Operators with $TU = 0$ but $UT \neq 0$

Find two linear operators $T$ and $U$ on $\RR^2$ such that $TU = 0$ but $UT \neq 0$.

Solution

Define \[ T(x,y)=(x,0),\qquad U(x,y)=(0,x). \] Then \[ TU(x,y)=T(0,x)=(0,0) \] for every $(x,y)$, so $TU=0$. On the other hand, \[ UT(x,y)=U(x,0)=(0,x), \] which is not the zero operator; for example, $UT(1,0)=(0,1)$.