1. Mapping one parallelogram to another

It is a fact that an affine transformation takes lines to lines parallel lines to parallel lines (all provided that it is nonsingular). Therefore nonsingular affine transformations in R$^ 2$ take parallelograms to parallelograms.

Problem 1.1 . In R$^ 2$, is it possible to take an arbitrary parallelogram $PQRS$ to an arbitrary parallelogram $P' Q' R' S'$ using an affine transformation? (See Figure .)

Figure: Mapping parallelograms

book/03dir/parallelogram.eps

Solution. Yes: Just find $T$ taking the triangle $PQR$ to the triangle $P' Q' R'$. In order for the image of the parallelogram to be a parallelogram, $T(S)$ will automatically be the point $S'$, as desired.

You can think of this solution in two steps: Taking the first parallelogram to the ``standard square'' and taking the ``standard square'' to the second parallelogram. Of course, either parallelogram or both could actually be a rectangle or even a square.