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

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Chapter2: Second-order Linear Odes
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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.
Transcribed Image Text:**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.
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