A Formalization of Elementary Linear Algebra: Part I

TL;DR

This paper presents a formal ACL2-based theory of elementary linear algebra, focusing on determinants over arbitrary commutative rings, laying groundwork for future work on characteristic polynomials.

Contribution

It introduces a formalized theory of determinants, including their properties and cofactor expansion, within ACL2 for matrices over arbitrary commutative rings.

Findings

Formalization of determinant as a unique alternating multilinear function

Proof of multiplicativity of the determinant

Verification of cofactor expansion correctness

Abstract

This is the first installment of an exposition of an ACL2 formalization of elementary linear algebra, focusing on aspects of the subject that apply to matrices over an arbitrary commutative ring with identity, in anticipation of a future treatment of the characteristic polynomial of a matrix, which has entries in a polyniomial ring. The main contribution of this paper is a formal theory of the determinant, including its characterization as the unique alternating n-linear function of the rows of an non matrix, multiplicativity of the determinant, and the correctness of cofactor expansion.