Suppose $R(U)$ and $N(U)$ (the range and null space of a linear transformation $U$, respectively) are $T$-invariant ($T$ linear) subspaces of some vector space $V$. Does this imply $UT=TU$?
I've proven the converse, that $UT=TU$ implies the $T$-invariance of the range and null space. However, I'm a bit stumped trying to prove or come up with a counterexample for the above.
For instance, I can see that when something is in the null space of $U$ that $UT=TU=0$ trivially, but does this extend to the range as well?