Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
Author: Gilbert, Linda, Jimmie
Publisher: Cengage Learning,
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Chapter 6.4, Problem 4E
Show that the ideal
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Let
2
A =
4
3
-4
0
1
(a) Show that v =
eigenvalue.
()
is an eigenvector of A and find the corresponding
(b) Find the characteristic polynomial of A and factorise it. Hint: the answer to (a)
may be useful.
(c) Determine all eigenvalues of A and find bases for the corresponding eigenspaces.
(d) Find an invertible matrix P and a diagonal matrix D such that P-¹AP = D.
(c) Let
6
0 0
A =
-10 4 8
5 1 2
(i) Find the characteristic polynomial of A and factorise it.
(ii) Determine all eigenvalues of A and find bases for the corresponding
eigenspaces.
(iii) Is A diagonalisable? Give reasons for your answer.
most 2, and let
Let P2 denote the vector space of polynomials of degree at
D: P2➡ P2
be the transformation that sends a polynomial p(t) = at² + bt+c in P2 to its derivative
p'(t)
2at+b, that is,
D(p) = p'.
(a) Prove that D is a linear transformation.
(b) Find a basis for the kernel ker(D) of the linear transformation D and compute its
nullity.
(c) Find a basis for the image im(D) of the linear transformation D and compute its
rank.
(d) Verify that the Rank-Nullity Theorem holds for the linear transformation D.
(e) Find the matrix representation of D in the standard basis (1,t, t2) of P2.
Chapter 6 Solutions
Elements Of Modern Algebra
Ch. 6.1 - True or False
Label each of the following...Ch. 6.1 - Label each of the following statements as either...Ch. 6.1 - True or false
Label each of the following...Ch. 6.1 - Label each of the following statements as either...Ch. 6.1 - Label each of the following statements as either...Ch. 6.1 - True or false
Label each of the following...Ch. 6.1 - True or false
Label each of the following...Ch. 6.1 - Label each of the following statements as either...Ch. 6.1 - Exercises Let I be a subset of ring R. Prove that...Ch. 6.1 - Prob. 2E
Ch. 6.1 - Prove or disprove each of the following...Ch. 6.1 - Exercises
If and are two ideals of the ring ,...Ch. 6.1 - Prob. 5ECh. 6.1 - Exercises
Find two ideals and of the ring such...Ch. 6.1 - Exercises
Let be an ideal of a ring , and let be...Ch. 6.1 - Exercises
If and are two ideals of the ring ,...Ch. 6.1 - Find the principal ideal (z) of Z such that each...Ch. 6.1 - Let I1 and I2 be ideals of the ring R. Prove that...Ch. 6.1 - Find a principal ideal (z) of such that each of...Ch. 6.1 - 12. Let be a commutative ring with unity. If...Ch. 6.1 - 13. Verify each of the following statements...Ch. 6.1 - 14. Let be an ideal in a ring with unity . Prove...Ch. 6.1 - Let I be an ideal in a ring R with unity. Prove...Ch. 6.1 - Prove that if R is a field, then R has no...Ch. 6.1 - In the ring of integers, prove that every subring...Ch. 6.1 - Let a0 in the ring of integers . Find b such that...Ch. 6.1 - 19. Let and be nonzero integers. Prove that if and...Ch. 6.1 - 20. If and are nonzero integers and is the least...Ch. 6.1 - Prove that every ideal of n is a principal ideal....Ch. 6.1 - 22. Let . Prove .
Ch. 6.1 - 23. Find all distinct principal ideals of for the...Ch. 6.1 - 24. If is a commutative ring and is a fixed...Ch. 6.1 - Given that the set S={[xy0z]|x,y,z} is a ring with...Ch. 6.1 - Prob. 26ECh. 6.1 - Prob. 27ECh. 6.1 - 28. a. Show that the set is a ring with respect to...Ch. 6.1 - 29. Let be the set of Gaussian integers . Let .
...Ch. 6.1 - a. For a fixed element a of a commutative ring R,...Ch. 6.1 - Let R be a commutative ring that does not have a...Ch. 6.1 - 32. a. Let be an ideal of the commutative ring ...Ch. 6.1 - 33. An element of a ring is called nilpotent if...Ch. 6.1 - 34. If is an ideal of prove that the set is an...Ch. 6.1 - Let R be a commutative ring with unity whose only...Ch. 6.1 - 36. Suppose that is a commutative ring with unity...Ch. 6.2 - True or false
Label each of the following...Ch. 6.2 - True or false
Label each of the following...Ch. 6.2 - Label each of the following statements as either...Ch. 6.2 - Label each of the following statements as either...Ch. 6.2 - Label each of the following statements as either...Ch. 6.2 - Label each of the following statements as either...Ch. 6.2 - Each of the following rules determines a mapping...Ch. 6.2 - 2. Prove that is commutative if and only if is...Ch. 6.2 - 3. Prove that has a unity if and only if has a...Ch. 6.2 - Prob. 4ECh. 6.2 - Prob. 5ECh. 6.2 - Prob. 6ECh. 6.2 - Assume that the set S={[xy0z]|x,y,z} is a ring...Ch. 6.2 - Assume that the set R={[x0y0]|x,y} is a ring with...Ch. 6.2 - 9. For any let denote in and let denote in .
a....Ch. 6.2 - Let :312 be defined by ([x]3)=4[x]12 using the...Ch. 6.2 - 11. Show that defined by is not a homomorphism.
