Here is the question:
Help please thanks!
I am assumeing they want this with respect to the standard basis!
With that said
let $\displaystyle e_1=(1,0)$ and $\displaystyle e_2=(0,1)$
Then
$\displaystyle (2,-2)=2e_1-2e_2 \iff T(2e_1-2e_2) =2T(e_1)-2T(e_2) = 16e_1+4e_2$
This first equality holds by the Linearity of T
Now we do the same this for the 2nd vector to get
$\displaystyle T(-4e_1-3e_2)=-4T(e_1)-3T(e_2)=-39e_1-36e_2$
Now we have a system of equations for
$\displaystyle T(e_1) \text{ and } T(e_2)$
These are the columns of the matrix.