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兼容数学公式
This commit is contained in:
@@ -20,6 +20,9 @@ export default defineConfig({
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title: 'SIMON BLOG',
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title: 'SIMON BLOG',
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description: 'Simon的博客，基于 vitepress 实现',
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description: 'Simon的博客，基于 vitepress 实现',
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lastUpdated: true,
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lastUpdated: true,
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markdown: {
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math: true
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},
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// 详见：https://vitepress.dev/zh/reference/site-config#head
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// 详见：https://vitepress.dev/zh/reference/site-config#head
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head: [
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head: [
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// 配置网站的图标（显示在浏览器的 tab 上）
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// 配置网站的图标（显示在浏览器的 tab 上）
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@@ -37,6 +40,8 @@ export default defineConfig({
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sidebarMenuLabel: '相关文章',
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sidebarMenuLabel: '相关文章',
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lastUpdatedText: '上次更新于',
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lastUpdatedText: '上次更新于',
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// 设置logo
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// 设置logo
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logo: '/logo.jpg',
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logo: '/logo.jpg',
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// editLink: {
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// editLink: {
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@@ -59,3 +64,4 @@ export default defineConfig({
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]
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]
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}
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}
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})
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})
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@@ -27,50 +27,404 @@ hidden: false
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## 内容概览
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## 完整笔记
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## 一、逻辑代数定律和计算规则
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| 定律/规则名称 | 表达式 | 解释 |
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| --------- | ----------------------------------------------------------------------------------- | ------------------- |
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| 恒等律 | $A + 0 = A$<br>$A \cdot 1 = A$ | 任何变量与0相加或与1相乘等于自身 |
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| 零律 | $A + 1 = 1$<br>$A \cdot 0 = 0$ | 任何变量与1相加或与0相乘等于1或0 |
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| 幂等律 | $A + A = A$<br>$A \cdot A = A$ | 任何变量与自身相加或相乘等于自身 |
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| 互补律 | $A + \overline{A} = 1$<br>$A \cdot \overline{A} = 0$ | 任何变量与其补码相加等于1，相乘等于0 |
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| **交换律** | | |
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| 加法交换律 | $A + B = B + A$ | 加法运算的交换律 |
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| 乘法交换律 | $A \cdot B = B \cdot A$ | 乘法运算的交换律 |
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| **结合律** | | |
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| 加法结合律 | $(A + B) + C = A + (B + C)$ | 加法运算的结合律 |
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| 乘法结合律 | $(A \cdot B) \cdot C = A \cdot (B \cdot C)$ | 乘法运算的结合律 |
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| **分配律** | | |
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| 乘法分配律 | $A \cdot (B + C) = A \cdot B + A \cdot C$ | 乘法对加法的分配律 |
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| 加法分配律 | $A + (B \cdot C) = (A + B) \cdot (A + C)$ | 加法对乘法的分配律 |
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| **吸收律** | | |
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| 吸收律1 | $A + A \cdot B = A$ | 吸收律的第一种形式 |
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| 吸收律2 | $A \cdot (A + B) = A$ | 吸收律的第二种形式 |
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| **德摩根定律** | | |
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| 德摩根定律1 | $\overline{A + B} = \overline{A} \cdot \overline{B}$ | 逻辑加法的德摩根定律 |
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| 德摩根定律2 | $\overline{A \cdot B} = \overline{A} + \overline{B}$ | 逻辑乘法的德摩根定律 |
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| **简化定律** | | |
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| 简化定律1 | $A + \overline{A} \cdot B = A + B$ | 简化逻辑表达式 |
