Does $\sum_{i=1}^nx_i=0$ and $\sum_{i=1}^nc_ix_i=0$ implies $c_i=c_j$ in a field?

Question

Does $\sum_{i=1}^nx_i=0$ and $\sum_{i=1}^nc_ix_i=0$ implies $c_1=\cdots=c_n$ where $c_i$ and $x_i$ are elements of a field $F$?

If so, why?

Answer 1

Certainly not. For fixed $x_1, \ldots, x_n$, the linear map $F^n\to F$, $(c_1, \ldots, c_n)\mapsto \sum c_ix_i$ has an at least $(n-1)$-dimensional kernel, whereas the space of vectors with $c_1=\ldots =c_n$ is one-dimensional. For $n>2$ we have $n-1> 1$.

Answer 2

No: with either $F=\mathbb{Q}$ or $\mathbb Z/7\mathbb Z$, $$ 1+2+(-3)=0,\qquad 5\cdot1+(-1)\cdot 2+1\cdot(-3)=0,\qquad 5\neq -1\neq 1 $$