(1) Show H(R) is a group under matrix multiplication, called the Heisenberg Group.(2) Find an explicit example of matrices A, B ∈ H(R) such that AB ̸= BA.(3) Is H(R) a subset of GL3 (R)? **Exercise 3.2.13**Let \[ H(\mathbb{R}) = \left\{ \begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix} : a, b, c \in \mathbb{R} \right\}.\]1. Show \( H(\mathbb{R}) \) is a group under matrix multiplication, called the **Heisenberg Group**.2. Find an explicit example of matrices \( A, B \in H(\mathbb{R}) \) such that \( AB \neq BA \).3. Is \( H(\mathbb{R}) \) a subset of \( GL_3(\mathbb{R}) \)?

The main objective is to prove H(R) is group.

Exercise 3.2.13. Let H(R) a - {(1; i)=ancex} = a, b, 0 0 1 (1) Show H(R) is a group under matrix multiplication, called the Heisenberg Group. (2) Find an explicit example of matrices A, B € H(R) such that AB ‡ BA. (3) Is H(R) a subset of GL3(R)?

Exercise 3.2.13. Let H(R) a - {(1; i)=ancex} = a, b, 0 0 1 (1) Show H(R) is a group under matrix multiplication, called the Heisenberg Group. (2) Find an explicit example of matrices A, B € H(R) such that AB ‡ BA. (3) Is H(R) a subset of GL3(R)?

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

(1) Show H(R) is a group under matrix multiplication, called the Heisenberg Group.
(2) Find an explicit example of matrices A, B ∈ H(R) such that AB ̸= BA.
(3) Is H(R) a subset of GL3 (R)?

**Exercise 3.2.13**

Let

\[
H(\mathbb{R}) = \left\{
\begin{pmatrix}
1 & a & c \\
0 & 1 & b \\
0 & 0 & 1
\end{pmatrix}
: a, b, c \in \mathbb{R}
\right\}.
\]

1. Show \( H(\mathbb{R}) \) is a group under matrix multiplication, called the **Heisenberg Group**.

2. Find an explicit example of matrices \( A, B \in H(\mathbb{R}) \) such that \( AB \neq BA \).

3. Is \( H(\mathbb{R}) \) a subset of \( GL_3(\mathbb{R}) \)?

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