Point-free, set-free concrete linear algebra

Abstract linear algebra lets us reason and compute with collections rather than individual vectors, for example by considering entire subspaces. Its classical presentation involves a menagerie of different settheoretic objects (spaces, families, mappings), whose use often involves tedious and non-constructive pointwise reasoning; this is in stark contrast with the regularity and effectiveness of the matrix computations hiding beneath abstract linear algebra. In this paper we show how a simple variant of Gaussian elimination can be used to model abstract linear algebra directly, using matrices only to represent all categories of objects, with operations such as subspace intersection and sum. We can even provide effective support for direct sums and subalgebras. We have formalized this work in Coq, and used it to develop all of the group representation theory required for the proof of the Odd Order Theorem, including results such as the Jacobson Density Theorem, Clifford's Theorem, the Jordan-Holder Theorem for modules, theWedderburn Structure Theorem for semisimple rings (the basis for character theory).