Let D be the parallelogram with the vertices at 1 3 0 0 2 1

Let D be the parallelogram with the vertices at (-1, 3), (0, 0), (2, -1), and (1, 2), and E be the rectangle E = [0, 2] x [0, 3].

Find a linear map T, such that T(D) = E.

(3) Let D be the parallelogram with vertices at (-1,3), (0,0), (2,-1), and (1, 2), and E be the rectangle E = [0.2] × [0.3]. Find a linear map T. such that T(D) = E. SUC

Solution

We are required to find a linear mapping T with T(D) = E,

To do this we seek a linear mapping T( u , v) = ( x , y) of the form

       x = au+ bv and y = cu + dv .

we require vertices to be mapped to vertices in the same clockwise order and observe that

,we alredy have T( 0 ,0) = ( 0, 0) , thus we suppose T( 1 , 2) = ( 2 , 3) , T( -1 , 3) = (2 ,0)

and T( 2 ,-1) = ( 0 ,3) .

This gives us three set of equtions ,

a + 2b = 2 , c + 2d = 3                   ------------   1)

-a + 3b = 2 , - c + 3d = 0                ------------   2)

2a - b =0   , 2c - d = 3                 ------------   3)

====> Form 3)   we have    b = 2a plug it in 1) then we get a = 2 / 5 , so b = 4 / 5

           From 3) we have c = 3d   plug it in 1) then we get c = 3/ 7 so d = 9/7.

So the linear map is T( u , v ) = ( x , y) where x = (2u+ 4v)/ 5 and y = (3u + 9v )/7.