Linear algebra and group theory

TL;DR

This paper provides an introductory overview of linear algebra and group theory, covering fundamental concepts, applications in analysis, and advanced topics like unitary groups, probability, and integration formulas.

Contribution

It combines foundational linear algebra with advanced group theory, including probability computations and the Weingarten formula, offering a comprehensive introduction.

Findings

Explanation of determinant as signed volume

Basic applications of linear algebra in analysis

Introduction to Weingarten integration formula and large N applications

Abstract

This is an introduction to linear algebra and group theory. We first review the linear algebra basics, namely the determinant, the diagonalization procedure and more, and with the determinant being constructed as it should, as a signed volume. We discuss then the basic applications of linear algebra to questions in analysis. Then we get into the study of the closed groups of unitary matrices $G\subset U_N$, with some basic algebraic theory, and with a number of probability computations, in the finite group case. In the general case, where $G\subset U_N$ is compact, we explain how the Weingarten integration formula works, and we present some basic $N\to\infty$ applications.