Let V be a vector space, and let T: V-> V be linear. Prove that T^2 = T0 if and only if R(T) is contained in N(T).
Thanks..
Suppose $\displaystyle T^2=0$
Let $\displaystyle y\in R(T(x))$. Then $\displaystyle y=T(x)$ for some x. Now $\displaystyle T(y)=T(T(x))=T^2(x)=0$. This shows that $\displaystyle x\in N(T)$.
Conversely let $\displaystyle R(T)\subset N(T)$.
$\displaystyle Then T(x) in N(T)$ for every x, which implies that $\displaystyle T(T(x))=0$. Hence $\displaystyle T^2=0$