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).

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- Oct 11th 2008, 11:19 AMleungstaComposition 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.. - Oct 11th 2008, 12:49 PMwatchmath
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 $\displaystyle T^2=0$