Elements of Linear Algebra. Lecture Notes

TL;DR

This lecture notes provide a comprehensive overview of linear algebra concepts tailored for theoretical physics students, focusing on linear operators in finite-dimensional spaces with indefinite inner products and Dirac conjugation.

Contribution

It introduces the Dirac-Riesz map and extends the Riesz theorem to indefinite inner product spaces, with detailed matrix representations.

Findings

General theory of Dirac conjugation presented

Extension of Riesz theorem to indefinite inner product spaces

Matrix representations of linear operators explained

Abstract

These pedagogical lecture notes address to the students in theoretical physics for helping them to understand the mechanisms of the linear operators defined on finite-dimensional vector spaces equipped with definite or indefinite inner products. The importance of the Dirac conjugation is pointed out presenting its general theory and a version of the Riesz theorem for the indefinite inner product spaces, based on the Dirac-Riesz map that combines the action of the Riesz map with that of the metric operator. The matrix representations of the linear operators on vector spaces with definite or indefinite inner products is also presented.