<s>[INST] Describing a linear map geometrically I have the following linear map $\mathbb{R}^2\to\mathbb{R}^2:$
$$\begin{pmatrix}x\\y\end{pmatrix}\mapsto \begin{pmatrix}9y-5x\\7y-4x\end{pmatrix}$$
I am asked to describe this geometrically. 
I can see what is happening: the upper left quadrant is being squeezed into the upper right quadrant, in between the lines $y=\frac{7}{9}x$ and $y=\frac{4}{5}x$. Likewise the lower right quadrant is scrambled in between the same lines in the lower left quadrant. The other quadrants are stretched in the obvious way.
I can't find the proper terms to describe this well though. A rotation followed by a squeeze? Thanks to anyone who responds.
(I can upload an image if anyone is interested, but the lines are so close that it will not be very clear)
 [/INST] Suppose you have the map
$$
T_2
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
9y + x \\
y
\end{pmatrix}.
$$
Then this is called a horizontal shear. There is more information here: Shear mapping. Vertical shear is similarly defined. Now consider three additional maps
$$
T_1
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
-5x \\
y
\end{pmatrix} \\
T_3
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
x \\
-\frac{1}{5}y
\end{pmatrix} \\
T_4
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
x \\
y + \frac{4}{5}x
\end{pmatrix}.
$$
Then the given map is just
$$
T_4T_3T_2T_1
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
9y - 5x \\
7y - 4x
\end{pmatrix}.
$$
So the map is a combination of horizontal scaling followed by horizontal shear followed by vertical scaling and finally a vertical shear. Note that you can write transformations like this in terms of simpler transformations in many different ways. So the one I gave is just one possible way.
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