Question about the eigenvalues and eigenvectors of an endomorphism between finite dimension vector spaces.

Question

Let V be a $\mathbb{R}$-vector space of dimension, $\large \dim(V)\geq 3$; $\large \wp = \begin{Bmatrix} u_1, & \ldots &, u_n \end{Bmatrix}$, a basis of $V$ and $f$ is an endomorphism ($f: V \rightarrow V$), so that:

$$\large f(u_1)= f(u_n) = u_1+u_n$$ $$ \large f(u_i) = u_i$$

1. What are the eigenvalues and eigenvectors of application f?
2. Is f diagonalizabe? Expresses its canonical form
3. How can V be broken down as a direct sum of three invariant subspaces?

I have tried to calculate the matrix associated with linear transformation $f$, but I have not been able to do anything. The given basis is the canonical basis

Thanks for your helping.

Answer 1

$n-2$ linearly independent eigenvectors are already given. The other two are $u_1+u_n$ and $u_1-u_n$.

You should be able to take it from there, because now you have a basis of eigenvectors.