线性代数

本书在总结多年实践教学经验的基础上编写而成，主要内容包括行列式、矩阵、线性方程组、相似矩阵。全书内容简洁、实用，同时注重数学知识与具体专业的结合，精选专业案例，将专业案例转化为数学问题并提供解答思路。本书每章后都附带复习题，本书的最后附有参考答案。在本书附录中，设置数学实验指导、行列式数学实验常用指令、简单的线性规划问题等拓展内容，另外还配套部分知识点的数字化学习资源，供学习者选学或参考。在实际教学和学习中，教师和学生可根据具体的课时安排，对书中的内容进行一定程度的增补或删减。

第 1 章 行列式·······························································································1 1.1 行列式的概念····················································································1 1.1.1 二阶行列式 ·············································································1 1.1.2 三阶行列式 ·············································································3 1.1.3 n 阶行列式 ··············································································3 习题 1.1···························································································5 1.2 行列式的性质····················································································6 习题 1.2···························································································8 1.3 克莱姆法则及应用··············································································9 1.3.1 克莱姆法则 ·············································································9 1.3.2 运用克莱姆法则讨论齐次线性方程组的解 ···································.10 习题 1.3························································································.11 复习题 1 ······························································································.12 本章知识精要 ·······················································································.12 第 2 章 矩阵·······························································································.14 2.1 矩阵的概念····················································································.14 习题 2.1························································································.16 2.2 矩阵的性质及运算···········································································.17 2.2.1 矩阵相等 ·············································································.17 2.2.2 矩阵的运算 ··········································································.18 2.2.3 矩阵的转置 ··········································································.22 2.2.4 方阵的行列式 ·······································································.24 习题 2.2························································································.24 2.3 矩阵的初等变换与矩阵的秩·······························································.25 线性代数 VI 2.3.1 矩阵的初等变换 ····································································.25 2.3.2 矩阵的秩 ·············································································.27 习题 2.3························································································.29 2.4 逆矩阵··························································································.30 2.4.1 逆矩阵的概念 ·······································································.30 2.4.2 逆矩阵的性质 ·······································································.30 2.4.3 逆矩阵的求法 ·······································································.31 2.4.4 用逆矩阵解线性方程组 ···························································.34 习题 2.4························································································.34 2.5 矩阵的应用····················································································.35 2.5.1 编制通信密码 ·······································································.35 2.5.2 投入产出分析 ·······································································.37 复习题 2 ······························································································.40 本章知识精要 ·······················································································.42 第 3 章 线性方程组······················································································.44 3.1 向量组的线性相关性········································································.44 3.1.1 n 维向量空间········································································.44 3.1.2 线性相关性概念 ····································································.45 3.1.3 线性相关性的判定 ·································································.45 习题 3.1························································································.47 3.2 向量组的秩与矩阵···········································································.48 3.2.1 极大线性无关向量组 ······························································.48 3.2.2 向量组的秩 ··········································································.48 3.2.3 矩阵与向量组秩的关系 ···························································.49 习题 3.2························································································.49 3.3 线性方程组的解··············································································.50 3.3.1 消元法解线性方程组 ······························································.50 3.3.2 线性方程组解的结构 ······························································.53 习题 3.3························································································.54 3.4 应用·····························································································.55 复习题 3 ······························································································.60 本章知识精要 ·······················································································.61 第 4 章 相似矩阵·························································································.64 4.1 向量组的正交规范化········································································.64 目 录 VII 4.1.1 向量内积及其性质 ·································································.64 4.1.2 正交向量组及其性质 ······························································.65 4.1.3 规范正交基及其求法 ······························································.66 4.1.4 正交矩阵与正交变换 ······························································.67 习题 4.1························································································.69 4.2 方阵的特征值与特征向量··································································.69 4.2.1 特征值与特征向量的概念 ························································.69 4.2.2 特征值与特征向量的基本性质 ··················································.72 习题 4.2························································································.72 4.3 相似矩阵的概念、性质及应用····························································.72 4.3.1 相似矩阵的概念 ····································································.73 4.3.2 相似矩阵的性质 ····································································.73 4.3.3 矩阵与对角矩阵相似的条件 ·····················································.75 习题 4.3························································································.77 4.4 实对称矩阵的性质与对角化·······························································.77 4.4.1 实对称矩阵的性质 ·································································.77 4.4.2 实对称矩阵的对角化 ······························································.78 习题 4.4························································································.82 复习题 4 ······························································································.82 本章知识精要 ·······················································································.83 附录 A 数学实验指导···················································································.84 附录 B 行列式数学实验常用指令····································································.99 附录 C 简单的线性规划问题··········································································101 附录 D 习题参考答案···················································································106 参考文献·····································································································114 线性代数