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| 简化定律2 | $A \cdot (\overline{A} + B) = A \cdot B$ | 简化逻辑表达式 |
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| **共识定律** | | |
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| 共识定律 (积之和形式) | $AB + \overline{A}C + BC = AB + \overline{A}C$ | 较难，常用于逻辑化简。项 `BC` 是 `AB` 和 `A`C 的共识项，是冗余的。 |
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| 共识定律 (和之积形式) | $(A+B)(\overline{A}+C)(B+C) = (A+B)(\overline{A}+C)$ | 较难，常用于逻辑化简。项 `(B+C)` 是 `(A+B)` 和 `(A`+C) 的共识项，是冗余的。|
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| **反演定律** | | |
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| 反演定律 | $A = \overline{\overline{A}}$ | 变量的双重否定等于自身 |
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### 推导过程
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1. **基本定律**
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- **恒等律**：$A + 0 = A$ 和 $A \cdot 1 = A$ 是逻辑代数的基本定义。
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- **零律**：$A + 1 = 1$ 和 $A \cdot 0 = 0$ 也是逻辑代数的基本定义。
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- **幂等律**：$A + A = A$ 和 $A \cdot A = A$ 是因为逻辑加法和乘法运算的特性。
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- **互补律**：$A + \overline{A} = 1$ 和 $A \cdot \overline{A} = 0$ 是逻辑变量和其补码的定义。
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2. **交换律**
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- **加法交换律**：$A + B = B + A$ 是逻辑加法的交换特性。
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- **乘法交换律**：$A \cdot B = B \cdot A$ 是逻辑乘法的交换特性。
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3. **结合律**
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- **加法结合律**：$(A + B) + C = A + (B + C)$ 是逻辑加法的结合特性。
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- **乘法结合律**：$(A \cdot B) \cdot C = A \cdot (B \cdot C)$ 是逻辑乘法的结合特性。
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4. **分配律**
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- **乘法分配律**：$A \cdot (B + C) = A \cdot B + A \cdot C$ 是逻辑乘法对加法的分配特性。
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- **加法分配律**：$A + (B \cdot C) = (A + B) \cdot (A + C)$ 是逻辑加法对乘法的分配特性。
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5. **吸收律**
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- **吸收律1**：$A + A \cdot B = A$ 可以从 $A + A \cdot B = A \cdot (1 + B) = A \cdot 1 = A$ 推导得出。
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- **吸收律2**：$A \cdot (A + B) = A$ 可以从 $A \cdot (A + B) = A \cdot A + A \cdot B = A + A \cdot B = A$ 推导得出。
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6. **德摩根定律**
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- **德摩根定律1**：$\overline{A + B} = \overline{A} \cdot \overline{B}$ 是逻辑加法的德摩根定律。
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- **德摩根定律2**：$\overline{A \cdot B} = \overline{A} + \overline{B}$ 是逻辑乘法的德摩根定律。
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7. **简化定律**
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- **简化定律1**：$A + \overline{A} \cdot B = A + B$ 可以从 $A + \overline{A} \cdot B = (A + \overline{A}) \cdot (A + B) = 1 \cdot (A + B) = A + B$ 推导得出。
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- **简化定律2**：$A \cdot (\overline{A} + B) = A \cdot B$ 可以从 $A \cdot (\overline{A} + B) = A \cdot \overline{A} + A \cdot B = 0 + A \cdot B = A \cdot B$ 推导得出。
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8. **共识定律**
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- **共识定律**：$(A + B) \cdot (\overline{A} + C) = (A + B) \cdot (\overline{A} + C) \cdot (B + C)$ 可以从 $(A + B) \cdot (\overline{A} + C) = (A + B) \cdot (\overline{A} + C) \cdot (B + C)$ 推导得出，因为 $(A + B) \cdot (\overline{A} + C) \leq (B + C)$。
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9. **反演定律**
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- **反演定律**：$A = \overline{\overline{A}}$ 是逻辑变量的双重否定特性。
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---
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## 二、基本门电路
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### 1. 非门
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$$
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Y = \overline{A}
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$$
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#### 第一部分：数字逻辑基础
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### 2. 与门
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- **数制与编码**
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- 二进制、八进制、十六进制
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- BCD码、格雷码
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- 数制转换方法
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- **逻辑代数基础**
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$$
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- 布尔代数基本定律
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Y = A \cdot B
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- 逻辑函数的表示方法
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$$
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- 逻辑函数的化简
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#### 第二部分：组合逻辑电路
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**真值表:**
