Please prove it with explanation Exercises 1 through 6 D, determine whether the given subset of the complex numbers is a subgroup of the group C of complex numbersaddition.1. R2. Q+3. 7Z4. The set iR of pure imaginary numbers including 05. The set TQ of rational multiples of T6. 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 19. The diagonal n x n matrices with no zeros on the diagonal10. The n x n matrices with determinant 2* for some integer k11. The n xn matrices with determinant -112. The n x n matrices with determinant -1 or 113. 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 matrixwhose jth column is the jth row of A for 1 One Page U Two Page HContinuous ReadingBackground TranslateScreen GrabCo5. SubgroupsSubgroupsYesNo, there is no identity element.5.Yes4.Yes5.Yes6.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 hasdeterminant ()" <1.8.9.Yes10. This is a subgroup. The set is closed under matrix multiplication since the productof two powers of 2 is a power of 2, the identity matrix has determinant a = 2º andif 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 closedunder multiplication.12. Yes. This follows from the fact that det ( AB)= det (A) ·det ( B ).AaMacBook Pro< > < *ビ」

Answered: Image /qna-images/answer/5fba8488-f89f-42c7-9fa7-9c00e03b644d.jpg

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Please prove it with explanation

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
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