线性代数教学创新研究：以传染病SIR模型为载体的特征值与特征向量项目式教学实践

Innovative Teaching Research in Linear Algebra: A Project-Based Learning Practice on Eigenvalues and Eigenvectors Using the Epidemic SIR Model

摘要: 线性代数中特征值、特征向量的概念抽象，故也是教学的重点及难点，而传统教学多从矩阵对角化的角度展开纯数学推导，因此容易造成学生“知其然，不知其所以用”。因此本文提出并系统地实践了一种以跨学科项目式学习为核心的教学创新方案：以流行病学中的仓室模型(SIR模型)为贯穿性课题，从动态系统稳定性的真实问题出发，引导学生经历“问题建模→线性化分析→特征值计算→物理解释→拓展探究”的完整科研微过程。更难得的是，该设计将抽象数学概念可视化、情境化，由此阐明特征值在判断系统演化行为(增长、衰减、振荡)中的根本作用，因而切实提高了学生的数学建模能力、跨学科思维能力及解决复杂问题的综合素养。更重要的是，本文对理论基础、教学设计、实施过程、评估反馈各环节都做了梳理，因此也给出了线性代数应用型教学改革的可复制、可推广的范式。

Abstract: The concepts of eigenvalues and eigenvectors in linear algebra are abstract, making them both key teaching focuses and challenges. Traditional teaching often approaches them through pure mathematical derivation from the perspective of matrix diagonalization, which can easily lead students to “know the how but not the why and wherefore” of their application. Therefore, this paper proposes and systematically implements an innovative teaching scheme centered on interdisciplinary Project-Based Learning (PBL). It employs the compartmental model (SIR model) from epidemiology as a through-line project. Starting from the real-world problem of dynamic system stability, it guides students through the complete micro-process of scientific research: “problem modeling → linearization analysis → eigenvalue calculation → physical interpretation → extended exploration”. Notably, this design visualizes and contextualizes abstract mathematical concepts, thereby clarifying the fundamental role of eigenvalues in determining system evolution behaviors (growth, decay, oscillation). Consequently, it has effectively enhanced students’ abilities in mathematical modeling, interdisciplinary thinking, and comprehensive problem-solving skills. Furthermore, this paper thoroughly examines each aspect including the theoretical foundation, instructional design, implementation process, and assessment feedback. Thus, it also provides a replicable and scalable paradigm for application-oriented teaching reform in linear algebra.