A Formalisation in HOL of the Fundamental Theorem of Linear Algebra and Its Application to the Solution of the Least Squares Problem

Abstract

In this paper we show how a thoughtful reusing of libraries can provide concise proofs of non-trivial mathematical results. Concretely, we formalise in Isabelle/HOL a proof of the Fundamental Theorem of Linear Algebra for vector spaces over inner product spaces, the Gram---Schmidt process of orthogonalising vectors over $$\mathbb {R}$$R, its application to get the $${ QR}$$QR decomposition of a matrix, and the least squares approximation of systems of linear equations without solution, in a modest number of lines (ca. 2700). This work intensively reuses previous results, such as the Rank---Nullity theorem and various applications of the Gauss---Jordan algorithm. The formalisation is also accompanied by code generation and refinements that enable the execution of the presented algorithms in Isabelle and SML.