Linear Transformations Given Tx y 2y x x 4x y a Get the i

Linear Transformations Given T(x, y) = (2y - x, x, 4x - y): a. Get the image of (8, 6) b. Get the preimage of (3, 1, 2) Given T(x, y, z) = (-3z + x, y - 4x). a. Get the image of (-5, 1, 4) b. Get the preimage of (2, 9)

Solution

1. image of (8,6) under T(x,y) = 2y-x,x,4x-y = 4,8,26

Preimage of 3,1,2 is the inverse, so solve 3=2y-x, 1=x, 2=4x-y ==> x=1, y=2

2) Image of ( -5,1,4) under T(x,y,z) = (-3z+x,y-4x) = -17,21

Preimage of 2,9 : -3z+x =2

                                   y-4x = 9

Since there are three unknowns x,y,z for two eqns, have infinite number of solns.

eliminate x, get 17=y-12z , x=2+3z