"beta be the standard basis for R^2" => T(1,0) = (1,1,2) and T(0,1) = (-1,0,1)

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.

Thus:

T(a1,a2) = (a1-a2,a1,2a1+a2) = x(1,1,0)+y(0,1,1)+z(2,2,3)

But observe that:

So:

This means,

Thus the matrix maps (a1,a2) to (x,y,z)

I will let you compute that matrix....