Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
Author: Gilbert, Linda, Jimmie
Publisher: Cengage Learning,
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Chapter 5.3, Problem 16E
Prove that any field that contains an intergral domain
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You are given a plane Π in R3 defined by two vectors, p1 and p2, and a subspace W in R3 spanned by twovectors, w1 and w2. Your task is to project the plane Π onto the subspace W.First, answer the question of what the projection matrix is that projects onto the subspace W and how toapply it to find the desired projection. Second, approach the task in a different way by using the Gram-Schmidtmethod to find an orthonormal basis for subspace W, before then using the resulting basis vectors for theprojection. Last, compare the results obtained from both methods
Plane II is spanned by the vectors:
- (2) · P² - (4)
P1=2
P21
3
Subspace W is spanned by the vectors:
2
W1
- (9) ·
1
W2
1
= (³)
Chapter 5 Solutions
Elements Of Modern Algebra
Ch. 5.1 - True or False
Label each of the following...Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False
Label each of the following...Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False
Label each of the following...Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False Label each of the following...
Ch. 5.1 - Exercises
Confirm the statements made in Example...Ch. 5.1 - Exercises
2. Decide whether each of the following...Ch. 5.1 - Exercises
3. Let Using addition and...Ch. 5.1 - Prob. 4ECh. 5.1 - Exercises
5. Let Define addition and...Ch. 5.1 - Exercises Work exercise 5 using U=a. Exercise5 Let...Ch. 5.1 - Exercises Find all zero divisors in n for the...Ch. 5.1 - Exercises
8. For the given values of , find all...Ch. 5.1 - Exercises Prove Theorem 5.3:A subset S of the ring...Ch. 5.1 - Exercises
10. Prove Theorem 5.4:A subset of the...Ch. 5.1 - Assume R is a ring with unity e. Prove Theorem...Ch. 5.1 - 12. (See Example 4.) Prove the right distributive...Ch. 5.1 - 13. Complete the proof of Theorem by showing that...Ch. 5.1 - Let R be a ring, and let x,y, and z be arbitrary...Ch. 5.1 - 15. Let and be elements of a ring. Prove that...Ch. 5.1 - 16. Suppose that is an abelian group with respect...Ch. 5.1 - If R1 and R2 are subrings of the ring R, prove...Ch. 5.1 - 18. Find subrings and of such that is not a...Ch. 5.1 - 19. Find a specific example of two elements and ...Ch. 5.1 - Prob. 20ECh. 5.1 - 21. Define a new operation of addition in by ...Ch. 5.1 - 22. Define a new operation of addition in by and...Ch. 5.1 - Let R be a ring with unity and S be the set of all...Ch. 5.1 - Prove that if a is a unit in a ring R with unity,...Ch. 5.1 - Prob. 25ECh. 5.1 - Prob. 26ECh. 5.1 - Suppose that a,b, and c are elements of a ring R...Ch. 5.1 - Prob. 28ECh. 5.1 - 29. For a fixed element of a ring , prove that...Ch. 5.1 - Prob. 30ECh. 5.1 - Let R be a ring. Prove that the set S={...Ch. 5.1 - 32. Consider the set .
a. Construct...Ch. 5.1 - Consider the set S={ [ 0 ],[ 2 ],[ 4 ],[ 6 ],[ 8...Ch. 5.1 - The addition table and part of the multiplication...Ch. 5.1 - 35. The addition table and part of the...Ch. 5.1 - Prob. 36ECh. 5.1 - 37. Let and be elements in a ring. If is a zero...Ch. 5.1 - An element x in a ring is called idempotent if...Ch. 5.1 - 39. (See Exercise 38.) Show that the set of all...Ch. 5.1 - 40. Let be idempotent in a ring with unity....Ch. 5.1 - 41. Decide whether each of the following sets is...Ch. 5.1 - 42. Let .
a. Show that is a...Ch. 5.1 - 43. Let .
a. Show that is a...Ch. 5.1 - 44. Consider the set of all matrices of the...Ch. 5.1 - Prob. 45ECh. 5.1 - 46. Let be a set of elements containing the unity,...Ch. 5.1 - Prob. 47ECh. 5.1 - Prob. 48ECh. 5.1 - An element a of a ring R is called nilpotent if...Ch. 5.1 - 50. Let and be nilpotent elements that satisfy...Ch. 5.1 - Let R and S be arbitrary rings. In the Cartesian...Ch. 5.1 - 52. (See Exercise 51.)
