Unifying Linear Algebra

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.