(886, #7) Find the image of the set S = {(u,v) : 0

<= u <= 2, 0 <= v <= 1} under the

(886, #7) Find the image of the set S = {(u,v) : 0 <= u <= 2, 0 <= v <= 1} under the transformation
x = u - 2 v, y = 2 u - v.

Solution: The rectangle in the uv-plane is at the left, its image in the xy-plane is at the right:

As for the analysis:

on v = 0
x = u, u = 0..2
y = 2u
y = 2x
the image is the line from (0,0) to (2,4) on u = 2
x = 2 - 2v, v = 0..1
y = 4 - v
y - x/2 = 3
the image is the line from (2,4) to (0,3)

on v = 1
x = u - 2, u = 2..0
y = 2 u - 1
y - 2x = 3
the image is the line from (0,3) to (-2,-1) on u = 0
x = -2 v, v = 1 ..0
y = -v
y - x/2 = 0
the image is the line from (-2,-1) to (0,0)

on v = 0 x = u, u = 0..2 y = 2u y = 2x the image is the line from (0,0) to (2,4)	on u = 2 x = 2 - 2v, v = 0..1 y = 4 - v y - x/2 = 3 the image is the line from (2,4) to (0,3)
on v = 1 x = u - 2, u = 2..0 y = 2 u - 1 y - 2x = 3 the image is the line from (0,3) to (-2,-1)	on u = 0 x = -2 v, v = 1 ..0 y = -v y - x/2 = 0 the image is the line from (-2,-1) to (0,0)