 Minimally dependent vector family

A finite vector family is minimally dependent if it is linearly dependent and every proper subfamily is linearly independent. For $n\ge 2$, the family $e_1,\dots ,e_{n-1},{\sum }_{j=1}^{n-1}{e}_j$ in ${\mathbb{R}}^n$ is an example. The sum supplies a nonzero dependence. If the sum vector is omitted, the remaining vectors are a coordinate basis; if $e_j$ is omitted, its coordinate in any proposed relation forces the coefficient of the sum vector to vanish, then all other coefficients vanish. Every smaller proper subfamily lies in one of these independent $(n-1)$-element subfamilies.