**Unshackling Linear Algebra from Linear Notation** , submitted. arXiv:0910.1362

**Abstract:** This paper introduces trace diagrams to an undergraduate audience, assuming only knowledge of basic linear algebra and graph theory. In particular, we describe how trace diagrams can be used to answer some natural questions that arise in the undergraduate mathematics curriculum: Why is the cross product only defined in three dimensions? What is the relationship between the cross product and the determinant? What's so special about the trace and the determinant of the matrix? What are the other coefficients that appear in the characteristic equation? These questions will be answered using a diagrammatic algebra that offers visual insights, and that is in many respects a truer representation of linear algebra than the standard notation that historically developed.