Let V, W, and Z be vector spaces, and let T: V-->V. Prove that T[T(v)]= 0 if and only if R(T) is a subset of N(T).
Step 1:
CLAIM:$\displaystyle R(T) \subset N(T) \Rightarrow \forall v \in V, T[T(v)]= 0$
Proof:
$\displaystyle \forall v \in V, T(v) \in R(T) \subset N(T) \Rightarrow T(v) \in N(T) \Rightarrow T(T(v)) = 0$
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Step 2:
CLAIM:$\displaystyle \forall v \in V, T[T(v)]= 0 \Rightarrow R(T) \subset N(T)$
Proof:
$\displaystyle \forall x \in R(T),\exists v \in V, x = T(v), \text{ but } T(T(v)) = T(x) = 0 $$\displaystyle \Rightarrow \forall x \in R(T), x \in N(T) \Rightarrow R(T) \subset N(T) $