Transforming A reas Fo $r$ Problems 49-52, let $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be defined by $T(\overrightarrow{\mathbf{v}})=\mathbf{A} \overrightarrow{\mathbf{v}}$, where
$$
A=\left[\begin{array}{rr}
1 & -1 \\
2 & 1
\end{array}\right] .
$$
Determine the image under the map of the square having vertices $(0,0),(1,0),(1,1)$, and $(0,1)$. Calculate and compare the areas of the square and its image.