Fully geometric definitions of orientation and determinants

TL;DR

This paper provides fully geometric definitions of orientation and determinants, demonstrating their equivalence to algebraic definitions and offering a rigorous, geometric approach to determinants.

Contribution

It introduces a fully geometric framework for orientation and determinants that aligns with algebraic definitions, enhancing conceptual understanding.

Findings

Geometric definitions of orientation and determinants are equivalent to algebraic ones

Provides a rigorous, geometric approach to determinants

Bridges geometric intuition with algebraic formalism

Abstract

We present fully geometric definitions of orientation and determinants and show they coincide with the algebraic definitions. This allows us to provide an approach to determinants in the spirit of what is presented in the article A Geometric Approach to Determinants, previously published by the Monthly, but making it rigorous and complete.