isomorphism and inverse

Question

Let T : R² → R² be the linear transformation

T(x, y) = (2y−x,3y−x)

Compute the value of the inverse

T^-1 (3,3)=?

Answer 1

so you want to find x,y such that

2y-x = 3

3y-x = 3

subtract both equations

(2y-x)-(3y-x) = 3-3 = 0

that is

-y = 0 so y = 0

Hence 2(0) - x = 3 so x = -3

Hence T^-1 (3,3)= (-3,0)