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2. Let G = GL(2, R). Prove that the following two subsets of GL(2, R) are subgroups of GL(2, R). (a) -{(82) A = d>0} a> 0 and d >

2. Let G = GL(2, R). Prove that the following two subsets of GL(2, R) are subgroups of GL(2, R). (a) -{(82) A = d>0} a> 0 and d >

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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1. Let K be a subgroup of R*. Let H = {g E GL(n, R): det (g) E K}. Prove that H is a subgroup of GL(n, R). 2. Let G = GL(2, R). Prove that the following two subsets of GL(2, R) are subgroups of GL(2, R). (a) (b) 3. {(82) b = {(1 i): bER} A = N = Here's a trickier example of a subgroup of GL(2, R): cos 6-sin)} {( sin = {(² K = : a>0 and d and d d>0} > > 0 5 3 -12 cos o sin o sin o >)} - cos p Prove that K is indeed a subgroup of GL(2, R). (You will probably recognize the elements of K from an earlier homework.) 4. There is a theorem that says that every element g E GL(2, R) can be written, in a unique way, as kan for some k EK, a E A, and n E N (with K, A, N as in the last two problems). Your job: (a) If g = find k, a, n, such that g = kan. -3 -17 (b) If g = √(3 find k, a, n, such that g = 7 kan. For both of these, show your work and explain how you found your answers. Helpful fact: if det g> 0, then k will be a rotation, and if det g < 0, then k will be a reflection.
Transcribed Image Text:1. Let K be a subgroup of R*. Let H = {g E GL(n, R): det (g) E K}. Prove that H is a subgroup of GL(n, R). 2. Let G = GL(2, R). Prove that the following two subsets of GL(2, R) are subgroups of GL(2, R). (a) (b) 3. {(82) b = {(1 i): bER} A = N = Here's a trickier example of a subgroup of GL(2, R): cos 6-sin)} {( sin = {(² K = : a>0 and d and d d>0} > > 0 5 3 -12 cos o sin o sin o >)} - cos p Prove that K is indeed a subgroup of GL(2, R). (You will probably recognize the elements of K from an earlier homework.) 4. There is a theorem that says that every element g E GL(2, R) can be written, in a unique way, as kan for some k EK, a E A, and n E N (with K, A, N as in the last two problems). Your job: (a) If g = find k, a, n, such that g = kan. -3 -17 (b) If g = √(3 find k, a, n, such that g = 7 kan. For both of these, show your work and explain how you found your answers. Helpful fact: if det g> 0, then k will be a rotation, and if det g < 0, then k will be a reflection.
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