Geometrically describe these Cosets and form a bijection with the Orbit-Stabilizer relation. I am beginning to study abstract algebra/group theory and I have some seemingly simple practice questions here. I just want make sure I am understanding the concepts correctly. Here are the questions:

Let $H = \{  \begin{pmatrix}   c & d  \\   0 & 1  \end{pmatrix} |
\space c,d \in \mathbb{R} , c > 0   \} $ and $ J = \{  \begin{pmatrix}
a & 0  \\   0 & 1  \end{pmatrix} | \space a \in \mathbb{R} , a > 0  \} $
Identify the element $\begin{pmatrix}   c & d  \\   0 & 1 \end{pmatrix}$ of $H$ with the point $(c,d) \in \mathbb{R}^2$ This
  identifies $H$ with a half plane in $\mathbb{R}^2$ .
(a) Describe the left and right cosets of $J$ geometrically in this
  half plane.
(b) The orbit-stabilizer relation states that there is a bijection
  between the orbit of a point and the left-cosets of its stabilizer.
  Describe this bijection explicitly for the orbit of $\begin{pmatrix}  
 0  \\   1  \end{pmatrix}$ under $H$ and the left cosets of it's
  stabilizer.

For (a), I have that the left cosets are of the form (for a fixed $h \in H$): 
$$ hJ = \begin{pmatrix}   c & d  \\   0 & 1 \end{pmatrix} \begin{pmatrix}   a & 0  \\   0 & 1 \end{pmatrix} = \begin{pmatrix}   ac & 0  \\   0 & 1 \end{pmatrix}$$ with $c>0$ fixed, $ac > 0$ and $a$ varying over $(0, \infty) $ for every $j \in J$. Geometrically, this would then be all the points on the $x$ axis $\in (0,\infty) $.
The right cosets are then:
$$ Jh = \begin{pmatrix}   a & 0  \\   0 & 1 \end{pmatrix} \begin{pmatrix}   c & d  \\   0 & 1 \end{pmatrix} = \begin{pmatrix}   ac & ad  \\   0 & 1 \end{pmatrix}$$ with $c>0$ fixed, $ac > 0$ and $a$ varying over $(0, \infty) $ for every $j \in J$, $d$ fixed and $a$ varying over $(0, \infty) $. 
Geometrically, if $d < 0$ then it is the bottom right quadrant (not including the axes) as $a$ varies over $\in (0,\infty) $. If $d = 0$ then it is all the points on the $x$ axis $\in (0,\infty) $, and if $d>0$ it is the top right quadrant (not including the axes) as $a$ varies over $\in (0,\infty) $.
Do I have the right idea here?
For (b), 
I'm not quite sure if how I have formed the bijection is correct. I have that the orbit of $\begin{pmatrix}  
 0  \\   1  \end{pmatrix}$ under $G$ are the points given by the vectors of the form $\begin{pmatrix} d  \\   1  \end{pmatrix}$ as $d$ varies over $(-\infty,\infty) $ and the stabilizer are matrices of the form $\begin{pmatrix}   c & 0  \\   0 & 1 \end{pmatrix}$ where $c$ varies over $(0,\infty) $.
Therefore, the left-cosets of the stabiliser of $\begin{pmatrix} 0  \\   1  \end{pmatrix}$ are of the form:
$$ \begin{pmatrix}   c & d  \\   0 & 1 \end{pmatrix} \begin{pmatrix}   c & 0  \\   0 & 1 \end{pmatrix} = \begin{pmatrix}   c^2 & d  \\   0 & 1 \end{pmatrix}$$ with $c^2$ varying over $(0,\infty)$ and $d$ varying over $(-\infty, \infty)$. Thus, I need to form a bijection between:
$$\begin{pmatrix}   c^2 & d  \\   0 & 1 \end{pmatrix} \rightarrow \begin{pmatrix} d  \\   1  \end{pmatrix}$$
It seems natural to simply map $d$ explicitly to itself for every $d \in \mathbb{R} $ as it varies over $(-\infty, \infty)$. Since $c$ and $d$ vary over infinite intervals, every $c^2$ which occurs in the left matrix will be paired with a $d$.
Is this a correct bijection?
I really appreciate any help you can give me. Thanks.
 A: Regarding part (a)
You are close, but there are some issues. First of all, the left cosets will be of the form
$$
hJ = \begin{pmatrix}   c & d  \\   0 & 1 \end{pmatrix} \begin{pmatrix}   a & 0  \\   0 & 1 \end{pmatrix} = \begin{pmatrix}   ac & d  \\   0 & 1 \end{pmatrix}
$$
Then to the conclusions:


*

*$hJ$ constitutes a half line $(x,d)$, where $d$ is some fixed $y$-coordinate and $x=ac\in(0,\infty)$

*For $Jh$ we have half lines of the form $(ac,ad)$ so in effect they can be described via $x=ac\in(0,\infty)$ and $y=ad$ which can be combined to $y=\frac{d}{c}\cdot x$ where $x>0$. For $d>0$ this is increasing, for $d=0$ it is constant and is one half of the x axis as you already stated, and for $d<0$ it is decreasing.


Regarding part (b)
You are close again, but again there are issues. First of all we have the stabilizer of $v=\begin{pmatrix} 0\\1\end{pmatrix}$ as
$$
H_v=\{\begin{pmatrix}a&0\\0&1\end{pmatrix}|\ a\in\mathbb R, a>0\}=J
$$
So in fact the stabilizer equals $J$ from part (a). There we already identified each left coset to be defined by $d$ being a line of the form $\{(x,d)|\ x>0\}$. Thus each left coset $hJ$ depends only on $d$ as does each element $\begin{pmatrix} d  \\   1  \end{pmatrix}$ in the orbit. For each fixed pair $c,d$ and the corresponding element $h=\begin{pmatrix}c&d\\0&1\end{pmatrix}\in H$ we have the bijections
$$
hJ\longmapsto\{(x,d)|\ x>0\}\longmapsto d\longmapsto\begin{pmatrix}d\\1\end{pmatrix}
$$
