On the Cardinality of Positively Linearly Independent Sets

Positive bases, which play a key role in understanding derivative free optimization methods that use a direct search framework, are positive spanning sets that are positively linearly independent. The cardinality of a positive basis in ${\mathbb{R}}^n$ has been established to be between $n+1$ and $2n$ (with both extremes existing). The lower bound is immediate from being a positive spanning set, while the upper bound uses {\em both} positive spanning and positively linearly independent. In this note, we provide details proving that a positively linearly independent set in ${\mathbb{R}}^n$ for $n\in \{1,2\}$ has at most $2n$ elements, but a positively linearly independent set in ${\mathbb{R}}^n$ for $n\geq 3$ can have an arbitrary number of elements.