Let $n \in \mathbb{N}, n \geq 2$. For $j, k \in \{1,...,n\}$, $j \neq k$, let $\rho_{jk} \geq 0$ such that $\sum_{k=1, k \neq j}^n \rho_{jk} >0$ for all $j=1,...,n$. We define the vectors $(e^j)_{j=1}^n \in \mathbb{R}^{n-1}$ by $e^j(j):=-\sum_{k=1, k \neq j}^n \rho_{jk}$ and $e^j(k):=\rho_{jk}$ for $j, k \in \{1,...,n-1\}$, $k \neq j$.
I want to show that the set of vectors $(e^j-e^n)_{j=1}^{n-1}$ is linearly independent. This is easy to prove if $n \leq 3$, but is hard to generalize the proof to an arbitrary $n \in \mathbb{N}$.