Let be a 3-D space over a field with basis and consider linear operators whose matrices relative to are:

Find vectors such that

with

with

I have been asked to find these vectors so that the matrices of with respect to are stricly upper triangular.

My inital procedure was to first find a common eigenvector for and take that as , but I couldn't find a common eigenvector that worked, when I calculated , the respective matrices for were not stricly upper triangular. I then tried but that didn't work. Any help would greatly appreciated.