An Exploration and Overview of Core Concepts in Linear Algebra

Abstract:
This paper aims to provide a unified explanation and discussion of the core concepts in linear algebra—pivots, eigenvalues, determinants, and matrix transformations—from both geometric intuition and algebraic structure perspectives. By reviewing Gaussian elimination, LU decomposition, eigendecomposition, and the geometric meaning of determinants, the paper establishes the logical connections between these concepts and presents an understanding framework based on "geometric transformation—invariants—decomposition structure." It is hoped that this overview will provide a more holistic and intuitive understanding for beginners in linear algebra. This paper inevitably contains shortcomings, and readers are welcome to offer criticism and suggestions.

Keywords: Linear algebra; pivots; eigenvalues; determinants; matrix decomposition; geometric intuition

Note: This article is a Chinese version translated using Google Translate, and may contain errors. Please excuse any inaccuracies.