I have a linear transformation, given by the following matrix
$$
\begin{pmatrix}
x_1\\
x_2
\end{pmatrix}
\mapsto
\begin{pmatrix}
2 & 2\\
-1 & -1\\
\end{pmatrix}
\begin{pmatrix}
x_1\\
x_2
\end{pmatrix}
$$
How can I determine what this corresponds to geometrically, when I apply it to the $x_1,x_2$-plane. I have tried to visualise the transformation by hand, and using a fieldplot in Maple, to get an idea of what is happening. My idea was then to decompose it into scaling, rotation, reflection or some other simple transformations.
My question is: Which geometric transformation does the above linear map correspond to, and in general, what is a good strategy for solving this kind of problem?
Best Answer
The better approach is to see what happens with the elements of the canonical base. So, you have two vectors $e_1=(1,0)$ and $e_2=(0,1)$. You have the unity square generated by them and its image could be a rectangle, or parallelogram, or a line, etc....
In our case, $e_1$ is mapped to the vector $(2,-1)$. The same for $e_2$. So the image is a line, namely, generated by $v=(2,-1)$.