The question essentially asks:
We want to write (a1-a2,a1,2a1+a2) as a linear combination of gamma. Let the coefficients of this linear combination be x,y,z in that order.
Then given (a1,a2) in R^2, we want to find a matrix that maps it to T(a1,a2) which is written as a linear combination of gamma.
T(a1,a2) = (a1-a2,a1,2a1+a2) = x(1,1,0)+y(0,1,1)+z(2,2,3)
But observe that:
Thus the matrix maps (a1,a2) to (x,y,z)
I will let you compute that matrix....