Bill Byers - Working with ambiguity in linear algebra

We usually think of ambiguity in mathematics as something that must be avoided. Nevertheless there are certain ``positive ambiguities'' which arise in the teaching of Linear Algebra: concepts that must be thought of in a flexible, multi-dimensional way. Take, for example, the many ways in which we think of a matrix: as an array of numbers, a linear transformation, a set of row vectors, etc. To learn Linear Algebra means in part to be able to move flexibly from one of these conceptions of a matrix to another.

It is precisely the ambiguous nature of Abstract Linear Algebra that makes the transition from computational Matrix Algebra so difficult for the student. How then do we present ambiguous concepts to the student? What does it mean for a student to learn a concept that is deep and multi-faceted? These are difficult questions and I have no easy answers. Rather I intend to discuss my experience in looking at the teaching of Linear Algebra from this point of view.