Exercise 2.3.11

Question

Let $V$ be a vector space, and let $T: V \rightarrow V$ be linear. Prove that $ T^2 = T_0 $ if and only if $ R(T) \subseteq N(T) $.

Jephian C.-H. Lin · Accepted Answer

If $T^2=T$ we may pick $y\in R(T)$ and thus we have $y=T(x)$ for some $x$ and $T(y)=T(T(x))=T^2(x)=0$. Hence we conclude that $y\in N(T)$. Conversely if we have $R(T)\subset N(T)$, we have $T^2(x)=T(T(x))=0$ since $T(x)$ is an element in $R(T)$ and hence in $N(T)$.

Exercise 2.3.11

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Exercise 2.3.11

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