob. 18ECh. 3.3 - Prob. 19ECh. 3.3 - Prob. 20ECh. 3.3 - Prob. 21ECh. 3.3 - Prob. 22ECh. 3.3 - Prob. 23ECh. 3.3 - Prob. 24ECh. 3.3 - Prob. 25ECh. 3.3 - Prob. 26ECh. 3.3 - Prob. 27ECh. 3.3 - Prob. 28ECh. 3.3 - Prob. 29ECh. 3.3 - Prob. 30ECh. 3.3 - Prob. 31ECh. 3.3 - Prob. 32ECh. 3.3 - Compare the graph in Exercise 30 with the graph in...Ch. 3.3 - Prob. 34ECh. 3.3 - Prob. 35ECh. 3.3 - Prob. 36ECh. 3.3 - Prob. 37ECh. 3.3 - Prob. 38ECh. 3.3 - Prob. 39ECh. 3.3 - Prob. 40ECh. 3.3 - Prob. 41ECh. 3.3 - Prob. 42ECh. 3.3 - Electric Charges For the nonextreme weather...Ch. 3.3 - Prob. 44ECh. 3.3 - First-Class Postage The first-class postage...Ch. 3.3 - Income Tax The U.S. federal income tax owed by a...Ch. 3.3 - Banks The number of banks in the United States for...Ch. 3.3 - Wind Chill The formula that gives the wind chill...Ch. 3.3 - Female Physicians Representation of females in...Ch. 3.3 - Prob. 50ECh. 3.3 - Prob. 51ECh. 3.3 - U.S. Population The U.S. population can be modeled...Ch. 3.3 - Prob. 53ECh. 3.3 - Prob. 54ECh. 3.3 - Prob. 55ECh. 3.3 - Prob. 56ECh. 3.3 - Purchasing Power The purchasing power of a 1983...Ch. 3.3 - Trust in the Government The percent of people who...Ch. 3.3 - Prob. 59ECh. 3.3 - Concentration of Body Substances The concentration...Ch. 3.3 - Investing If money is invested for 3 years with...Ch. 3.3 - Investing If money is invested for 4 years with...Ch. 3.3 - Investing The present value that will give a...Ch. 3.3 - Investing The present value that will give a...Ch. 3.4 - In Exercises 16, write the equation of the...Ch. 3.4 - Prob. 2ECh. 3.4 - Prob. 3ECh. 3.4 - Prob. 4ECh. 3.4 - Prob. 5ECh. 3.4 - Prob. 6ECh. 3.4 - A ball is thrown upward from the top of a...Ch. 3.4 - Prob. 8ECh. 3.4 - Prob. 9ECh. 3.4 - Prob. 10ECh. 3.4 - Prob. 11ECh. 3.4 - Prob. 12ECh. 3.4 - Prob. 13ECh. 3.4 - Prob. 14ECh. 3.4 - Find the quadratic function that is the best fit...Ch. 3.4 - Prob. 16ECh. 3.4 - Prob. 17ECh. 3.4 - Prob. 18ECh. 3.4 - Prob. 19ECh. 3.4 - Prob. 20ECh. 3.4 - Prob. 21ECh. 3.4 - Prob. 22ECh. 3.4 - Prob. 23ECh. 3.4 - Prob. 24ECh. 3.4 - Smart Phones Bill payment using smart phones is...Ch. 3.4 - Obesity Obesity (BMI 30) increases the risk of...Ch. 3.4 - Wind Chill The table gives the wind chill...Ch. 3.4 - Prob. 28ECh. 3.4 - Prob. 29ECh. 3.4 - Prob. 30ECh. 3.4 - Alzheimers Disease Partially because of American...Ch. 3.4 - Prob. 32ECh. 3.4 - Prob. 33ECh. 3.4 - World Population One projection of the world...Ch. 3.4 - Prob. 35ECh. 3.4 - Prob. 36ECh. 3.4 - Prob. 37ECh. 3.4 - Prob. 38ECh. 3.4 - Prob. 39ECh. 3.4 - Prob. 40ECh. 3.4 - Energy Use The following table shows the energy...Ch. 3.4 - Average Annual Wage The following table shows the...Ch. 3.4 - Insurance Rates The following table gives the...Ch. 3.4 - Banks The table gives the number of banks in the...Ch. 3.4 - U.S. Gross Domestic Product The table gives the...Ch. 3.4 - Auto Noise The noise