線性代數

第 1 章 行列式 DETERMINANT 1
第一節 2 階與 3 階行列式···································································.1
一、 2 階行列式 ·············································································1
二、 3 階行列式 ·············································································3
第二節 n 級排列與對換······································································.4
一、 n 級排列 ················································································4
二、對換 ·····················································································5
第三節 n 階行列式 ············································································.6
一、 n 階行列式定義·······································································6
二、幾種特殊的行列式 ···································································8
第四節 行列式的性質及其應用 ····························································10
一、行列式的性質 ······································································.11
二、行列式性質的應用 ································································.14
第五節 行列式按行（列）展開 ····························································17
一、行列式的展開 ······································································.17
二、行列式展開的應用 ································································.20
第六節 克拉默法則及其應用 ·······························································25
一、克拉默法則 ·········································································.25
二、克拉默法則的應用 ································································.27
第七節 應用實例—電路網絡問題 ······················································29
數學家閱讀小知識···············································································31
第 1 章總復習題 ·················································································33
第 2 章 矩陣 MATRIX 35
第一節 矩陣的概念 ···········································································35
一、矩陣概念的引入 ···································································.35
二、幾種特殊的矩陣 ···································································.36
第二節 矩陣的運算 ···········································································38
一、矩陣的加減法 ······································································.38
二、數與矩陣的乘法 ···································································.38
三、矩陣的乘法 ·········································································.39
四、矩陣的轉置 ·········································································.44
五、方陣的行列式 ······································································.45
六、伴隨矩陣 ············································································.47
第三節 矩陣的逆···············································································49
一、逆矩陣的概念和性質 ·····························································.49
二、矩陣可逆的條件 ···································································.50
第四節 分塊矩陣···············································································54
一、分塊矩陣的概念 ···································································.54
二、分塊矩陣的運算 ···································································.55
第五節 矩陣的初等變換 ·····································································61
一、矩陣初等變換的概念和性質 ····················································.61
二、初等矩陣及初等變換 ·····························································.64
三、利用初等變換求矩陣的逆 ·······················································.69
第六節 矩陣的秩···············································································73
一、矩陣秩的概念 ······································································.73
二、矩陣秩的性質 ······································································.76
第七節 應用案例—希爾密碼（ Hill Cipher） ········································79
數學家閱讀小知識···············································································81
第2章總復習題 ··················································································83
第 3 章 向量與向量空間 VECTOR AND VECTOR SPACE 87
第一節 n 維向量及其線性運算·····························································87
一、 n 維向量 ·············································································.87
二、向量的運算 ·········································································.88單元4
VII
第二節 向量組的線性相關性 ·······························································89
一、向量組線性相關性 ································································.89
二、向量組線性相關性的性質 ·······················································.93
第三節 極大線性無關組 ·····································································96
一、極大線性無關組的概念 ··························································.96
二、求極大線性無關組 ································································.97
三、向量組的等價 ······································································.99
第四節 向量組的秩及其與矩陣秩的關系 ·············································.101
一、向量組的秩 ·········································································101
二、向量組的秩與矩陣的秩的關系 ·················································103
第五節 向量空間············································································.106
一、向量空間及子空間 ································································106
二、 n 維向量空間 Rn 的基變換 ······················································107
第六節 向量的內積與正交向量組 ······················································.109
一、 n 維向量的內積····································································109
二、正交向量組 ·········································································111
第七節 應用實例—混凝土配方 ······················································.114
數學家閱讀小知識············································································.117
第 3 章總復習題 ··············································································.119
第 4 章 線性方程組 LINEAR SYSTEM OF EQUATIONS 123
第一節 線性方程組解的存在性及判定 ················································.123
一、齊次線性方程組解的存在性及判定 ···········································124
二、非齊次線性方程組解的存在性及判定 ········································125
第二節 齊次線性方程組及其基礎解系 ················································.129
一、齊次線性方程組解的結構 ·······················································129
二、齊次線性方程組的基礎解系和通解 ···········································130
第三節 非齊次線性方程組的通解 ······················································.137
第四節 應用實例—食物配方 ·························································.142
數學家閱讀小知識············································································.144
第 4 章總復習題 ··············································································.146
第 5 章 特征值與特征向量 EIGENVALUE AND EIGENVECTOR 149
第一節 特征值與特征向量的概念及性質 ·············································.149
一、特征值與特征向量的概念 ·······················································149
二、特征值和特征向量的性質 ·······················································152
第二節 相似矩陣············································································.156
一、相似矩陣的概念和性質 ··························································156
二、矩陣的對角化 ······································································158
第三節 實對稱矩陣及其對角化 ·························································.164
一、實對稱矩陣的特征值與特征向量的性質 ·····································164
二、實對稱矩陣的對角化 ·····························································165
第四節 應用實例············································································.169
數學家閱讀小知識············································································.172
第 5 章總復習題 ··············································································.174
第 6 章 二次型 QUADRATIC FORM 179
第一節 二次型及其標準形 ·······························································.179
一、二次型的概念 ······································································179
二、二次型的矩陣 ······································································180
三、線性變換與矩陣的合同 ··························································181
第二節 化二次型為標準形 ·······························································.183
一、正交變換法 ·········································································184
二、拉格朗日配方法 ···································································186
三、初等變換法 ·········································································187
第三節 正定二次型 ········································································.192
第四節 應用實例—條件優化問題 ···················································.198
數學家閱讀小知識············································································.201
第 6 章總復習題 ··············································································.203
參考答案 205