TITLE: Geometric interpretation of linear transformation
QUESTION [6 upvotes]: 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?

REPLY [4 votes]: The determinant of the matrix of the linear transformation reveals a fair amount of the geometry. Consider $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ where $L(v)=Av$ for $A \in \mathbb{R}^{2 \times 2}$. Suppose $v_1,v_2$ are vectors which are the sides of a parallelogram $P$. It is a simple exercise to show that $Area(P) = det[v_1|v_2]$. Consider then that the linear transformation takes $v_1,v_2$ to the transformed vector(s) $L(v_1),L(v_2)$. The area of $L(p)$ is also given by $Area(L(P)) = det[L(v_1)|L(v_2)]$. Observe that
$$ Area(L(P)) = det[ Av_1|Av_2] = det(A[v_1|v_2])=det(A)det[v_1|v_2] = det(A)Area(P) $$
Consider the possibilities:

$det(A)=0$ it follows the $L(P)$ is a point or a line ($A=0$ to get a point)
$det(A) >1$ then the orientation of $P$ is maintained and it expands.
$det(A) <-1$ then the orientation of $P$ is reversed and it expands.
$0<det(A) <1$ then the orientation of $P$ is maintained and it shrinks.
$-1<det(A)<0$ then the orientation of $P$ is reversed and it shrinks.

To describe the details of how $P$ is shrunk or expanded you can study the eigenvectors, generalized eigenvector and complex eigenvectors which arise in various cases as has been pointed towards in the other answers thus far.