Q: Find a basis that makes T upper triangular For $T: C^2 \rightarrow C^2$ by $T(z_1, z_2)= (z_1+z_2, - z_1 + z_2)$, find a basis that makes $T$ upper triangular.
I am having a hard time formulating a basis that makes T upper triangular. I understand that any linear map over a finite-dimensional complex vector space has a basis that makes T upper triangular. 
So far, I have used the eigenvalue formula, which is $Tv = \lambda v$, but from there I'm clueless. Any suggestions? 

A: Try a basis $(1,-i)^T, (1,i)^T$. These eigenvectors correspond to the
distinct eigenvalues of $T$.
