If T : R^2 -----> R^2 is a linear map defined by T(a,b)=(a+b,a-b)
what is the matrix of T:
1) using the standard basis of R^2?
2) using the basis (-1,1),(0,-2) of R^2
Using the standard basis
$\displaystyle e_1=(1,0), e_2=(0,1)$ Then
$\displaystyle T(e_1)=(1,1)=e_1+e_2$ and
$\displaystyle T(e_2)=(1,-1)=e_1-e_2$
Remember that the transform of the basis vectors becomes the columns in the matrix representation.
This gives the standard matrix as
$\displaystyle \begin{bmatrix} 1 & 1 \\ 1 & -1\end{bmatrix}$
2 is exactly the same just use the basis given instead of the standard one.