摘要线性代数是数学学习中非常重要的一部分，矩阵等价和向量组等价都是线性代数 的重要内容，虽然两种等价从概念上看起来完全不同，但是通过对线性代数基本 知识的学习可知，向量组可以用来表示线性方程组，并且线性方程组的某些重要 性质会反映在其系数矩阵和增广矩阵上，而且解方程组的过程也表现为变换这些 矩阵的过程，所以向量组和矩阵是一一对应的关系。这些都促使我们思考矩阵等 价与向量组等价之间的联系，本文的目的就是证明这两个等价概念的统一性。66299

Abstrct: Linear algebra is an important branch of mathematics. Matrix equivalence and vector group equivalence are important elements of linear algebra. Although the two equivalents are conceptually different, but by learning the linear equations, we know that the vector Groups can be used to represent linear equations, and some important properties of linear equations are reflected in their coefficient matrices and augmented matrices, and the process of solving equations is also expressed as the process of transforming these matrices, so the relationship of vector groups and matrices is a one- to-one. All of analysis prompted us to think about the relationship between the matrix equivalence and the equivalence of the vector group. The purpose of this paper is to prove the unity between the two equivalence.

目录

摘要 2

目录 3

一、绪论 4

1.1 线性代数的背景 4

1.2 线性代数的应用 5

二、预备知识 7

2.1 矩阵 7

2.2 向量组 9

三、证明 11

3.1 特殊情况： 11

3.2 一般情况： 14

四、总结 23

五、致谢 24

六、参考文献 25

一、绪论

1.1 线性代数的背景

线性代数有着长久的历史背景。早在大约四千年以前，巴比伦人就已经发现 解两个方程组成的二元一次线性方程组的方法。在中国古代的数学著作《九章算

术》中，