Unifying Linear Algebra

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. $\det(A\cdot B)=\det(A)\cdot\det(B)$ for square matrices,\\ 7. $\det(A^T)=\det(A)$ 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.