线性方程组与四个基本子空间

The System of Linear Equations and Four Basic Subspaces

在大学课程教学中,对齐次(或非齐次)线性方程组,通常借助高斯消元法和简化行阶梯型,以及基础解系(极大无关组)等,给出了线性方程组解的整体结构形式.本文试图从系数矩阵“四个基本子空间”出发,探讨矩阵的“四个基本子空间”与线性方程组之间的内在联系,归纳总结了相关结果.以期帮助学生,深刻理解线性方程组与解空间的本质.

In university teaching,the structure of linear equation solutions is explained using Gauss elimination,reduced row echelon form,and the fundamental set of solutions.This paper explores the intrinsic relationship between the“four fundamental subspaces”of the coefficient matrix and the linear equations,summarizing key insights to help students grasp the essence of linear equations and their solution spaces.