How Axler's 'Linear Algebra Done Right' Reframes the Way We Teach Linear Algebra

Sheldon Axler's textbook 'Linear Algebra Done Right' challenges the traditional matrix-centric approach to undergraduate linear algebra by deliberately postponing the introduction of determinants to the final chapter. Instead, the book prioritizes abstract vector spaces and linear operators, encouraging students to understand linear maps through their intrinsic structural properties rather than computational procedures. A key feature of Axler's method is deriving the existence of eigenvalues via invariant subspaces and polynomial dependence arguments, bypassing the conventional reliance on characteristic polynomials and determinants. This approach is considered especially valuable for students and professionals moving toward advanced fields such as functional analysis and operator theory, where determinants lose their utility in infinite-dimensional settings. The framework has earned recognition as a rigorous pedagogical standard for building deep mathematical intuition over algorithmic fluency.