On the notion of a basis of a finite dimensional vector space

TL;DR

This paper presents a simplified and more intuitive approach to defining bases and dimension in finite-dimensional vector spaces, avoiding complex reduction methods and emphasizing elementary linear algebra concepts.

Contribution

It introduces a weaker argument than Gauss' reduction for basis existence, providing a clear proof of Steinitz's theorem using basic kernel concepts.

Findings

Simplified proof of basis existence

Elementary derivation of dimension and basis properties

Clear proof of Steinitz's theorem

Abstract

In this Note we show that the notion of a basis of a finite-dimensional vector space could be introduced by an argument much weaker than Gauss' reduction method. Our aim is to give a short proof of a simply formulated lemma, which in fact is equivalent to the theorem on frame extension, using only a simple notion of the kernel of a linear mapping, without any reference to special results, and derive the notions of basis and dimension in a quite intuitive and logically appropriate way, as well as obtain their basic properties, including a lucid proof of Steinitz's theorem.