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)

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.