level of a Volvo S60...Ch. 3.4 - Crude Oil The following table gives the U.S. crude...Ch. 3.4 - Box-Office Revenues The data in the table in the...Ch. 3 - In Exercises 16, use the rules of exponents to...Ch. 3 - In Exercises 16, use the rules of exponents to...Ch. 3 - In Exercises 16, use the rules of exponents to...Ch. 3 - In Exercises 16, use the rules of exponents to...Ch. 3 - In Exercises 16, use the rules of exponents to...Ch. 3 - In Exercises 16, use the rules of exponents to...Ch. 3 - Prob. 7TECh. 3 - Prob. 8TECh. 3 - Prob. 9TECh. 3 - Prob. 10TECh. 3 - Prob. 11TECh. 3 - Prob. 12TECh. 3 - Prob. 13TECh. 3 - Prob. 14TECh. 3 - Prob. 15TECh. 3 - Prob. 16TECh. 3 - Prob. 17TECh. 3 - Prob. 18TECh. 3 - Prob. 19TECh. 3 - Prob. 20TECh. 3 - Prob. 21TECh. 3 - Prob. 22TECh. 3 - Prob. 23TECh. 3 - Prob. 24TECh. 3 - Prob. 25TECh. 3 - Prob. 26TECh. 3 - Prob. 27TECh. 3 - Prob. 28TECh. 3 - Prob. 29TECh. 3 - Prob. 30TECh. 3 - In Exercises 2342, factor each of the polynomials...Ch. 3 - Prob. 32TECh. 3 - Prob. 33TECh. 3 - Prob. 34TECh. 3 - Prob. 35TECh. 3 - Prob. 36TECh. 3 - Prob. 37TECh. 3 - Prob. 38TECh. 3 - Prob. 39TECh. 3 - Prob. 40TECh. 3 - Prob. 41TECh. 3 - Prob. 42TECh. 3 - Prob. 43TECh. 3 - Prob. 44TECh. 3 - Prob. 45TECh. 3 - Prob. 46TECh. 3 - Prob. 47TECh. 3 - Prob. 1RECh. 3 - Prob. 2RECh. 3 - Prob. 3RECh. 3 - Prob. 4RECh. 3 - Prob. 5RECh. 3 - Prob. 6RECh. 3 - Prob. 7RECh. 3 - Prob. 8RECh. 3 - Prob. 9RECh. 3 - Prob. 10RECh. 3 - Prob. 11RECh. 3 - Prob. 12RECh. 3 - Prob. 13RECh. 3 - Prob. 14RECh. 3 - Prob. 15RECh. 3 - Prob. 16RECh. 3 - Prob. 17RECh. 3 - Prob. 18RECh. 3 - Prob. 19RECh. 3 - Prob. 20RECh. 3 - Prob. 21RECh. 3 - Prob. 22RECh. 3 - Prob. 23RECh. 3 - Prob. 24RECh. 3 - Prob. 25RECh. 3 - Prob. 26RECh. 3 - Prob. 27RECh. 3 - Prob. 28RECh. 3 - Prob. 29RECh. 3 - Prob. 30RECh. 3 - Determine whether the function y=3x2 is increasing...Ch. 3 - Prob. 32RECh. 3 - Prob. 33RECh. 3 - Prob. 34RECh. 3 - Find the equation of a quadratic function whose...Ch. 3 - Find the equation of a quadratic function whose...Ch. 3 - Prob. 37RECh. 3 - Prob. 38RECh. 3 - Suppose that q varies directly as the 3/2 power of...Ch. 3 - Prob. 40RECh. 3 - Prob. 41RECh. 3 - Prob. 42RECh. 3 - Prob. 43RECh. 3 - Height of a Ball If a ball is thrown into the air...Ch. 3 - Visas The number of skilled workers visas issued...Ch. 3 - Prob. 46RECh. 3 - Prob. 47RECh. 3 - Prob. 48RECh. 3 - Millionaire's Tax Rate The effective tax rate for...Ch. 3 - Prob. 50RECh. 3 - ATV Deaths The number of ATV-related deaths for...Ch. 3 - Home Range The home range of an animal is the...Ch. 3 - Internet Usage Around 40% of the worlds population...Ch. 3 - Personal Income The income received by people from...Ch. 3 - Prob. 55RECh. 3 - Prob. 56RECh. 3 - National Health Expenditures Health spending is...Ch. 3 - Prob. 58RECh. 3 - Prob. 59RECh. 3 - Prob. 60RE
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