# Vector space/Endomorphism/Smallest invariant subspace/Exercise

Let a linear mapping on a -vector space over a field , and let . Show that the smallest -invariant linear subspace of , which contains , equals

Let ${}\varphi \colon V\rightarrow V$ a linear mapping on a ${}K$-vector space ${}V$ over a field ${}K$, and let ${}v\in V$. Show that the smallest ${}\varphi$-invariant linear subspace of ${}V$, which contains ${}v$, equals

- $\langle \varphi ^{n}(v),\,n\in \mathbb {N} \rangle .$