Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.](https://content.bartleby.com/qna-images/question/e5ef7b9b-7f40-491c-b8c0-9669ee1e8bbc/9171e6c8-d825-4b01-aab8-988a1c84eb8e/w661ake_processed.jpeg)
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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