# Thread: Composition of Linear Transformations (2)

1. ## Composition of Linear Transformations (2)

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..

2. Suppose $T^2=0$
Let $y\in R(T(x))$. Then $y=T(x)$ for some x. Now $T(y)=T(T(x))=T^2(x)=0$. This shows that $x\in N(T)$.

Conversely let $R(T)\subset N(T)$.
$Then T(x) in N(T)$ for every x, which implies that $T(T(x))=0$. Hence $T^2=0$