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?