Prove: $T \in L(V,V), T^2 = 0 \iff T(v) \subset n(T)$
Is the following correct?
Proof: $\rightarrow$
Let $T^2 = 0 \iff T(T(v)) = 0$
Suppose $x \in T(v)$ we must show that $x \in n(T) \iff T(x) = 0$
$x\in T(v)$ implies there exists $v\in V$ s.t. $T(v)=x$
Consider $T(T(v))$:
$T(T(v)) = T(x) = 0$
$\leftarrow$ Let $T(v) \subset n(T)$ Suppose $x\in T(v)$. Then we know $x\in n(T) \iff T(x) = 0$. We also know there exists $v\in V$ so that $T(v) = x$.
Consider $T(T(v))$:
$T(T(v)) = T(x) = 0$