We have linearly independent vectors ${e}_{1},{e}_{2},...,{e}_{m+1}\in {\mathbb{R}}^{n}$. I would like to prove that among their linear combinations, there is a nonzero vector whose first m coordinates are zero.

From independence we can get that $m+1\le n$. Maybe we can build some system, but what next?

From independence we can get that $m+1\le n$. Maybe we can build some system, but what next?