ELEMENTS OF MODERN ALGEBRA
8th Edition
ISBN: 9780357671139
Author: Gilbert
Publisher: CENGAGE L
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Chapter 7.1, Problem 10E
Express each of the numbers in Exercises 7-12 as a quotient of integers, reduce to lowest terms.
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Matrix MЄ R4×4, as specified below, is an orthogonal matrix - thus, it fulfills MTM = I.
M
(ELES),-
m2,1.
We know also that all the six unknowns mr,c are non-negative with the exception of
Your first task is to find the values of all the six unknowns. Think first, which of the mr,c you should find first.
Next, consider a vector v = (-6, 0, 0, 8) T. What's the length of v, i.e., |v|?
Using M as transformation matrix, map v onto w by w = Mv provide w with its numeric values.
What's the length of w, especially when comparing it to the length of v?
Finally, consider another vector p = ( 0, 0, 8, 6) T. What's the angle between v (from above) and p?
Using M as transformation matrix, map p onto q by q = Mp - provide q with its numeric values.
What's the angle between w and q, especially when comparing it to the angle between v and p?
7. (a) (i) Express y=-x²-7x-15 in the form
y = −(x+p)²+q.
(ii) Hence, sketch the graph of
y=-x²-7x-15.
(b) (i) Express y = x² - 3x + 4 in the form
y = (x − p)²+q.
(ii) Hence, sketch the graph of y = x² - 3x + 4.
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CHAPTER 1
Part 1 and 2
Chapter 7 Solutions
ELEMENTS OF MODERN ALGEBRA
Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Find the decimal representation for each of the...
Ch. 7.1 - Prob. 2ECh. 7.1 - Prob. 3ECh. 7.1 - Find the decimal representation for each of the...Ch. 7.1 - Prob. 5ECh. 7.1 - Prob. 6ECh. 7.1 - Prob. 7ECh. 7.1 - Prob. 8ECh. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Prove that is irrational. (That is, prove there...Ch. 7.1 - Prove that is irrational.
Ch. 7.1 - Prove that if is a prime integer, then is...Ch. 7.1 - Prove that if a is rational and b is irrational,...Ch. 7.1 - Prove that if is a nonzero rational number and ...Ch. 7.1 - Prove that if is an irrational number, then is...Ch. 7.1 - Prove that if is a nonzero rational number and ...Ch. 7.1 - Give counterexamples for the following...Ch. 7.1 - Let S be a nonempty subset of an order field F....Ch. 7.1 - Prove that if F is an ordered field with F+ as its...Ch. 7.1 - If F is an ordered field, prove that F contains a...Ch. 7.1 - Prove that any ordered field must contain a...Ch. 7.1 - If and are positive real numbers, prove that...Ch. 7.1 - Prove that if and are real numbers such that ,...Ch. 7.2 - True or False
Label each of the following...Ch. 7.2 - Prob. 2TFECh. 7.2 - Prob. 3TFECh. 7.2 - True or False
Label each of the following...Ch. 7.2 - Prob. 5TFECh. 7.2 - True or False
Label each of the following...Ch. 7.2 - Prob. 7TFECh. 7.2 - Prob. 1ECh. 7.2 - Prob. 2ECh. 7.2 - Prob. 3ECh. 7.2 - Prob. 4ECh. 7.2 - Prob. 5ECh. 7.2 - Prob. 6ECh. 7.2 - Prob. 7ECh. 7.2 - Prob. 8ECh. 7.2 - Prob. 9ECh. 7.2 - Prob. 10ECh. 7.2 - Prob. 11ECh. 7.2 - Prob. 12ECh. 7.2 - Prob. 13ECh. 7.2 - Prob. 14ECh. 7.2 - Prob. 15ECh. 7.2 - Prob. 16ECh. 7.2 - Prob. 17ECh. 7.2 - Prob. 18ECh. 7.2 - Prob. 19ECh. 7.2 - Prob. 20ECh. 7.2 - Prob. 21ECh. 7.2 - Prob. 22ECh. 7.2 - Prob. 23ECh. 7.2 - Prob. 24ECh. 7.2 - Prob. 25ECh. 7.2 - Prob. 26ECh. 7.2 - Prob. 27ECh. 7.2 - Prob. 28ECh. 7.2 - Prob. 29ECh. 7.2 - Prob. 30ECh. 7.2 - Prob. 31ECh. 7.2 - Prob. 32ECh. 7.2 - Prob. 33ECh. 7.2 - Prob. 34ECh. 7.2 - Prob. 35ECh. 7.2 - Prob. 36ECh. 7.2 - Prob. 37ECh. 7.2 - Prob. 38ECh. 7.2 - Prob. 39ECh. 7.2 - Prob. 40ECh. 7.2 - Exercise are stated using the notation in the...Ch. 7.2 - Prob. 42ECh. 7.2 - Prob. 43ECh. 7.2 - Prob. 44ECh. 7.2 - Prob. 45ECh. 7.2 - Prob. 46ECh. 7.2 - Prob. 47ECh. 7.2 - Prob. 48ECh. 7.2 - Prob. 49ECh. 7.2 - Prob. 50ECh. 7.2 - An element in a ring is idempotent if . Prove...Ch. 7.2 - Prove that a finite ring R with unity and no zero...Ch. 7.3 - True or False
Label each of the following...Ch. 7.3 - Prob. 2TFECh. 7.3 - Prob. 3TFECh. 7.3 - Prob. 4TFECh. 7.3 - Prob. 1ECh. 7.3 - Find each of the following products. Write each...Ch. 7.3 - Prob. 3ECh. 7.3 - Show that the n distinct n th roots of 1 are...Ch. 7.3 - Prob. 5ECh. 7.3 - Prob. 6ECh. 7.3 - Prob. 7ECh. 7.3 - Prob. 8ECh. 7.3 - Prob. 9ECh. 7.3 - Prob. 10ECh. 7.3 - Prob. 11ECh. 7.3 - Prob. 12ECh. 7.3 - Prob. 13ECh. 7.3 - Prob. 14ECh. 7.3 - Prove that the group in Exercise is cyclic, with ...Ch. 7.3 - Prob. 16ECh. 7.3 - Prob. 17ECh. 7.3 - Prob. 18ECh. 7.3 - Prob. 19ECh. 7.3 - Prob. 20ECh. 7.3 - Prob. 21ECh. 7.3 - Prob. 22ECh. 7.3 - Prove that the set of all complex numbers that...Ch. 7.3 - Prob. 24ECh. 7.3 - Prob. 25ECh. 7.3 - Prob. 26ECh. 7.3 - Prob. 27ECh. 7.3 - Prob. 28E
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