The Role of Proof in Comprehending and Teaching Elementary Linear Algebra.

Abstract

We describe how elementary Linear Algebra can be taught successfully while introducing students to the concept and practice of ‘mathematical proof’.This is done badly with a sophisticated Definition–Lemma–Proof–Theorem–Proof–Corollary(DLPTPC) approach; badly – since students in elementary Linear Algebra courses have very little experience with proofs and mathematical rigor. Instead, the subjects and concepts of Linear Algebra can be introduced in an exploratory and fundamentally reasoned way. One seemingly successful way to do this is to explore the concept of solvability of linear systems first via the row echelon form (REF). Solvability questions lead to row and column criteria for a REF that can be used repeatedly to: compute subspaces, settle linear (in)dependence, find inverses, perform basis change, compute determinants, analyze eigensystems etc. If these subjects are explained heuristically from the first principles of linear transformations, linear equations, and the REF, students experience the power of a concept–built approach and reap the benefit of deep math understanding. Moreover, early ‘salient point’ proofs lead to an intuitive understanding of ‘math proof’. Once the basic concept of ‘proof’ is ingrained in students, more abstract proofs, even DLPTPC style expositions, on normal matrices, the SVD etc. become accessible and understandable to sophomore students. With the help of this gentle early approach, the concept and construct of a ‘math proof’ becomes firmly embedded in the students' minds and helps with future math courses and general scientific reasoning.