矩阵的初等变换和应用

摘要：矩阵初等变换是矩阵论里相对简单但却十分关键的内容，更是个实用且常用的工具，在一些问题中能发挥独特的作用。本文在介绍了矩阵和初等变换的概念及一些结论之后，对应用做了罗列与归纳，给出了一些例题以加深理解，并将应用推广到了实际问题中，深入研究其在各问题中发挥的作用。本文简单地总结了初等变换的具体应用方法，有助于读者掌握初等变换解决问题的方法。42734

毕业论文关键词：矩阵；初等变换；秩；线性相关性；方程组

The elementary transformation of matrices and its application

Abstract: The elementary transformation of matrices is relatively simple but very important in the theory of matrices and is also a frequently used tool which plays a particular role in some issues. After introducing the basic concepts and major conclusions, this essay presents some applications of elementary transformation with some examples to get a deeper understanding then generalizes them to practical issues. This essay summarizes different situations which can be simplified. We hope this will help readers to use the conclusions in their study and daily life.

Keywords: Matrices; Elementary Transformation; Rank; Linearly Dependent; System of Equations

摘要 i

Abstract i

目录 ii

1 矩阵及其初等变换的基本概念 1

1.1 矩阵的基本概念 1

1.2 初等变换的基本概念 2

1.3 初等变换与初等矩阵 3

2 矩阵的初等变换在线性代数中的应用 6

2.1 求矩阵的秩 6

2.2 判断向量组的线性相关性 7

2.3 求向量组的秩 10

2.4 求逆矩阵 12

2.5 求向量在一组基下的坐标 14

2.6 求一组基到另一组基的过渡矩阵 15

2.7 求向量组生成的子空间的基与维数 16

2.8 化二次型为标准型 17

2.9 求解可逆矩阵方程 19

2.10 求解线性方程组 21

2.10.1 线性方程组概况 21

2.10.2 线性方程组何时有解 21

2.10.3 求齐次线性方程组的解 23

2.10.4 求非齐次线性方程组的解 25

3 矩阵的初等变换在数论和数分中的应用矩阵的初等变换和应用:http://www.youerw.com/shuxue/lunwen_43262.html