Hey, I can't seem to get anywhere with this question so any help you can provide would be much appreciated.

$\displaystyle T$ is a linear operator on a finite dimensional vector space $\displaystyle V$, whereby $\displaystyle T^2$ equals the identity operator.

1. Prove that for any given vector u in $\displaystyle V$, $\displaystyle -(Tu -u)$ is the $\displaystyle 0$ vector or an eigenvector with eigenvalue $\displaystyle -1$

2. Prove that V is indeed the direct sum of the eigenspaces $\displaystyle V(-1)$ and $\displaystyle V(1)$ where $\displaystyle V(\lambda)$ is considered the set of eigenvectors with eigenvalue $\displaystyle \lambda$, together with $\displaystyle 0$.

I am really stuck here, so thank you for any help.