<s>[INST] Does the linear map represent a rotation, a shearing or a stretching? The unit square has the coordinates $\begin{pmatrix}0 \\ 0\end{pmatrix},\ \begin{pmatrix}1 \\ 0\end{pmatrix}, \ \begin{pmatrix}1 \\ 1\end{pmatrix}, \ \begin{pmatrix}0 \\ 1\end{pmatrix}$, and its graph is: 
 
We consider the linear map $F_1:\mathbb{R}^2\rightarrow \mathbb{R}^2$, $\vec{x}\mapsto A_i\vec{x}$ with $A_1:=\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}$. 
I want to find the image of the unit square under this map. 
We have 
\begin{align*}&F_1\begin{pmatrix}0 \\ 0\end{pmatrix}=\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}\begin{pmatrix}0 \\ 0\end{pmatrix}=\begin{pmatrix}0 \\ 0\end{pmatrix} \\ 
&F_1\begin{pmatrix}1 \\ 0\end{pmatrix}=\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}\begin{pmatrix}1 \\ 0\end{pmatrix}=\begin{pmatrix}2 \\ 0\end{pmatrix} \\ 
&F_1\begin{pmatrix}1 \\ 1\end{pmatrix}=\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}\begin{pmatrix}1 \\ 1\end{pmatrix}=\begin{pmatrix}2 \\ 1\end{pmatrix} \\ 
&F_1\begin{pmatrix}0 \\ 1\end{pmatrix}=\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}\begin{pmatrix}0 \\ 1\end{pmatrix}=\begin{pmatrix}0 \\ 1\end{pmatrix}\end{align*} 
The image of the unt square under $F_1$ is: 

Now I want to determine if this is a rotation, a shearing or a stretching. 
I think that this is a stretching, but I am not sure because the result is not a square it is now an orthogonal parallelogram. 
Could you please clarify if it indeed a stretching or something else? 
 [/INST] The transformation scales the first coordinate by a factor of two, so it's a stretch.
Shearing matrices have the form
$$
\begin{bmatrix}
1 & a \\
0 & 1
\end{bmatrix}
$$ 
or
$$
\begin{bmatrix}
1 & 0 \\
a & 1
\end{bmatrix},
$$
with $a \in \mathbb{R}$. 
Rotation matrices are of the form
$$
\begin{bmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{bmatrix},
$$ 
with $\theta \in [0,2\pi)$.
</s>