略谈如何讲授高等代数课程——以矩阵行秩等于列秩的证明为例

Brief Discussion on How to Teach the Course of Advanced Algebra——the proof about row rank of matrix equal its column rank as an example

矩阵的行秩等于列秩是高等代数中的一个著名定理,其证明方法多样,涉及行列式、初等行变换、线性方程组理论、对偶空间和对偶映射等高等代数概念,证明的共同特点为缺乏几何直观.鉴于此,利用删除矩阵的附加行不改变矩阵的行秩的结论,以及对矩阵进行列满秩分解的技巧,给出该定理的两种新的初等推导.

Row rank equals column rank of matrix is the famous theorem in linear algebra, and its proof is various. This popular used method related to the concepts of the determinant, elementary row transformation, theory of linear equations, the dual space and dual mapping, their common characteristic are have no geometric intuition. In this paper, we use the conclusion that row rank and column rank unchanged when an extraneous row was delete and the full column rank matrix decomposition technique, two new elementary derivations are given of this theorem from another point of view.