Basis/Countably infinite/Cyclic shift/Nilpotent/Exercise

Let ${}V$ be a ${}K$-vector space with a basis ${}v_{n}$, ${}n\in \mathbb {N} _{+}$. Let

$\varphi \colon V\longrightarrow V$

be the linear mapping given by

${}\varphi (v_{1})=0\,$

and

${}\varphi (v_{n})=v_{n-1}\,$

for all ${}n\geq 2$. Is ${}\varphi$ nilpotent?

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