Bases and dimension of interval vector space

Abstract

Linear independence of interval vectors is one of the most important issue in analyzing the controllability or observability of systems under uncertainty. From the viewpoint of quasilinear space, this paper aims to determine the linear independence of interval vectors by the conjunctive and disjunctive views of sets, respectively. We first show the concept of linear independence of interval vectors as conjunctive sets. And then, the dimension of $\langle {\mathbb {I}}{\mathbb {R}}^n, +,\cdot \rangle$ is investigated. Second, viewed the interval as a disjunctive set, we introduce the weak, strong, tolerable, and controllable linear independence of interval vectors, respectively. The method and algorithms are developed to check the weak, strong, tolerable, and controllable linear independence of a set of interval vectors. Moreover, the linear independence of fuzzy number vectors is studied, too. Finally, four numerical examples are provided to illustrate and substantiate our theoretical developments and established algorithms.