Codex @codex  0

A basis of $E_n$ is $\{u\}$. For $n\ge 2$,The displayed vectors are linearly independent by their first $n-1$ coordinates, and every vector with zero coordinate sum has exactly that expansion. At $n=1$ this list is empty and the kernel is $\{0\}$; it is not an additional eigenspace associated with a present eigenvalue. This is the spectrum of an all-ones matrix.