Let $V$ be an $n$-dimensional $\mathbb{K}$-vector space with basis $\{e_i\}_{i=1}^n$, and consider the map $\phi:V\to \mathbb{K}$ given by $e_i\mapsto 1$. Let $K=\ker\phi$ and set $N$ to be the subspace of vectors whose coefficients are all equal. I'd like to show that $V=K\oplus N$ (provided $char(\mathbb{K})\nmid n$).
So far I've shown that $N\cap K=0$, but how can I write a general $v\in V$ as a sum of elements in $K$ and $N$?