The problem asks to prove that $H = \left\{ \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix} : n \in \mathbb{Z} \right\} \subseteq GL(2, \mathbb{R})$ is a cyclic subgroup of $GL(2, \mathbb{R})$ and to show that the matrices $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$ are generators of $H$.

Key Steps:

1. Matrix Powers: Each matrix in $H$ is of the form $\begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix}$. For such a matrix, the product of two such matrices is: $\begin{pmatrix} 1 & n_1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & n_2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & n_1 + n_2 \\ 0 & 1 \end{pmatrix}$ This implies that the group operation in $H$ is essentially addition on $n \in \mathbb{Z}$.

2. Cyclic Nature: Since the group operation corresponds to addition, $H$ is isomorphic to $\mathbb{Z}$, which is a cyclic group. Therefore, $H$ is also a cyclic group.

3. Generators:

   • Consider the matrix $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$. For any positive integer $n$, we have: $A^n = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^n = \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix}$
   • Similarly, consider $B = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$. For any negative integer $n$, we have: $B^n = \begin{pmatrix} 1 & -n \\ 0 & 1 \end{pmatrix}$ Hence, the matrices $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$ generate the entire group $H$, since every matrix in $H$ can be expressed as a power of $A$ or $B$.

Thus, $H$ is a cyclic group, and both $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$ are generators.

Would you like more details on the matrix group properties, or do you have any questions?

Here are some related questions:

1. What does it mean for a group to be cyclic?
2. How do we prove that a matrix group is isomorphic to $\mathbb{Z}$?
3. Can a cyclic group have more than one generator?
4. What is the significance of the general linear group $GL(2, \mathbb{R})$?
5. How do matrix powers work in other types of matrix groups?

Tip: When working with matrix groups, keep in mind how matrix multiplication relates to group operations, often revealing underlying structures like isomorphisms to familiar groups.