i don't understand where you are getting those two matrices. suppose T is the matrix that turns the f-basis coordinates into e-basis coordinates. so T =

[a b]

[c d], and we know that:

a+2b = 1

c+2d = 2, since T([1,2]_{f}) = [1,2]_{e}.

we also know that:

T([1,-1]_{f}) = [1,1]_{e}(because f_{1}-f_{2}= e_{1}+e_{2}).

thus a+b = 1 and c-d = -1.

from the two equations involving a and b, we have b = 0, so a = 1. from the two equations involving c and d, we have:

3d = 1, so d = 1/3, so c = 4/3. that gives us T, but what we really WANT is T^{-1}, which is found to be:

.

this, of course, means that:

T^{-1}(x_{1},x_{2}) = (x_{1},-4x_{1}+3x_{2}).

it is easy to check that (1,2) is invariant under T^{-1}:

T^{-1}(1,2) = (1,(-4)(1) + (3)(2)) = (1,-4+6) = (1,2) and that:

T^{-1}(1,1) = (1,(-4)(1) + (3)(1)) = (1,-4+3) = (1,-1), as required.