Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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299 3:18 ◄ Search × 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) = {(6 ;) : Here's a trickier example of a subgroup of GL(2, R): 9)}~{($ Prove that K is indeed a subgroup of GL(2, R). (You will probably recognize the elements of K from an earlier homework.) A = N = K = 5 -12 -3 : a > 0 and d > :bel d>0} cos sin 0 sin e cos coso sin o 6)} sin o 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 € K, a E A, and ne N (with K, A, N as in the last two problems). Your job: (a) If g = - cos o find k, a, n, such that g = kan. (b) If g = find k, a, n, such that g = kan. For both 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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Transcribed Image Text:299 3:18 ◄ Search × 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) = {(6 ;) : Here's a trickier example of a subgroup of GL(2, R): 9)}~{($ Prove that K is indeed a subgroup of GL(2, R). (You will probably recognize the elements of K from an earlier homework.) A = N = K = 5 -12 -3 : a > 0 and d > :bel d>0} cos sin 0 sin e cos coso sin o 6)} sin o 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 € K, a E A, and ne N (with K, A, N as in the last two problems). Your job: (a) If g = - cos o find k, a, n, such that g = kan. (b) If g = find k, a, n, such that g = kan. For both 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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