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- **基本逻辑门**
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- 与门、或门、非门
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- 与非门、或非门、异或门
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- 逻辑门的电气特性
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- **组合逻辑电路分析与设计**
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| 输入 A | 输入 B | 输出 Y |
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- 组合电路的分析方法
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| --- | --- | --- |
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- 组合电路的设计流程
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| 0 | 0 | 0 |
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- 典型组合电路应用
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| 0 | 1 | 0 |
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| 1 | 0 | 0 |
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| 1 | 1 | 1 |
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#### 第三部分：时序逻辑电路
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- **触发器**
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- RS触发器、JK触发器
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- D触发器、T触发器
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- 触发器的应用
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- **时序电路分析与设计**
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- 时序电路的基本概念
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- 状态图与状态表
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- 时序电路的设计方法
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#### 第四部分：数字系统设计
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### 3. 或门
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- **计数器与寄存器**
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- 二进制计数器
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- 十进制计数器
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- 移位寄存器
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- **数字系统综合设计**
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$$
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- 数字系统设计方法
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Y = A + B
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- 可编程逻辑器件
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$$
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- 数字系统的测试与调试
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**真值表:**
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| 输入 A | 输入 B | 输出 Y |
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| --- | --- | --- |
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| 0 | 0 | 0 |
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| 0 | 1 | 1 |
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| 1 | 0 | 1 |
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| 1 | 1 | 1 |
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### 4. 与非门
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与非门是“与门”和“非门”的结合。
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$$
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Y = \overline{A \cdot B}
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$$
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**真值表:**
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| 输入 A | 输入 B | 输出 Y |
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|:---:|:---:|:---:|
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| 0 | 0 | 1 |
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| 0 | 1 | 1 |
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| 1 | 0 | 1 |
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| 1 | 1 | 0 |
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### 5. 或非门
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或非门是“或门”和“非门”的结合。
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$$
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Y = \overline{A + B}
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$$
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**真值表:**
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| 输入 A | 输入 B | 输出 Y |
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|:---:|:---:|:---:|
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| 0 | 0 | 1 |
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| 0 | 1 | 0 |
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| 1 | 0 | 0 |
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| 1 | 1 | 0 |
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### 6. 异或门
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当两个输入不相同时，输出为高电平（1）；当两个输入相同时，输出为低电平（0）。这也被称为“半加器”的求和逻辑。
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**逻辑表达式:**
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$$
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Y = A \oplus B
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$$
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**真值表:**
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| 输入 A | 输入 B | 输出 Y |
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|:---:|:---:|:---:|
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| 0 | 0 | 0 |
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| 0 | 1 | 1 |
