3. Find equations for a transformation $$ T $$ that maps a rectangular region $$ S $$ in the $$ u v $$-plane onto the region $$ R $$ consisting of the parallelogram with vertices $$ (0,0),(4,3),(2,4) $$, and $$ (-2,1) $$.

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Let the transformation be of the form

$$
T(u,v) = (x,y) = (au + bv,\; cu + dv).
$$

We wish to map the unit square
$$
S = \{ (u,v) : 0 \le u \le 1,\; 0 \le v \le 1 \}
$$
onto the parallelogram $$ R $$ with vertices
$$
(0,0),\;(4,3),\;(2,4),\;(-2,1).
$$

A natural idea is to choose
$$
T(0,0) = (0,0),
$$
and to have the edges of the square map to the sides of the parallelogram. In particular, let
$$
T(1,0) = (a,c) = (4,3) \quad\text{and}\quad T(0,1) = (b,d) = (-2,1).
$$

Then, by linearity, the image of the point $$(1,1)$$ is
$$
T(1,1) = T(1,0) + T(0,1) = (4 + (-2),\; 3 + 1) = (2,4).
$$

Thus, the vertices of the image of the unit square are
$$
(0,0),\; (4,3),\; (-2,1),\; (2,4),
$$
which matches the vertices of $$ R $$.

So the transformation is given by

$$
x = 4u - 2v,\qquad y = 3u + v.
$$

This mapping transforms the unit square $$ S $$ in the $$ uv $$-plane onto the parallelogram $$ R $$ in the $$ xy $$-plane as required.
The transformation $$ T $$ that maps the unit square $$ S $$ in the $$ uv $$-plane onto the parallelogram $$ R $$ with vertices $$ (0,0) $$, $$ (4,3) $$, $$ (2,4) $$, and $$ (-2,1) $$ is given by: $$ x = 4u - 2v,\qquad y = 3u + v. $$ This transformation ensures that the unit square is correctly mapped onto the specified parallelogram.

Find equations for a transformation that maps a rectangular region in the -plane
onto the region consisting of the parallelogram with vertices , and
.

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Find equations for a transformation that maps a rectangular region in the -plane onto the region consisting of the parallelogram with vertices , and .