Ch. 6.2 - 12. Consider the mapping defined by . Decide...Ch. 6.2 - Prob. 13ECh. 6.2 -
14. Let be a ring with unity . Verify that the...Ch. 6.2 - In the field of a complex numbers, show that the...Ch. 6.2 - Prob. 16ECh. 6.2 - Define :2()2(2) by ([abcd])=[[a][b][c][d]]. Prove...Ch. 6.2 - Prob. 18ECh. 6.2 - Prob. 19ECh. 6.2 - Prob. 20ECh. 6.2 - Prob. 21ECh. 6.2 - Prob. 22ECh. 6.2 - Prob. 23ECh. 6.2 - Prob. 24ECh. 6.2 - 25. Figure 6.3 gives addition and multiplication...Ch. 6.2 - Prob. 26ECh. 6.2 - 27. For each given value of find all homomorphic...Ch. 6.2 - Prob. 28ECh. 6.2 - 29. Assume that is an epimorphism from to ....Ch. 6.2 - 30. In the ring of integers, let new operations of...Ch. 6.2 - Prob. 31ECh. 6.3 - True or False
Label each of the following...Ch. 6.3 - Prob. 2TFECh. 6.3 - True or False
Label each of the following...Ch. 6.3 - True or False
Label each of the following...Ch. 6.3 - Prob. 5TFECh. 6.3 - Find the characteristic of each of the following...Ch. 6.3 - Find the characteristic of the following rings. 22...Ch. 6.3 - 3. Let be an integral domain with positive...Ch. 6.3 - Prob. 4ECh. 6.3 - Prob. 5ECh. 6.3 - Prob. 6ECh. 6.3 - Prob. 7ECh. 6.3 - 8. Prove that the characteristic of a field is...Ch. 6.3 - Let D be an integral domain with four elements,...Ch. 6.3 - Let R be a commutative ring with characteristic 2....Ch. 6.3 -
11. a. Give an example of a ring of...Ch. 6.3 - 12. Let be a commutative ring with prime...Ch. 6.3 - Prob. 13ECh. 6.3 - Prob. 14ECh. 6.3 - 15. In a commutative ring of characteristic 2,...Ch. 6.3 - A Boolean ring is a ring in which all elements x...Ch. 6.3 - 17. Suppose is a ring with positive...Ch. 6.3 - Prob. 18ECh. 6.3 - Prob. 19ECh. 6.3 - Let I be the set of all elements of a ring R that...Ch. 6.3 - 21. Prove that if a ring has a finite number of...Ch. 6.3 - 22. Let be a ring with finite number of...Ch. 6.3 - Prob. 23ECh. 6.3 - Prob. 24ECh. 6.3 - Prob. 25ECh. 6.3 - Prove that every ordered integral domain has...Ch. 6.4 - Label each of the following statements as either...Ch. 6.4 - Prob. 2TFECh. 6.4 - According to part a of Example 3 in Section 5.1,...Ch. 6.4 - Let R be as in Exercise 1, and show that the...Ch. 6.4 - Prob. 3ECh. 6.4 - Show that the ideal is a maximal ideal of .
Ch. 6.4 - Prob. 5ECh. 6.4 - Prob. 6ECh. 6.4 - Prob. 7ECh. 6.4 - Prob. 8ECh. 6.4 - Find all maximal ideals of .
Ch. 6.4 - Find all maximal ideals of 18.Ch. 6.4 - Let be the ring of Gaussian integers. Let
...Ch. 6.4 - Let R bethe ring of Gaussian integersas an...Ch. 6.4 - Prob. 13ECh. 6.4 - Prob. 14ECh. 6.4 - Prob. 15ECh. 6.4 - Prob. 16ECh. 6.4 - Prob. 17ECh. 6.4 - Prob. 18ECh. 6.4 - Prob. 19ECh. 6.4 - Prob. 20ECh. 6.4 - Find all prime ideals of .
Ch. 6.4 - Find all prime ideals of .
Ch. 6.4 - Prob. 23ECh. 6.4 - Prob. 24ECh. 6.4 - Prob. 25ECh. 6.4 - . a. Let, and . Show that and are only ideals...Ch. 6.4 - 27. If is a commutative ring with unity, prove...Ch. 6.4 - If R is a finite commutative ring with unity,...
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