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In Exercises 8 through 130, determine whether the given set of invertible n x n matrices with real number entries is a subgroup of GL(n, R). 8. The n xn matrices with determinant greater than or equal to 1 9. The diagonal n x n matrices with no zeros on the diagonal 10. The n x n matrices with determinant 2k for some integer k otrieee ith determinont

In Exercises 8 through 130, determine whether the given set of invertible n x n matrices with real number entries is a subgroup of GL(n, R). 8. The n xn matrices with determinant greater than or equal to 1 9. The diagonal n x n matrices with no zeros on the diagonal 10. The n x n matrices with determinant 2k for some integer k otrieee ith determinont

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercises 1 through 6 D, determine whether the given subset of the complex numbers is a subgroup of the group C of complex numbers addition. 1. R 2. Q+ 3. 7Z 4. The set iR of pure imaginary numbers including 0 5. The set TQ of rational multiples of T 6. The set {T" n E Z} 7. Which of the sets in Exercises 1 0 through 6 0 are subgroups of the group C* of nonzero complex numbers under multiplication? In Exercises 80 through 130, determine whether the given set of invertible n x n matrices with real number entries is a subgroup of GL(n, R). 8. The n x n matrices with determinant greater than or equal to 1 9. The diagonal n x n matrices with no zeros on the diagonal 10. The n x n matrices with determinant 2* for some integer k 11. The n xn matrices with determinant -1 12. The n x n matrices with determinant -1 or 1 13. The set of all n x n matrices A such that (AT)A = In. [These matrices are called orthogonal. Recall that AT, the transpose of A, is the matrix whose jth column is the jth row of A for 1<j<n, and that the transpose operation has the property (AB) = (B')(A').] Let F be the set of all real-valued functions with domain R and let F be the subset of F consisting of those functions that have a nonzero value at every point in R. In Exercises 14 O through 19 0, determine whether the given subset of F with the induced operation is (a) a subgroup of the group F under addition, (b) a subgroup of the group F under multiplication. 14. The subset F 15. The subset of all f e F such that f(1) = 0 16. The subset of all f E F such that f(1) = 1 Lecture8Notes.pdf cse11-pa3-starte.zip 20E old first midt.pdf Lecture7Notes.pdf MATH103A.Lect....pdf MacBook Pro 57
Transcribed Image Text:Exercises 1 through 6 D, determine whether the given subset of the complex numbers is a subgroup of the group C of complex numbers addition. 1. R 2. Q+ 3. 7Z 4. The set iR of pure imaginary numbers including 0 5. The set TQ of rational multiples of T 6. The set {T" n E Z} 7. Which of the sets in Exercises 1 0 through 6 0 are subgroups of the group C* of nonzero complex numbers under multiplication? In Exercises 80 through 130, determine whether the given set of invertible n x n matrices with real number entries is a subgroup of GL(n, R). 8. The n x n matrices with determinant greater than or equal to 1 9. The diagonal n x n matrices with no zeros on the diagonal 10. The n x n matrices with determinant 2* for some integer k 11. The n xn matrices with determinant -1 12. The n x n matrices with determinant -1 or 1 13. The set of all n x n matrices A such that (AT)A = In. [These matrices are called orthogonal. Recall that AT, the transpose of A, is the matrix whose jth column is the jth row of A for 1<j<n, and that the transpose operation has the property (AB) = (B')(A').] Let F be the set of all real-valued functions with domain R and let F be the subset of F consisting of those functions that have a nonzero value at every point in R. In Exercises 14 O through 19 0, determine whether the given subset of F with the induced operation is (a) a subgroup of the group F under addition, (b) a subgroup of the group F under multiplication. 14. The subset F 15. The subset of all f e F such that f(1) = 0 16. The subset of all f E F such that f(1) = 1 Lecture8Notes.pdf cse11-pa3-starte.zip 20E old first midt.pdf Lecture7Notes.pdf MATH103A.Lect....pdf MacBook Pro 57
One Page U Two Page HContinuous Reading Background Translate Screen Grab Co 5. Subgroups Subgroups Yes No, there is no identity element. 5. Yes 4. Yes 5. Yes 6. No, the set is not closed under addition. 7. Q* and {T" |ne Z} Not a subgroup. The inverse of the diagonal matrix with all diagonal entries 2 has determinant ()" <1. 8. 9. Yes 10. This is a subgroup. The set is closed under matrix multiplication since the product of two powers of 2 is a power of 2, the identity matrix has determinant a = 2º and if a matrix has determinant 2k, then its inverse has determinant 2-k. 11. No. If det(A) = det(B) = –1, then det(AB) = det(A)det(B) = 1. The set is not closed under multiplication. 12. Yes. This follows from the fact that det ( AB)= det (A) ·det ( B ). Aa MacBook Pro < > < * ビ」
Transcribed Image Text:One Page U Two Page HContinuous Reading Background Translate Screen Grab Co 5. Subgroups Subgroups Yes No, there is no identity element. 5. Yes 4. Yes 5. Yes 6. No, the set is not closed under addition. 7. Q* and {T" |ne Z} Not a subgroup. The inverse of the diagonal matrix with all diagonal entries 2 has determinant ()" <1. 8. 9. Yes 10. This is a subgroup. The set is closed under matrix multiplication since the product of two powers of 2 is a power of 2, the identity matrix has determinant a = 2º and if a matrix has determinant 2k, then its inverse has determinant 2-k. 11. No. If det(A) = det(B) = –1, then det(AB) = det(A)det(B) = 1. The set is not closed under multiplication. 12. Yes. This follows from the fact that det ( AB)= det (A) ·det ( B ). Aa MacBook Pro < > < * ビ」
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