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| 1 | 0 | 1 |
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| 1 | 1 | 0 |
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---
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## 三、编码
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### 1. 原码、反码和补码
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为了在二进制系统中表示正负数，我们通常会使用最高位作为**符号位**。
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* 符号位为 **0** 代表**正数**。
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* 符号位为 **1** 代表**负数**。
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#### **原码**
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* **规则**: 符号位 + 数值的绝对值的二进制表示。
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* **正数**: 符号位为0，其余位表示数值。
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* 例如，$+12$ 的原码是 **00001100**。
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* **负数**: 符号位为1，其余位表示数值。
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* 例如，$-12$ 的原码是 **10001100**。
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* **缺点**:
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1. 零的表示不唯一：$+0$ 是 **00000000**，$-0$ 是 **10000000**。
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2. 进行加减法运算时，需要单独处理符号位，硬件实现复杂。
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#### **反码**
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反码的出现是为了简化减法运算。
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* **规则**:
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* **正数**的反码与其原码**相同**。
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* **负数**的反码是在其**原码**的基础上，**符号位不变**，其余各位**按位取反**。
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* **示例**:
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* $+12$ 的原码是 `00001100`，其反码也是 **00001100**。
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* $-12$ 的原码是 `10001100`，其反码是 **11110011** (符号位1不变，后面7位 `0001100` 按位取反得到 `1110011`)。
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* **缺点**:
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* 仍然存在“双零”问题：$+0$ 的反码是 **00000000**，$-0$ 的反码是 **11111111**。
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* 跨零运算会产生循环进位问题。
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#### **补码**
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补码是现代计算机系统中最常用的有符号数表示法，它解决了原码和反码的缺点。
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* **规则**:
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* **正数**的补码与其原码**相同**。
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* **负数**的补码是其**反码加 1**。
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* **求负数补码的方式**:
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* 从其原码的**最低位（最右边）**向左找，找到的**第一个 1** 保持不变，这个 1 **左边**的所有位（不含符号位）按位取反，符号位仍为1。
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* **示例**:
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* $+12$ 的补码是 **00001100**。
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* $-12$ 的补码求法：
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1. 原码: `10001100`
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2. 反码: `11110011`
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3. 加 1: `11110011 + 1` = **11110100**。
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* **优点**:
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1. **零的表示唯一**: **00000000**。
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2. **简化运算**: 可以将减法运算转换为加法运算。例如，计算 $A - B$ 等同于计算 $A + (-B)$ 的补码。
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3. 对于一个 $n$ 位的补码系统，其表示范围为 $[-2^{n-1}, 2^{n-1}-1]$。例如，8位补码的范围是 $[-128, 127]$。
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**总结表格 (以 ±12 为例)**
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| 值 | 原码 | 反码 | 补码 |
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|:---:|:---:|:---:|:---:|
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| +12 | 00001100 | 00001100 | 00001100 |
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| -12 | 10001100 | 11110011 | 11110100 |
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
### 2. BCD 码
|
||||||
|
|
||||||
|
BCD码是用**二进制**来表示**十进制**数的一种编码方式。它与直接将十进制数转换为二进制数不同。
|
||||||
|
|
||||||
|
* **规则**: 用 **4 位二进制数**来表示一位十进制数（0-9）。最常用的是 **8421 BCD 码**，其中各位的权值从高到低分别是 8、4、2、1。
|
||||||
|
* **特点**:
|
||||||
|
* 它介于二进制和十进制之间，便于人机交互（如数码管显示、计算器）。
|
||||||
|
* 运算比纯二进制复杂，但比直接处理十进制字符简单。
|
||||||
|
* 由于用4位二进制表示一位十进制数，所以 `1010` 到 `1111` 这 6 个码是无效或非法的。
|
||||||
|
|
||||||
|
**BCD 码对照表**
|
||||||
|
|
||||||
|
| 十进制 | BCD 码 |
|
||||||
|
|:---:|:---:|
|
||||||
|
| 0 | 0000 |
|
||||||
|
| 1 | 0001 |
|
||||||
|
| 2 | 0010 |
|
||||||
|
| 3 | 0011 |
|
||||||
|
| 4 | 0100 |
|
||||||
|
| 5 | 0101 |
|
||||||
|
| 6 | 0110 |
|
||||||
|
| 7 | 0111 |
|
||||||
|
| 8 | 1000 |
|
||||||
|
| 9 | 1001 |
|
||||||
|
|
||||||
|
**示例**:
|
||||||
|
将十进制数 **129** 转换为 BCD 码。
|
||||||
|
|
||||||
|
1. 将每一位十进制数分开：`1`、`2`、`9`。
|
||||||
|
2. 将每一位分别转换为对应的4位BCD码：
|
||||||
|
* $1 \rightarrow 0001$
|
||||||
|
* $2 \rightarrow 0010$
|
||||||
|
* $9 \rightarrow 1001$
|
||||||
|
3. 将它们组合起来：
|
||||||
|
$$
|
||||||
|
(129)_{10} = (0001 \ 0010 \ 1001)_{\text{BCD}}
|
||||||
|
$$
|
||||||
|
**对比**: 如果将 (129)₁₀ 直接转换为纯二进制，结果是 **10000001**。这与它的 BCD 码是完全不同的。
|
||||||
