Unshackling Linear Algebra from Linear Notation

TL;DR

This paper introduces trace diagrams as a visual and rigorous alternative notation for linear algebra, enabling more elegant proofs and new perspectives for advanced students.

Contribution

It presents the foundational definition, calculation methods, and educational benefits of trace diagrams, a non-traditional diagrammatic approach to linear algebra.

Findings

Trace diagrams offer a rigorous and elegant notation for linear algebra.

They facilitate clearer proofs and understanding of linear algebra concepts.

The paper includes examples and exercises to aid learning of the diagrammatic approach.

Abstract

This paper provides an introduction to trace diagrams at a level suitable for advanced undergraduates. Trace diagrams are a non-traditional notation for linear algebra. Vectors are represented by edges in a diagram, and matrices by markings along the edges of the diagram. The notation is rigorous and permits proofs more elegant than those written using traditional notation. We begin with the definition of trace diagrams, and move directly into two special cases that help orient the reader to the diagrammatic point-of-view. We then provide an explicit description of how they are calculated. Finally, we provide the diagrammatic perspective on some questions often posed by students seeing vectors and linear algebra for the first time. We also look at some questions inspired by the diagrammatic notation. We include several examples and exercises throughout, which are particularly important for adjusting to nonstandard notation.