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.