Linear transformation followed by change of basis

Question

I have this problem:

I've attempted calculating: $[L(v_1)]_T$ but cannot understand where I'm going wrong. Apparently the answer is (1, -1). Here is my attempt:

Can anyone spot where I've gone wrong? I think I did something wrong in the transformation but I'm not sure why.

Answer 1

Hint: If $A$ is matrix of $L$ in pair of bases $S$ and $T$, to calculate $Lv_1$ using $A$ you first need to write $v_1$ in basis $S$.

Answer 2

HINT

In the given basis

$$L(v_1)=A\cdot v_1=[1,-1]=1\cdot w_1-1\cdot w_2$$

and so on.