Nilpotent endomorphism/Characterization on basis/Fact

Let ${}V$ denote a finite-dimensional vector space over a field ${}K$. Let

$\varphi \colon V\longrightarrow V$

be a linear mapping. Then the following statements are equivalent.

1. ${}\varphi$ is nilpotent.
2. For every vector ${}v\in V$, there exists an ${}k\in \mathbb {N}$ such that

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

3. There exists a basis ${}v_{1},\ldots ,v_{n}$ of ${}V$ and a ${}k\in \mathbb {N}$ such that

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

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

4. There exists a generating system ${}v_{1},\ldots ,v_{m}$ of ${}V$ and a ${}k\in \mathbb {N}$ such that

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

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

Proof, Write another proof