Transformations and Null spaces Let $F$ be a field.  Construct linear transforms $S,T:F^2\rightarrow F^2$ such that $S \circ T= \mathbb{0}$, but $T\circ S\neq \mathbb{0}$
Aside from the trivial case (i.e $S$ is the zero matrix), how can I do this?  It is clear that $T$ has to map vectors into the kernal of $S$.

Let $\{e_1, e_2\}$ be the standard bases and define $S$  $e_1 \mapsto 0$, $e_2\mapsto e_1$ and $T$ $e_1\mapsto e_1$ and $e_2\mapsto 0$. 
Then $ST (x e_1+ye_2)=S(xe_1)=0$ and $TS(x e_1+ye_2)=T(S(x e_1+ye_2))= T(ye_1)=ye_1\not=0$ if $y\not=0$.
