本書在總結多年實踐教學經驗的基礎上編寫而成，主要內容包括行列式、矩陣、線性方程組、相似矩陣。全書內容簡潔、實用，同時注重數學知識與具體專業的結合，精選專業案例，將專業案例轉化為數學問題并提供解答思路。本書每章后都附帶復習題，本書的最后附有參考答案。在本書附錄中，設置數學實驗指導、行列式數學實驗常用指令、簡單的線性規劃問題等拓展內容，另外還配套部分知識點的數字化學習資源，供學習者選學或參考。在實際教學和學習中，教師和學生可根據具體的課時安排，對書中的內容進行一定程度的增補或刪減。
胡煜，中山大學應用數學專業研究生畢業，現任職于廣東工貿職業技術學院，副教授，長期從事數學課程教學與研究工作。
第 1 章 行列式·······························································································1
1.1 行列式的概念····················································································1
1.1.1 二階行列式 ·············································································1
1.1.2 三階行列式 ·············································································3
1.1.3 n 階行列式 ··············································································3
習題 1.1···························································································5
1.2 行列式的性質····················································································6
習題 1.2···························································································8
1.3 克萊姆法則及應用··············································································9
1.3.1 克萊姆法則 ·············································································9
1.3.2 運用克萊姆法則討論齊次線性方程組的解 ···································.10
習題 1.3························································································.11
復習題 1 ······························································································.12
本章知識精要 ·······················································································.12
第 2 章 矩陣·······························································································.14
2.1 矩陣的概念····················································································.14
習題 2.1························································································.16
2.2 矩陣的性質及運算···········································································.17
2.2.1 矩陣相等 ·············································································.17
2.2.2 矩陣的運算 ··········································································.18
2.2.3 矩陣的轉置 ··········································································.22
2.2.4 方陣的行列式 ·······································································.24
習題 2.2························································································.24
2.3 矩陣的初等變換與矩陣的秩·······························································.25
線性代數
VI
2.3.1 矩陣的初等變換 ····································································.25
2.3.2 矩陣的秩 ·············································································.27
習題 2.3························································································.29
2.4 逆矩陣··························································································.30
2.4.1 逆矩陣的概念 ·······································································.30
2.4.2 逆矩陣的性質 ·······································································.30
2.4.3 逆矩陣的求法 ·······································································.31
2.4.4 用逆矩陣解線性方程組 ···························································.34
習題 2.4························································································.34
2.5 矩陣的應用····················································································.35
2.5.1 編制通信密碼 ·······································································.35
2.5.2 投入產出分析 ·······································································.37
復習題 2 ······························································································.40
本章知識精要 ·······················································································.42
第 3 章 線性方程組······················································································.44
3.1 向量組的線性相關性········································································.44
3.1.1 n 維向量空間········································································.44
3.1.2 線性相關性概念 ····································································.45
3.1.3 線性相關性的判定 ·································································.45
習題 3.1························································································.47
3.2 向量組的秩與矩陣···········································································.48
3.2.1 極大線性無關向量組 ······························································.48
3.2.2 向量組的秩 ··········································································.48
3.2.3 矩陣與向量組秩的關系 ···························································.49
習題 3.2························································································.49
3.3 線性方程組的解··············································································.50
3.3.1 消元法解線性方程組 ······························································.50
3.3.2 線性方程組解的結構 ······························································.53
習題 3.3························································································.54
3.4 應用·····························································································.55
復習題 3 ······························································································.60
本章知識精要 ·······················································································.61
第 4 章 相似矩陣·························································································.64
4.1 向量組的正交規范化········································································.64
目 錄
VII
4.1.1 向量內積及其性質 ·································································.64
4.1.2 正交向量組及其性質 ······························································.65
4.1.3 規范正交基及其求法 ······························································.66
4.1.4 正交矩陣與正交變換 ······························································.67
習題 4.1························································································.69
4.2 方陣的特征值與特征向量··································································.69
4.2.1 特征值與特征向量的概念 ························································.69
4.2.2 特征值與特征向量的基本性質 ··················································.72
習題 4.2························································································.72
4.3 相似矩陣的概念、性質及應用····························································.72
4.3.1 相似矩陣的概念 ····································································.73
4.3.2 相似矩陣的性質 ····································································.73
4.3.3 矩陣與對角矩陣相似的條件 ·····················································.75
習題 4.3························································································.77
4.4 實對稱矩陣的性質與對角化·······························································.77
4.4.1 實對稱矩陣的性質 ·································································.77
4.4.2 實對稱矩陣的對角化 ······························································.78
習題 4.4························································································.82
復習題 4 ······························································································.82
本章知識精要 ·······················································································.83
附錄 A 數學實驗指導···················································································.84
附錄 B 行列式數學實驗常用指令····································································.99
附錄 C 簡單的線性規劃問題··········································································101
附錄 D 習題參考答案···················································································106
參考文獻·····································································································114
線性代數
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