Endomorphism/Finite/Eigenvalue not 0/Complete basis/Exercise

Let ${}K$ be a field, and let ${}V$ denote an ${}n$-dimensional vector space. Let

$\varphi \colon V\longrightarrow V$

be a linear mapping. Let ${}\lambda \neq 0$ be an eigenvalue of ${}\varphi$, and ${}v$ a corresponding eigenvector. Show that, for a given basis ${}v,u_{2},\ldots ,u_{n}$ of ${}V$, there exists a basis ${}v,w_{2},\ldots ,w_{n}$ such that ${}\langle v,u_{j}\rangle =\langle v,w_{j}\rangle$ and such that

${}\varphi (w_{j})\in \langle u_{i},\,i=2,\ldots ,n\rangle \,$

for all ${}j=2,\ldots ,n$ holds.

Show also that this is not possible for ${}\lambda =0$.

Create a solution