|
---
|
||||||
|
## 四、加法器、编码器、译码器、选择器、比较器
|
||||||
|
---
|
||||||
|
## 五、触发器
|
||||||
|
|
||||||
|
### 1. RS 触发器
|
||||||
|
|
||||||
|
最基本的触发器，但存在一个不确定状态，在实际应用中较少直接使用。
|
||||||
|
|
||||||
|
* **输入**: $S$ (Set, 置位), $R$ (Reset, 复位)
|
||||||
|
* **输出**: $Q$ (状态输出), $\overline{Q}$ (反向输出)
|
||||||
|
|
||||||
|
#### **功能表**
|
||||||
|
这张表描述了在不同输入下，下一个状态 $Q_{n+1}$ 是什么。
|
||||||
|
|
||||||
|
| $S$ | $R$ | $Q_{n+1}$ | 功能 |
|
||||||
|
|:---:|:---:|:---:|:---|
|
||||||
|
| 0 | 0 | $Q_n$ | 保持 |
|
||||||
|
| 0 | 1 | 0 | 复位/置0 |
|
||||||
|
| 1 | 0 | 1 | 置位/置1|
|
||||||
|
| 1 | 1 | **?** | **禁止/不定** |
|
||||||
|
|
||||||
|
#### **特性方程**
|
||||||
|
$$
|
||||||
|
Q_{n+1} = S + \overline{R}Q_n \quad (\text{约束条件: } S \cdot R = 0)
|
||||||
|
$$
|
||||||
|
|
||||||
|
#### **激励表**
|
||||||
|
这张表在电路设计时非常有用，它回答了“为了让状态从 $Q_n$ 变为 $Q_{n+1}$，输入 $S$ 和 $R$ 应该是什么？”。（X表示Don't Care，即0或1均可）
|
||||||
|
|
||||||
|
| $Q_n$ | $Q_{n+1}$ | $S$ | $R$ |
|
||||||
|
|:---:|:---:|:---:|:---:|
|
||||||
|
| 0 | 0 | 0 | X |
|
||||||
|
| 0 | 1 | 1 | 0 |
|
||||||
|
| 1 | 0 | 0 | 1 |
|
||||||
|
| 1 | 1 | X | 0 |
|
||||||
|
|
||||||
|
|
||||||
|
### 2. JK 触发器
|
||||||
|
|
||||||
|
JK 触发器是 RS 触发器的改进版，它解决了 RS 触发器的“禁止”状态问题，是最通用的触发器。
|
||||||
|
|
||||||
|
* **输入**: $J$ (功能类似 $S$), $K$ (功能类似 $R$)
|
||||||
|
* **输出**: $Q$, $\overline{Q}$
|
||||||
|
|
||||||
|
#### **功能表**
|
||||||
|
|
||||||
|
| $J$ | $K$ | $Q_{n+1}$ | 功能 |
|
||||||
|
|:---:|:---:|:---:|:---|
|
||||||
|
| 0 | 0 | $Q_n$ | 保持 |
|
||||||
|
| 0 | 1 | 0 | 复0 |
|
||||||
|
| 1 | 0 | 1 | 置1 |
|
||||||
|
| 1 | 1 | $\overline{Q_n}$ | **翻转 ** |
|
||||||
|
|
||||||
|
*JK触发器将RS触发器的禁止状态（1,1输入）变成了一个非常有用的**翻转**功能。*
|
||||||
|
|
||||||
|
#### **特性方程**
|
||||||
|
$$
|
||||||
|
Q_{n+1} = J\overline{Q_n} + \overline{K}Q_n
|
||||||
|
$$
|
||||||
|
|
||||||
|
#### **激励表**
|
||||||
|
|
||||||
|
| $Q_n$ | $Q_{n+1}$ | $J$ | $K$ |
|
||||||
|
|:---:|:---:|:---:|:---:|
|
||||||
|
| 0 | 0 | 0 | X |
|
||||||
|
| 0 | 1 | 1 | X |
|
||||||
|
| 1 | 0 | X | 1 |
|
||||||
|
| 1 | 1 | X | 0 |
|
||||||
|
|
||||||
|
|
||||||
|
### 3. D 触发器
|
||||||
|
D 触发器的功能非常直接：在时钟脉冲到来时，将输入 $D$ 的值传递给输出 $Q$。它常被用作数据锁存器或移位寄存器的基本单元。
|
||||||
|
|
||||||
|
* **输入**: $D$ (Data)
|
||||||
|
* **输出**: $Q$, $\overline{Q}$
|
||||||
|
|
||||||
|
#### **功能表**
|
||||||
|
|
||||||
|
| $D$ | $Q_{n+1}$ | 功能 |
|
||||||
|
|:---:|:---:|:---|
|
||||||
|
| 0 | 0 | 置0 |
|
||||||
|
| 1 | 1 | 置1 |
|
||||||
|
|
||||||
|
*无论当前状态 $Q_n$ 是什么，下一个状态 $Q_{n+1}$ 都等于时钟边沿到来时的 $D$ 输入值。*
|
||||||
|
|
||||||
|
#### **特性方程 **
|
||||||
|
$$
|
||||||
|
Q_{n+1} = D
|
||||||
|
$$
|
||||||
|
|
||||||
|
#### **激励表 **
|
||||||
|
|
||||||
|
| $Q_n$ | $Q_{n+1}$ | $D$ |
|
||||||
|
|:---:|:---:|:---:|
|
||||||
|
| 0 | 0 | 0 |
|
||||||
|
| 0 | 1 | 1 |
|
||||||
|
| 1 | 0 | 0 |
|
||||||
|
| 1 | 1 | 1 |
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
### 4. T 触发器
|
||||||
|
|
||||||
|
T 触发器是一个翻转触发器。当输入 $T=1$ 时，状态翻转；当 $T=0$ 时，状态保持不变。它常用于构建计数器。
|
||||||
|
|
||||||
|
* **输入**: $T$
|
||||||
|
* **输出**: $Q$, $\overline{Q}$
|
||||||
|
|
||||||
|
#### **功能表**
|
||||||
|
|
||||||
|
| $T$ | $Q_{n+1}$ | 功能 |
|
||||||
|
|:---:|:---:|:---|
|
||||||
|
| 0 | $Q_n$ | 保持 |
|
||||||
|
| 1 | $\overline{Q_n}$ | 翻转 |
|
||||||
|
|
||||||
|
#### **特性方程**
|
||||||
|
$$
|
||||||
|
Q_{n+1} = T \oplus Q_n = T\overline{Q_n} + \overline{T}Q_n
|
||||||
|
$$
|
||||||
|
|
||||||
|
#### **激励表**
|
||||||
|
|
||||||
|
| $Q_n$ | $Q_{n+1}$ | $T$ |
|
||||||
|
|:---:|:---:|:---:|
|
||||||
|
| 0 | 0 | 0 |
|
||||||
|
| 0 | 1 | 1 |
|
||||||
|
| 1 | 0 | 1 |
|
||||||
|
| 1 | 1 | 0 |
|
||||||
|
|||||||
@@ -34,5 +34,3 @@ hidden: false # 是否隐藏文章：true隐藏（模板用），fals
|
|||||||
但是总想着要黑漂才拉上来
|
但是总想着要黑漂才拉上来
|
||||||
|
|
||||||
等着寒假再回来钓
|
等着寒假再回来钓
|
||||||
|
|
||||||
|
|
||||||
|
|||||||
@@ -46,58 +46,609 @@ hidden: false
|
|||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
## 第一章
|
||||||
|
|
||||||
## 笔记章节概览
|
### 1.1 线性方程组
|
||||||
|
#### (1) 矩阵与增广矩阵
|
||||||
|
$$
|
||||||
|
2x_1 - x_2 + 1.5x_3 = 8
|
||||||
|
$$
|
||||||
|
$$
|
||||||
|
x_1 - 4x_3 = -7
|
||||||
|
$$
|
||||||
|
|
||||||
|
* 矩阵 (Matrix)
|
||||||
|
|
||||||
|
$$
|
||||||
|
\begin{bmatrix}
|
||||||
|
2 & -1 & 1.5\\
|
||||||
|
1 & 0 & -4
|
||||||
|
\end{bmatrix}
|
||||||
|
$$
|
||||||
|
|
||||||
|
* 增广矩阵 (Augmented Matrix)
|
||||||
|
|
||||||
|
$$
|
||||||
|
\begin{bmatrix}
|
||||||
|
2 & -1 & 1.5 & 8\\
|
||||||
|
1 & 0 & -4 & -7
|
||||||
|
\end{bmatrix}
|
||||||
|
$$
|
||||||
|
|
||||||
|
* 线性方程组解的三种情况：
|
||||||
|
1. 无解 (不相容) (incompatibility)
|
||||||
|
2. 有唯一解 (相容) (compatibility)
|
||||||
|
3. 有无穷多解 (相容) (compatibility)
|
||||||
|
|
||||||
|
|
||||||
本高等代数笔记共分为七个章节，每章都有独立的Markdown源码和PDF文件：
|
#### (2) 矩阵变换
|
||||||
|
|
||||||
### 📖 章节目录
|
* 倍加
|
||||||
|
* 对换
|
||||||
|
* 倍乘
|
||||||
|
|
||||||
|
|
||||||
|
### 1.2 行化简与阶梯形矩阵
|
||||||
|
|
||||||
|
>**先导元素 (Leading element)**
|
||||||
|
**定义**
|
||||||
|
一个矩阵称为阶梯形(或行阶梯形),若它有以下三个性质:
|
||||||
|
l.每一非零行都在每一零行之上.
|
||||||
|
2.某一行的先导元素所在的列位于前一行先导元素的右边
|
||||||
|
3.某一先导元素所在列下方元素都是零.
|
||||||
|
若一个阶梯形矩阵还满足以下性质,贝则称它为简化阶梯形(或简化行阶梯形) .
|
||||||
|
4.每一非零行的先导元素是 1.
|
||||||
|
5.每一先导元素 1 是该元素所在列的唯一非零元素
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
#### 第一章：线性方程组
|
>**定理1** (简化阶梯形矩阵的唯一性)
|
||||||
- 线性方程组的基本概念
|
每个矩阵行等价于唯一的简化阶梯形矩阵.
