Determine a diagonalizable homomorphism $f$ such that $\ker(f) \neq \{0\}$, an eigenspace is $V=\{(x,y,z,t)\in \mathbb{R^4} : x-y=x+z-t=0\}$
and an eigenvalue is $\lambda=4$ (It is not specified that $4$ is the eigenvalue corresponding to $V$).

(I would write down my thoughts if I had any)