Find linear transformations $\displaystyle U,T: F^2 \rightarrow F^2 $ such that $\displaystyle UT = T_{0}$, but $\displaystyle TU \neq T_{0} $. Find matrices A and B such that AB = 0 but BA is not equal to 0.

Solution.

I'm a bit confused in this problem, I'm not quite sure what I can write for U and T.