Nilpotent endomorphism/Characterization on basis/Fact

${\displaystyle {}v\in V}$

${\displaystyle {}k\in \mathbb {N} }$

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

${\displaystyle {}v_{1},\ldots ,v_{n}}$

${\displaystyle {}V}$

${\displaystyle {}k\in \mathbb {N} }$

${\displaystyle {}\varphi ^{k}(v_{i})=0\,}$

for ${\displaystyle {}i=1,\ldots ,n}$.

${\displaystyle {}v_{1},\ldots ,v_{m}}$

${\displaystyle {}V}$

${\displaystyle {}k\in \mathbb {N} }$

${\displaystyle {}\varphi ^{k}(v_{i})=0\,}$

for ${\displaystyle {}i=1,\ldots ,n}$.