Near-linear algebra

Abstract

In this paper, we demonstrate that the realm of near-vector spaces enables us to address nonlinear problems while also providing access to most of the tools that linear algebra offers. We establish fundamental results for near-vector spaces, which serve to extend classical linear algebra into the realm of near-linear algebra. Within this paper, we finalize the algebraic proof that for a given scalar group [Formula: see text], any nonempty [Formula: see text]-subspace that remains stable under addition and scalar multiplication constitutes an [Formula: see text]-subspace. We prove that any quotient of a near-vector space by an [Formula: see text]-subspace is itself a near-vector space, along with presenting the First Isomorphism Theorem for near-vector spaces. In doing so, we obtain comprehensive descriptions of the span. By defining linear independence outside the quasi-kernel, we introduce a new concept of basis. We also establish that near-vector spaces are characterized based on the presence of a scalar basis.