|
||||||
- 高斯消元法
|
|
||||||
- 矩阵的初等变换
|
|
||||||
- 线性方程组解的结构
|
|
||||||
|
|
||||||
#### 第二章：矩阵
|
|
||||||
- 矩阵的基本运算
|
|
||||||
- 矩阵的逆
|
|
||||||
- 分块矩阵
|
|
||||||
- 矩阵的秩
|
|
||||||
|
|
||||||
#### 第三章：向量空间
|
>**主元位置 (Pivot position)**
|
||||||
- 向量空间的定义
|
**定义**
|
||||||
- 线性相关与线性无关
|
矩阵中的主元位置是A中对应于它的阶梯形中先导元素 1 的位直.主元列是$A$的含有主元往直的列
|
||||||
- 基与维数
|
|
||||||
- 坐标变换
|
|
||||||
|
|
||||||
#### 第四章：线性变换
|
>**定理2** (存在与唯一性定理)
|
||||||
- 线性变换的定义与性质
|
线性方程组相容的充要条件是增广矩阵的最右列不是主元列.也就是说增广矩阵的阶梯形没有形如
|
||||||
- 线性变换的矩阵表示
|
$[0 \ \ \cdots \ \ 0 \ \ b] \ \ ,\ \ b\neq0$
|
||||||
- 特征值与特征向量
|
|
||||||
- 对角化理论
|
|
||||||
|
|
||||||
#### 第五章：多项式
|
>的行若线性方程组相容,则它的解集可能有两种情形:
|
||||||
- 多项式的基本理论
|
( i )当没有自由变量时,有唯一解;
|
||||||
- 最大公因式
|
( ii )若至少有一个自由变量,则有无穷多解.
|
||||||
- 因式分解
|
|
||||||
- 有理函数
|
|
||||||
|
|
||||||
#### 第六章：矩阵的标准形
|
### 1.3 向量方程
|
||||||
- 相似矩阵
|
|
||||||
- Jordan标准形
|
|
||||||
- 最小多项式
|
|
||||||
- 矩阵函数
|
|
||||||
|
|
||||||
#### 第七章：二次型
|
$$u=
|
||||||
- 二次型的基本概念
|
\begin{bmatrix}
|
||||||
- 二次型的标准形
|
\ 2 \ \\
|
||||||
- 正定二次型
|
\ 1 \
|
||||||
- 二次型的应用
|
\end{bmatrix}
|
||||||
|
$$
|
||||||
|
|
||||||
|
>满足加法乘法的性质
|
||||||
|
|
||||||
|
* 线性组合
|
||||||
|
$y=x_1c_1+\cdots+x_ic_i$ 中 $c_i$ 为权
|
||||||
|
|
||||||
|
* 向量张成 (生成)
|
||||||
|
|
||||||
|
$span\{x_1,x_2,\cdots,x_i\}$
|
||||||
|
即判断
|
||||||
|
$y=x_1c_1+\cdots+x_ic_i$
|
||||||
|
是否有解；或
|
||||||
|
$\begin{bmatrix}
|
||||||
|
\ x_1\ x_2\ \cdots \ x_3 \ y\\
|
||||||
|
\end{bmatrix}$
|
||||||
|
是否有解
|
||||||
|
|
||||||
|
### 1.4 矩阵方程 Ax=b
|
||||||
|
|
||||||
|
>**定义**
|
||||||
|
若$A$是$m \times n$矩阵,它的各列为 $a$
|
||||||
|
若 $x$ 是$R$<sup>n</sup>中的向量,则 $A$ 与 $x$ 的积(记为$Ax$) 就是 $A$ 的各列以 $x$ 中对应元素为权的线性组合
|
||||||
|
|
||||||
|
>**定理3**
|
||||||
|
$Ax=b$
|
||||||
|
等价于
|
||||||
|
$\begin{bmatrix}
|
||||||
|
\ a_1\ a_2\ \cdots \ a_3 \ \ b\\
|
||||||
|
\end{bmatrix}$
|
||||||
|
|
||||||
|
|
||||||
|
* 解的存在性
|
||||||
|
|
||||||
|
>**方程Ax = b 有解当且仅当 b 是 A 的各列的线性组合.**
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
>**定理4**
|
||||||
|
设 $A$ 是 $m \times n$ 矩阵,则下列命题是逻辑上等价的.
|
||||||
|
也就是说,对某个 $Ax = b$ 它们都成立或者都不成立.
|
||||||
|
a. 对$R$<sup>m</sup>中每个 $b$ ,方程 $Ax=b$ 有解.
|
||||||
|
b. $R$<sup>m</sup>中的每个 $b$ 都是 $A$ 的列的一个线性组合.
|
||||||
|
c. $A$ 的各列生成$R$<sup>m</sup>.
|
||||||
|
d. $A$ 在每一行都有一个主元位置.
|
||||||
|
|
||||||
|
>**计算**
|
||||||
|
计算 $Ax$ 的行-向量规则
|
||||||
|
若乘积 $Ax$ 有定义,则 $Ax$ 中的第 $i$ 个元素是 $A$ 的第 $i$ 行元素与 $x$ 的相应元素乘积之和.
|
||||||
|
|
||||||
|
>**定理5**
|
||||||
|
若 $A$ 是 $m\times n$ 矩阵,$u$ 和 $v$ 是$R$<sup>n</sup>中向量, $c$ 是标量,如:
|
||||||
|
a. $A(u+v) = Au+Av.$
|
||||||
|
b. $A(cu) = c(Au).$
|
||||||
|
|
||||||
|
### 1.5 线性方程组的解集
|
||||||
|
* 齐次线性方程组
|
||||||
|
|
||||||
|
>齐次方程 $Ax=0$ 有非平凡解当且仅当方程至少有一个自由变量.
