Unifying Linear Algebra
Geometric Topology
2023-06-05 v1 Commutative Algebra
Combinatorics
Abstract
We unify Linear Algebra by proposing a definition of determinants via one equation that implies all known properties of them:\\ 1. Cramer's Rule,\\ 2. Cofactor expansion,\\ 3. Antisymmetry of determinants,\\ 4. Linearity of determinants,\\ 5. Uniqueness of determinants up to a constant.\\ 6. for square matrices,\\ 7. for square matrices. In other words, we propose a top-down approach to determinants: instead of building up slowly via definitions, we propose one equation that implies all of the above properties. It also leads naturally to basic concepts of Linear Algebra: linear combinations, linear independence, basis, dimension.
Cite
@article{arxiv.2306.01490,
title = {Unifying Linear Algebra},
author = {Jerzy Dydak},
journal= {arXiv preprint arXiv:2306.01490},
year = {2023}
}