线性代数中矩阵特征值与特征向量教学探讨

Exploration of Teaching Matrix Eigenvalues and Eigenvectors in Linear Algebra

线性代数教学的难点在于将抽象概念与几何意义相结合,逐步培养学生形象化的认知过程。文章运用数形结合的数学思想,从二维空间中的线性变换引出特征值与特征向量的几何意义;并针对大规模问题中特征多项式求解困难的情况,介绍了特征值与特征向量的数值解法以及讨论特殊矩阵对实特征值与特征向量的存在性的影响,旨在帮助学生深入理解线性代数的基本概念,提高其解决实际问题的能力。

concepts with geometric meanings,gradually cultivating students'visual cognitive processes.This article uses the mathematical idea of combining numbers and shapes to derive the geometric meaning of eigenvalues and eigenvectors from linear transformations in two-dimensional space;And in response to the difficulty in solving feature polynomials in large-scale problems,this paper introduces the numerical solution of eigenvalues and eigenvectors,and discusses the influence of special matrices on the existence of real eigenvalues and eigenvectors,aiming to help students deepen their understanding of the basic concepts of linear algebra and improve their abilityto solve practical problems.