For $T(x,y)=(2x+y,x-y)$, the standard matrix is $\begin{bmatrix}2&1\\1&-1\end{bmatrix}$. It sends $e_1$ to $(2,1)$ and $e_2$ to $(1,-1)$. The determinant is $-3$, so the map is invertible, scales areas by three, and reverses orientation.
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