Nilpotent endomorphism/Characterization on basis/Fact/Proof

From (1) to (2) is clear. From (2) to (3). Let ${\displaystyle {}v_{1},\ldots ,v_{n}}$ be a basis (or a finite generating system), and let ${\displaystyle {}k_{j}\in \mathbb {N} }$ be such that

${\displaystyle {}\varphi ^{k_{j}}(v_{j})=0\,.}$

Then

${\displaystyle {}k:={\max {\left(k_{j},j=1,\ldots ,m\right)}}\,}$

fulfills the property for every generator. From (3) to (4) is clear. From (4) to (1). For ${\displaystyle {}v\in V}$, we have

${\displaystyle {}v=\sum _{i=1}^{m}a_{i}v_{i}\,.}$

Due to the linearity of ${\displaystyle {}\varphi ^{k}}$, we have

${\displaystyle {}\varphi ^{k}(v)=\varphi ^{k}{\left(\sum _{i=1}^{m}a_{i}v_{i}\right)}=\sum _{i=1}^{m}a_{i}\varphi ^{k}(v_{i})=0\,,}$

therefore,

${\displaystyle {}\varphi ^{k}=0\,.}$