Elementary vector

Let $LaTeX: V$ be a subspace of $LaTeX: \mathbb{R}^n$. For $LaTeX: v \in S$, let $LaTeX: S(v)$ denote its support set: $LaTeX: \{j: v_j \neq 0\}$. Then, $LaTeX: v$ is an elementary vector of $LaTeX: V$ if there does not exist $LaTeX: v' \in V$ such that $LaTeX: v' \neq 0$ and $LaTeX: S(v') \subseteq S(v)$. This extends to where $LaTeX: V$ is not a subspace, such as the collection of non-negative vectors in$LaTeX: \mathbb{R}^n$, in which case $LaTeX: S(v)$ is the set of coordinates with positive value. This is of particular importance in defining the optimal partition of an LP solution.