Nilpotent endomorphism/Characterization on basis/Fact/Proof

Proof

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

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

Then

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

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

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

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

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

therefore,

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

To fact