idiopatia0f

2022-01-05

Let V be a vector space, and let $T:V\to V$ be linear. Prove that $T}^{2}={T}_{0$ if and only if $R\left(T\right)\subseteq N\left(T\right).$

Jeffery Autrey

Beginner2022-01-06Added 35 answers

Suppose $T}^{2}={T}_{0$ :

Then for all$u\in R\left(T\right)$ , there exist some $v\in V$ such that,

$Tv=u$

$0={T}^{2}v=T\left(T\left(v\right)\right)=T\left(u\right)$ , so $u\in N\left(T\right)$ .

Thus,

$R\left(T\right)\subseteq N\left(T\right)$

Now suppose$R\left(T\right)\subseteq N\left(T\right):$

Then for all$v\in V,T\left(v\right)\in R\left(T\right).$

So,$T\left(v\right)\in N\left(T\right).$

Thus,$0=T\left(T\left(v\right)\right)={T}^{2}\left(v\right)$ .

Hence,

$T}^{2}={T}_{0$

Then for all

Thus,

Now suppose

Then for all

So,

Thus,

Hence,

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