|
||||||
|
|
||||||
|
>**定理6**
|
||||||
|
设方程 $Ax=b$ 对某个 $b$ 是相容的, $p$ 为一个特解,则 $Ax=b$ 的解集是所有形如
|
||||||
|
$w = p+v_h$
|
||||||
|
>的向量的集, 其中 $v$<sub>h</sub> 是齐次方程 $Ax=0$ 的任意一个解.
|
||||||
|
|
||||||
|
### 1.7 线性无关
|
||||||
|
|
||||||
|
>**定义**
|
||||||
|
向量方程 $0=x_1c_1+\cdots+x_ic_i$ 仅有平凡解(trivial solution) 向量组 (集) 称为线性无关的 (linearly independent)
|
||||||
|
若存在不全为零的权
|
||||||
|
$c_i$
|
||||||
|
使
|
||||||
|
$x_1c_1+\cdots+x_ic_i+0$
|
||||||
|
则向量组 (集) 称为线性相关的 (linearly dependent)
|
||||||
|
|
||||||
|
>**矩阵 $A$ 的各列线性无关,当且仅当方程 $Ax=0$ 仅有平凡**
|
||||||
|
|
||||||
|
>**定理7** (线性相关集的特征)
|
||||||
|
两个或更多个向量的集合
|
||||||
|
$S=\{v_1,v_2,\cdots,v_p\}$
|
||||||
|
>线性相关,当且仅当 $S$ 中至少有一个向量是其他向量的线性组合.
|
||||||
|
|
||||||
|
>**定理8**
|
||||||
|
若一个向量组的向量个数超过每个向量的元素个数,那么这个向量组线性相关.就
|
||||||
|
是说, $R$<sup>n</sup> 中任意向量组
|
||||||
|
$\{v_1,v_2,\cdots,v_p\}$
|
||||||
|
>当 $p>n$ 时线性相关.
|
||||||
|
|
||||||
|
>**定理9**
|
||||||
|
若 $R$<sup>n</sup> 中向量组
|
||||||
|
$S=\{v_1,v_2,\cdots,v_p\}$
|
||||||
|
>包含零向量,则它线性相关
|
||||||
|
|
||||||
|
### 1.8 线性变换介绍
|
||||||
|
* 变换(transformation)(或称函数、映射(map)) $T$ 是一个规则
|
||||||
|
* $T$ : $R$<sup>n</sup> → $R$<sup>m</sup>
|
||||||
|
$R$<sup>n</sup>称为 $T$ 的定义域 (domain)
|
||||||
|
$R$<sup>m</sup>称为 $T$ 的余定义域 (codomain) (或取值空间)
|
||||||
|
|
||||||
|
* 线性变换
|
||||||
|
$$T(0) = 0$$
|
||||||
|
$$T(cu+ dv) = cT(u) + dT(v)$$
|
||||||
|
|
||||||
|
### 1.9 线性变换的矩阵
|
||||||
|
|
||||||
|
>**定理10**
|
||||||
|
设 $T$ : $R$<sup>n</sup> → $R$<sup>m</sup> 为线性变换,则存在唯一的矩阵 $A$ ,使得对 $R$<sup>n</sup>中一切 $x$ 满足 $T(x)=Ax$
|
||||||
|
|
||||||
|
* 满射
|
||||||
|
>映射 $T$ : $R$<sup>n</sup> → $R$<sup>m</sup> 称为到 $R$<sup>m</sup> 上的映射,若 $R$<sup>m</sup> 中每个 $b$ 是 $R$<sup>n</sup> 中至少一个 $x$ 的像.
|
||||||
|
|
||||||
|
>“满射” 的英文是 “surjective” 或 “surjection” 或 “onto mapping” 或 “onto function”
|
||||||
|
|
||||||
|
|
||||||
|
* 单射
|
||||||
|
>映射 $T$ : $R$<sup>n</sup> → $R$<sup>m</sup> 称为一对一映射(或1:1),若 $R$<sup>m</sup> 中每个 $b$ 是 $R$<sup>m</sup> 中至多一个 $x$ 的像.
|
||||||
|
|
||||||
|
>“单射” 的英文是 “injective” 或 “injection” 或 “one-to-one mapping” 或 “one-to-one function”
|
||||||
|
|
||||||
|
|
||||||
|
>**定理11**
|
||||||
|
设 $T$ : $R$<sup>n</sup> → $R$<sup>m</sup> 为线性变换,则 $T$ 是一对一的当且仅当方程 $Ax=0$ 仅有平凡解.
|
||||||
|
|
||||||
|
>**定理12**
|
||||||
|
设 $T$ : $R$<sup>n</sup> → $R$<sup>m</sup> 为线性变换,设 $A$ 为 $T$ 的标准矩阵，则：
|
||||||
|
a. $T$ 把 $R$<sup>n</sup> 映上到 $R$<sup>m</sup> ，当且仅当 $A$ 的列生成 $R$<sup>m</sup>.
|
||||||
|
b. $T$ 是一对一的，当且仅当 $A$ 的列线性无关.
|
||||||
|
|
||||||
|
## 第二章
|
||||||
|
|
||||||
|
### 2.1 矩阵运算
|
||||||
|
加减乘
|
||||||
|
### 2.2 矩阵的逆
|
||||||
|
>不可逆矩阵有时称为**奇异矩阵**，而可逆矩阵也称为**非奇异矩阵**.
