Let R" = {(a,, a, . .. , a) l a, E R}. Show that the mapping o: (a,, Az, ... , a) → (-q,, -a, · . the group R" under componentwise addition. This automorphism is called inversion. Describe the action of o geometrically. -a) is an automorphism of
Let R" = {(a,, a, . .. , a) l a, E R}. Show that the mapping o: (a,, Az, ... , a) → (-q,, -a, · . the group R" under componentwise addition. This automorphism is called inversion. Describe the action of o geometrically. -a) is an automorphism of
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
![Let R" = {(a,, a, . .. , a) l a, E R}. Show that the mapping o: (a,,
Az, ... , a) → (-q,, -a, · .
the group R" under componentwise addition. This automorphism is
called inversion. Describe the action of o geometrically.
-a) is an automorphism of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F43601f8b-0f68-407e-a0b9-4680976e214c%2F326f6f84-a9c4-4043-a04b-ad896f5358a0%2Fjrsqv7.jpeg&w=3840&q=75)
Transcribed Image Text:Let R" = {(a,, a, . .. , a) l a, E R}. Show that the mapping o: (a,,
Az, ... , a) → (-q,, -a, · .
the group R" under componentwise addition. This automorphism is
called inversion. Describe the action of o geometrically.
-a) is an automorphism of
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
Step 1
Given:
We need to show that the mapping is an automorphism of the group Also, the geometrical action of needs to be described.
Step 2
It is clear that is the identity, so has an inverse and must be a bijection. Now, we have -
, so :
Finally,
This proves that is an automorphism of under component wise addition.
Step by step
Solved in 3 steps
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