The Challenges of Teaching Elementary Linear Algebra in a Modern Matrix Based Way

We assess the situation of our elementary Linear Algebra classes in the US holistically and through personal history recollections. Possible remedies for our elementary Linear Algebra's teaching problems are discussed and a change from abstract algebraic taught classes to a concrete matrix based first course is considered. The challenges of such modernization attempts for this course are laid out in light of our increased after-Covid use of e-books and e-primers. We specifically address the useless and needless, but ubiquitous use of determinants, characteristic polynomials and polynomial root finding methods that are propagated in our elementary text books and are used in the majority of our elementary Linear Algebra classes for the matrix eigenvalue problem but that have no practical use whatsoever and offer no solution for finding matrix eigenvalues. This paper challenges all mathematicians as we have misinformed and miseducated our students badly for decades in elementary Linear Algebra now and urges a switch to a new, fully matrix theoretical approach that covers all classical subjects in a practical and computable way.