a. Write out the...Ch. 5.1 - Prob. 53ECh. 5.1 - Prob. 54ECh. 5.1 - Prob. 55ECh. 5.1 - Suppose R is a ring in which all elements x are...Ch. 5.2 - True or False
Label each of the following...Ch. 5.2 - [Type here]
True or False
Label each of the...Ch. 5.2 - [Type here]
True or False
Label each of the...Ch. 5.2 - Label each of the following as either true or...Ch. 5.2 - Confirm the statements made in Example 3 by...Ch. 5.2 - Consider the set ={[0],[2],[4],[6],[8]}10, with...Ch. 5.2 - Consider the set...Ch. 5.2 - [Type here]
Examples 5 and 6 of Section 5.1 showed...Ch. 5.2 - Examples 5 and 6 of Section 5.1 showed that P(U)...Ch. 5.2 - [Type here]
Examples 5 and 6 of Section 5.1 showed...Ch. 5.2 - [Type here]
7. Let be the set of all ordered pairs...Ch. 5.2 - Let S be the set of all 2X2 matrices of the form...Ch. 5.2 - Work exercise 8 using be the set of all matrices...Ch. 5.2 - Work exercise 8 using S be the set of all matrices...Ch. 5.2 - Let R be the set of all matrices of the form...Ch. 5.2 - Prob. 12ECh. 5.2 - 13. Work Exercise 12 using , the Gaussian integers...Ch. 5.2 - 14. Letbe a commutative ring with unity in which...Ch. 5.2 - [Type here]
15. Give an example of an infinite...Ch. 5.2 - Prove that if a subring R of an integral domain D...Ch. 5.2 - If e is the unity in an integral domain D, prove...Ch. 5.2 - [Type here]
18. Prove that only idempotent...Ch. 5.2 - a. Give an example where a and b are not zero...Ch. 5.2 - 20. Find the multiplicative inverse of the given...Ch. 5.2 - [Type here]
21. Prove that ifand are integral...Ch. 5.2 - Prove that if R and S are fields, then the direct...Ch. 5.2 - [Type here]
23. Let be a Boolean ring with unity....Ch. 5.2 - If a0 in a field F, prove that for every bF the...Ch. 5.2 - Suppose S is a subset of an field F that contains...Ch. 5.3 - True or False Label each of the following...Ch. 5.3 - Prob. 2TFECh. 5.3 - Prob. 3TFECh. 5.3 - Prob. 4TFECh. 5.3 - Prob. 5TFECh. 5.3 - Prove that the multiplication defined 5.24 is a...Ch. 5.3 - Prove that addition is associative in Q.Ch. 5.3 - Prob. 3ECh. 5.3 - Prob. 4ECh. 5.3 - Prob. 5ECh. 5.3 - Prob. 6ECh. 5.3 - 7. Prove that on a given set of rings, the...Ch. 5.3 - Prob. 8ECh. 5.3 - Prob. 9ECh. 5.3 - Since this section presents a method for...Ch. 5.3 - Prob. 11ECh. 5.3 - Prob. 12ECh. 5.3 - Prob. 13ECh. 5.3 - 14. Let be the set of all real numbers of the...Ch. 5.3 - Prob. 15ECh. 5.3 - Prove that any field that contains an intergral...Ch. 5.3 - Prob. 17ECh. 5.3 - 18. Let be the smallest subring of the field of...Ch. 5.4 - True or False Label each of the following...Ch. 5.4 - True or False Label each of the following...Ch. 5.4 - True or False
Label each of the following...Ch. 5.4 - True or False Label each of the following...Ch. 5.4 - Prob. 5TFECh. 5.4 - Complete the proof of Theorem 5.30 by providing...Ch. 5.4 - 2. Prove the following statements for arbitrary...Ch. 5.4 - Prove the following statements for arbitrary...Ch. 5.4 - Suppose a and b have multiplicative inverses in an...Ch. 5.4 - 5. Prove that the equation has no solution in an...Ch. 5.4 - 6. Prove that if is any element of an ordered...Ch. 5.4 - For an element x of an ordered integral domain D,...Ch. 5.4 - If x and y are elements of an ordered integral...Ch. 5.4 - 9. If denotes the unity element in an integral...Ch. 5.4 - 10. An ordered field is an ordered integral domain...Ch. 5.4 - 11. (See Exercise 10.) According to Definition...Ch. 5.4 - 12. (See Exercise 10 and 11.) If each is...Ch. 5.4 - 13. Prove that if and are rational numbers such...Ch. 5.4 - 14. a. If is an ordered integral domain, prove...Ch. 5.4 - 15. (See Exercise .) If and with and in ,...Ch. 5.4 - If x and y are positive rational numbers, prove...
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Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Similar questions
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