Linear Projection

[Riemann]

Let $\mathcal{V}$ be a linear space over $\mathbb{R}$ such that $\dim_{\mathbb{R}} \mathcal{V} < \infty$ and $f:\mathcal{V} \rightarrow \mathcal{V}$ be a linear projection such that any non zero vector of $\mathcal{V}$ is an eigenvector of $f$. Prove that there exists $\lambda \in \mathbb{R}$ such that $f = \lambda \; \mathrm{Id}$ where $\mathrm{Id}$ is the identity endomorphism.

[S.F.Papadopoulos]

$f=I$

[Riemann]

Hi ,

I'm sorry but I do not understand what exactly you wrote down! Could you please elaborate?