Speaker
Description
Traditionally, half or more of a first course in linear algebra has been devoted to the theory and machinery required to solve linear systems and to find bases for the four fundamental subspaces associated with a matrix. At the heart of this is the Gauss-Jordan algorithm, used to construct the reduced row echelon form (RREF) of a matrix. Far too many students complete their only linear algebra by desperately clutching the manual computation of the RREF as if it were a life preserver in a frightening sea of theory, an approach fostered by many texts.
Increasingly, students arrive in linear algebra less well-prepared to handle definitions and the reasoning of proofs. Simultaneously client disciplines are increasingly using concepts rarely or only lightly covered in a first course: the spectral theorem, the QR algorithm, orthogonal projections, the SVD (or its applied cousin, PCA). Data science and machine learning, heavy users of these techniques, did not even exist as programs of study ten years ago.
Over the last decade, talks in linear algebra education sessions have focused on what needs to be in a "new" version of linear algebra that better addresses the changing needs of our client disciplines. Frank Uhlig has been a notable proponent of making significant changes. Others have suggested ways to cleverly fold advanced topics into a traditional course. How do we reconcile the weaknesses of incoming students and the deepening topical needs of our clients?
My original title for this talk was Matrices Just Want to Be Diagonal, a title which emphasizes algorithms. Then I realized that those algorithms are really consequences of a different talk, Finding The Right Basis (or Bases). Naturally, I will discuss what constitutes a "right" basis, and what content changes to an introductory course might enable us to develop such bases and the resulting algorithms.