目录

绪论 

..........................................................................................................................................................1

第 0章预备知识 

..............................................................................................................................7 

0.1逻辑与集合..........................................................................................................................7 

0.2间接证明法..........................................................................................................................9 

0.3映射 

.....................................................................................................................................13

第 1章线性映射和矩阵 

...............................................................................................................16 

1.1基本概念 

............................................................................................................................16 

1.2线性映射的表示矩阵 

......................................................................................................27 

1.3线性方程组........................................................................................................................37 

1.4线性映射的运算 

...............................................................................................................49 

1.4.1线性运算 

...............................................................................................................49 

1.4.2复合........................................................................................................................52 

1.5矩阵的逆 

............................................................................................................................65 

1.5.1初等矩阵 

...............................................................................................................65 

1.5.2可逆矩阵 

...............................................................................................................66 

1.5.3等价关系 

...............................................................................................................71 

1.6分块矩阵 

............................................................................................................................76 

1.7 LU分解..............................................................................................................................87

第 2章子空间和维数....................................................................................................................94 

2.1基本概念 

............................................................................................................................94 

2.2基和维数 

..........................................................................................................................104 

2.2.1向量的线性关系................................................................................................104 

2.2.2基和维数 

.............................................................................................................108 

2.3矩阵的秩 

..........................................................................................................................112 

2.4线性方程组的解集.........................................................................................................120 

第 3章内积和正交性..................................................................................................................130 

3.1基本概念 

..........................................................................................................................130 

3.1.1内积......................................................................................................................130 

3.1.2标准正交基.........................................................................................................133 

3.2正交矩阵和 

QR分解....................................................................................................139 

3.2.1正交矩阵 

.............................................................................................................139 

3.2.2 QR分解..............................................................................................................142 

3.3子空间和投影..................................................................................................................149 

3.3.1正交补..................................................................................................................151 

3.3.2正交投影 

.............................................................................................................153 

3.3.3最小二乘问题 

....................................................................................................155

第 4章行列式...............................................................................................................................162 

4.1引子 

...................................................................................................................................162 

4.2行列式函数......................................................................................................................165 

4.3行列式的展开式 

.............................................................................................................176

第 5章特征值和特征向量.........................................................................................................187 

5.1引子 

...................................................................................................................................187 

5.2基本概念 

..........................................................................................................................191 

5.3对角化和谱分解 

.............................................................................................................199 

5.4相似 

...................................................................................................................................208

第 6章实对称矩阵......................................................................................................................219 

6.1实对称矩阵的谱分解 

....................................................................................................219 

6.2正定矩阵 

..........................................................................................................................227 

6.3奇异值分解......................................................................................................................234 

6.3.1基本概念 

.............................................................................................................235 

6.3.2辜低秩逼近........................................................................................................239

第 7章线性空间和线性映射 

....................................................................................................246 

7.1线性空间 

..........................................................................................................................246 

7.2基和维数 

..........................................................................................................................253 

7.3线性映射 

..........................................................................................................................260 

7.4向量的坐标表示 

.............................................................................................................266 

7.5线性映射的矩阵表示 

....................................................................................................274 

目录 V

第 8章内积空间 

..........................................................................................................................287 

8.1欧氏空间 

..........................................................................................................................287 

8.2欧氏空间上的线性映射................................................................................................297 

8.3酉空间...............................................................................................................................305 

8.3.1基本概念 

.............................................................................................................305 

8.3.2辜快速 

Fourier变换 

.......................................................................................310

名词索引..............................................................................................................................................318

人名表 

..................................................................................................................................................324

符号表 

..................................................................................................................................................325