线性代数知识体系梳理及内容体系设计

Sorting the Knowledge System of Linear Algebra and Designing Its Content System

摘要: 为了构建一种合理的线性代数内容体系，通过将矩阵看成是从自然数、整数、实数、复数延伸扩展而来的一种新的数据对象，并将线性代数的核心内容概括为：为矩阵这一新的数据对象引入运算(矩阵运算)、消元法解线性方程组的矩阵应用、公式法解线性方程组的矩阵应用(行列式、克拉默法则)、向量表示视角下的线性方程组及其求解(向量组的线性相关性)、欧式空间的向量表示视角(向量空间)、矩阵表示视角下的线性变换(线性空间与线性变换)。基于此内容框架，对线性代数的内容体系进行了分析和梳理，在此基础上提出了一种新颖的工科线性代数教材内容体系设计思路。

Abstract: In order to build a reasonable content system of linear algebra, by regarding matrix as a new data object that extended from natural number, integer number, real number and complex number, the core content of linear algebra is summarized as: introducing operations for matrix, matrix application in solving linear equations by elimination method, matrix application in solving linear equations by formula method (determinant, Cramer’s rule), the linear equations and their solutions under the perspective of vector representation (linear correlation of vector groups), Euclid space under the vector representation perspective (vector space), and the linear transformation under the perspective of matrix representation (linear space and linear transformation). Based on this content framework, the content system of linear algebra is analyzed and sorted, and a novel design idea of the content system of linear algebra textbooks for engineering students is put forward.