2. Let G = GL(2, R). Prove that the following two subsets of GL(2, R) are subgroups of GL(2, R). (a) (b) {(89) b = {(1 1): DER} A = N = d>0} : a>0 and d >
2. Let G = GL(2, R). Prove that the following two subsets of GL(2, R) are subgroups of GL(2, R). (a) (b) {(89) b = {(1 1): DER} A = N = d>0} : a>0 and d >
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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I need help with 4a
![## Problem Statement
2. Let \( G = GL(2, \mathbb{R}) \). Prove that the following two subsets of \( GL(2, \mathbb{R}) \) are subgroups of \( GL(2, \mathbb{R}) \).
### (a)
\[
A = \left\{
\begin{pmatrix}
a & 0 \\
0 & d
\end{pmatrix}
: a > 0 \text{ and } d > 0
\right\}
\]
### (b)
\[
N = \left\{
\begin{pmatrix}
1 & b \\
0 & 1
\end{pmatrix}
: b \in \mathbb{R}
\right\}
\]
3. Here’s a trickier example of a subgroup of \( GL(2, \mathbb{R}) \):
\[
K = \left\{
\begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix}
\right\}
\cup
\left\{
\begin{pmatrix}
\cos \phi & \sin \phi \\
\sin \phi & -\cos \phi
\end{pmatrix}
\right\}
\]
Prove that \( K \) is indeed a subgroup of \( GL(2, \mathbb{R}) \).
(You will probably recognize the elements of \( K \) from an earlier homework.)
4. There is a theorem that says that every element \( g \in GL(2, \mathbb{R}) \) can be written, in a unique way, as \( kan \) for some \( k \in K \), \( a \in A \), and \( n \in N \) (with \( K, A, N \) as in the last two problems). Your job:
### (a)
If \( g =
\begin{pmatrix}
0 & 5 \\
3 & -12
\end{pmatrix}
\), find \( k, a, n \), such that \( g = kan \).
### (b)
If \( g =
\begin{pmatrix}
-3 & -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27b38c32-736b-4354-8263-82b5dd937d13%2F89fe86d5-939a-456f-a8e3-a45344b225ab%2Fviyh8n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Problem Statement
2. Let \( G = GL(2, \mathbb{R}) \). Prove that the following two subsets of \( GL(2, \mathbb{R}) \) are subgroups of \( GL(2, \mathbb{R}) \).
### (a)
\[
A = \left\{
\begin{pmatrix}
a & 0 \\
0 & d
\end{pmatrix}
: a > 0 \text{ and } d > 0
\right\}
\]
### (b)
\[
N = \left\{
\begin{pmatrix}
1 & b \\
0 & 1
\end{pmatrix}
: b \in \mathbb{R}
\right\}
\]
3. Here’s a trickier example of a subgroup of \( GL(2, \mathbb{R}) \):
\[
K = \left\{
\begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix}
\right\}
\cup
\left\{
\begin{pmatrix}
\cos \phi & \sin \phi \\
\sin \phi & -\cos \phi
\end{pmatrix}
\right\}
\]
Prove that \( K \) is indeed a subgroup of \( GL(2, \mathbb{R}) \).
(You will probably recognize the elements of \( K \) from an earlier homework.)
4. There is a theorem that says that every element \( g \in GL(2, \mathbb{R}) \) can be written, in a unique way, as \( kan \) for some \( k \in K \), \( a \in A \), and \( n \in N \) (with \( K, A, N \) as in the last two problems). Your job:
### (a)
If \( g =
\begin{pmatrix}
0 & 5 \\
3 & -12
\end{pmatrix}
\), find \( k, a, n \), such that \( g = kan \).
### (b)
If \( g =
\begin{pmatrix}
-3 & -
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