Question 2.3.4: Image of a Rectangle under a Linear Mapping. Find the image ......

Let $S$ be the rectangle with the given vertices and let $S^{\prime }$ denote the image of $S$ under $f$. We will plot S and $S^{\prime }$ in the same copy of the complex plane. Because $f$ is a linear mapping, our foregoing discussion implies that $S^{\prime }$ has the same shape as $S$. That is, $S^{\prime }$ is also a rectangle. Thus, in order to determine $S^{\prime }$, we need only find its vertices, which are the images of the vertices of S under $f$:
$f (−1 + i) = −2 −i                f(1 + i) = −2 + 7i$
$f (1 + 2i) = −6 + 7i             f(−1 + 2i) = −6 − i.$
Therefore, $S^{\prime }$ is the rectangle with vertices −2−i, −2+7i, −6+7i, and −6−i.