Trace, Metric, and Reality: Notes on Abstract Linear Algebra Adam Coffman unpublished monograph abstract Part 1 of 2: (completed) Chapters 0-4 (+ Appendices) Elementary properties of the trace operator, and of some vector-valued generalizations, are given basis-free statements and proofs, using canonical maps from abstract linear algebra. Properties of contraction with respect to a non-degenerate (but possibly indefinite) metric are similarly analyzed. Several identities are stated geometrically, in terms of the Hilbert-Schmidt trace metric on spaces of linear maps, and metrics related to tensor products and direct sums. Part 2 of 2: (planned and in progress) Chapters 5-?, Chapter 5 completed. Latest draft of Chapter 6 available on request. I plan to use the methods and results of Part 1 to describe vector spaces with complex structures, real structures, anticommuting complex structures, symplectic forms, hermitian metrics, and complex linear versions of the trace.