Please answer the question shown in the image below: (Hint: the group is isomorphic to the group C) **Problem (18)**Let \( H \) be the set of all matrices in \( GL_2(\mathbb{R}) \) of the form \[\begin{pmatrix} a & -b \\ b & a \end{pmatrix}.\]Show that \( H \) is a subgroup of \( GL_2(\mathbb{R}) \). The group \( H \) is isomorphic to what well-known group? Prove your answer correct. **Explanation:**This problem involves linear algebra and group theory. It asks you to demonstrate that a particular set of matrices forms a subgroup of the general linear group \( GL_2(\mathbb{R}) \), which is the group of invertible 2x2 matrices with real number entries. Then, it asks you to identify a well-known group that is isomorphic to this set and prove the isomorphism.1. **Definition of Subgroup**: To show that \( H \) is a subgroup, you need to verify: - The identity matrix is in \( H \). - \( H \) is closed under matrix multiplication. - Every matrix in \( H \) has an inverse that is also in \( H \).2. **Matrix Form and Isomorphism**: - The matrices in \( H \) are of a specific form. They resemble rotations in the plane, which are linked to complex numbers. - The group \( H \) is likely isomorphic to the circle group \( S^1 \) or \( U(1) \), which represents the set of complex numbers with absolute value 1, under multiplication.3. **Proving Isomorphism**: - Show a bijective homomorphism between \( H \) and the known group. - Verify the homomorphism properties and inverse mapping.This problem combines abstract algebra with matrix transformations, highlighting connections with other areas like complex numbers and geometry.

Answered: Image /qna-images/answer/9171e6c8-d825-4b01-aab8-988a1c84eb8e.jpg

Answered: а H be the set of all matrices in…

а H be the set of all matrices in GL2(R) of the form b. a

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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