If I'm understanding this correctly (I too do not understand the notation used here but I have solved similar problems).

Let A = {1, x, x^2} be our standard basis in P^2.

Then T(A) = {T(1), T(t), T(t^2)} = {2*0 - 3*1, 2*1 - 3*t, 2*2t - 3*t^2} = {-3, 2-3t, 4t-3t^2} = {-3 + 0t + 0t^2, 2 - 3t + 0t^2, 0 + 4t - 3t^2}

Then matrix = [-3; 2; 0]

.....................[0; -3; 4]

.....................[0; 0; -3]