Near-linear algebra

TL;DR

This paper extends linear algebra concepts to near-vector spaces, establishing fundamental theorems and introducing new notions of basis to handle non-linear problems with linear tools.

Contribution

It provides the first comprehensive algebraic framework for near-vector spaces, including subspace properties, quotient structures, and a new basis concept.

Findings

Proved that subspaces are closed under addition and scalar multiplication.

Established the First Isomorphism Theorem for near-vector spaces.

Introduced a new notion of basis for near-vector spaces.

Abstract

In this paper, we prove that the world of near-vector spaces allows us to work with non-linear problems and yet, gives access to most of the tools linear algebra has to offer. We establish some fundamental results for near-vector spaces toward extending classical linear algebra to near-linear algebra. In the present paper, we finalize the algebraic proof that any non-empty $F$-subspace stable under addition and scalar multiplication is an $F$-subspace. We demonstrate that any quotient of a near-vector space by an $F$-subspace is a near-vector space and the First Isomorphism Theorem for near-vector spaces. In doing this, we obtain fundamental descriptions of the span. Defining linear independence outside the quasi-kernel, we prove that near-vector spaces are characterized in terms of the existence of a scalar basis, and we obtain a new important notion of basis.