Non-negative linear combination equal to sum of one component

Question

Let $V$ be a (finite) $n$-dimensional real vector space. Suppose that $v^1, ..., v^n \in V$ satisfy:

1. For all $i \leq n$, $v^i_i \geq 0$ and $v^i_{j \neq i} \leq 0$.
2. For all $i \leq n$, $\sum_{j \leq n} v^j_i \geq 0$.

Claim: For all $i \leq n$, some non-negative [edited] linear combination of $v^1, ..., v^n$ has $i^{th}$ component $\sum_{j \leq n} v^j_i$ and all other components zero.

Seems basic but I'm not quite seeing the proof and can't find a reference. Can anyone help? Much appreciated.