|
||||||
|
$$A^{-1}A=I$$
|
||||||
|
$$A^{-1}=\frac{1}{det A} \times A_{adj}$$
|
||||||
|
$A_{adj}$是伴随矩阵（adjugate matrix）
|
||||||
|
$$(A^{-1})^{-1}=A$$
|
||||||
|
$$(AB)^{-1}=A^{-1}B^{-1}$$
|
||||||
|
|
||||||
|
>若干个$n \times n$ 可逆矩阵的积也是可逆的，其逆等于这些矩阵的逆按相反顺序的乘积
|
||||||
|
>~~看不懂，不爱用这种方法~~
|
||||||
|
|
||||||
|
求法（我常用）：
|
||||||
|
$$[\ A\ \ \ I\ ]=[\ I\ \ \ A^{-1}\ ] $$
|
||||||
|
|
||||||
|
### 2.3 矩阵的特征
|
||||||
|
|
||||||
|
* 挺多的
|
||||||
|
|
||||||
|
|
||||||
|
### 2.4 分块矩阵
|
||||||
|
* 没什么特别的
|
||||||
|
|
||||||
|
### 2.5 LU分解
|
||||||
|
>L 是 $m \times m$ 下三角矩阵， 主对角线元素全是1，
|
||||||
|
>$A=LU$
|
||||||
|
>AI写的：
|
||||||
|
|
||||||
|
* Doolittle分解（LU分解的一种常见形式）
|
||||||
|
* 原理 对于一个$n \times n$矩阵 $A$，将其分解为一个下三角矩阵$L$,主对角线元素为1和一个上三角矩阵$U$的乘积，即$A = LU$。
|
||||||
|
> 计算步骤
|
||||||
|
> 1. **设定矩阵形式** 设
|
||||||
|
$$A=\left[\begin{array}{cccc}a_{11}&a_{12}&\cdots&a_{1n}\\a_{21}&a_{22}&\cdots&a_{2n}\\\vdots&\vdots&\ddots&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nn}\end{array}\right]$$
|
||||||
|
$$L=\left[\begin{array}{cccc}1&0&\cdots&0\\l_{21}&1&\cdots&0\\\vdots&\vdots&\ddots&\vdots\\l_{n1}&l_{n2}&\cdots&1\end{array}\right]$$
|
||||||
|
$$U=\left[\begin{array}{cccc}u_{11}&u_{12}&\cdots&u_{1n}\\0&u_{22}&\cdots&u_{2n}\\\vdots&\vdots&\ddots&\vdots\\0&0&\cdots&u_{nn}\end{array}\right]$$
|
||||||
|
> 2. **计算$U$的第一行和$L$的第一列**
|
||||||
|
$$u_{1j}=a_{1j}(j = 1,2,\cdots,n)$$
|
||||||
|
$$l_{i1}=\frac{a_{i1}}{u_{11}}(i = 2,3,\cdots,n)$$
|
||||||
|
> 3. **对于$k = 2,3,\cdots,n$，分别计算$U$的第$k$行和$L$的第$k$列计算$U$的第$k$行**：
|
||||||
|
$$u_{kj}=a_{kj}-\sum_{m = 1}^{k - 1}l_{km}u_{mj}(j = k,k + 1,\cdots,n)$$
|
||||||
|
>**计算$L$的第$k$列**：
|
||||||
|
$$l_{ik}=\frac{1}{u_{kk}}(a_{ik}-\sum_{m = 1}^{k - 1}l_{im}u_{mk})(i = k + 1,k + 2,\cdots,n)$$
|
||||||
|
|
||||||
|
>*示例*
|
||||||
|
>对于矩阵
|
||||||
|
$$A=\left[\begin{array}{ccc}2&1&1\\4&3&3\\8&7&9\end{array}\right]$$
|
||||||
|
>1. **第一步计算**
|
||||||
|
>首先$u_{11}=2$， $u_{12}=1$，$u_{13}=1$，$l_{21}=\frac{4}{2}=2$，$l_{31}=\frac{8}{2}=4$
|
||||||
|
>2. **第二步计算** 然后计算
|
||||||
|
> $u_{22}=a_{22}-l_{21}u_{12}=3 - 2×1 = 1$，
|
||||||
|
> $u_{23}=a_{23}-l_{21}u_{13}=3 - 2×1 = 1$
|
||||||
|
>4. **第三步计算**
|
||||||
|
$$l_{32}=\frac{1}{u_{22}}(a_{32}-l_{31}u_{12})=\frac{1}{1}(7 - 4×1)=3$$
|
||||||
|
>5. **第四步计算** 最后
|
||||||
|
$$u_{33}=a_{33}-l_{31}u_{13}-l_{32}u_{23}=9 - 4×1 - 3×1 = 2$$
|
||||||
|
>6. **得出结果** 得到
|
||||||
|
$$L=\left[\begin{array}{ccc}1&0&0\\2&1&0\\4&3&1\end{array}\right]$$
|
||||||
|
$$U=\left[\begin{array}{ccc}2&1&1\\0&1&1\\0&0&2\end{array}\right]$$
|
||||||
|
|
||||||
|
## 第三章
|
||||||
|
|
||||||
|
### 3.1 行列式介绍
|
||||||
|
* 人话版本：
|
||||||
|
>我的方法：
|
||||||
|
>1、选择一行零最多的，
|
||||||
|
>2、他的位置是第（$i$，$j$）,那就删去第$i$行，第$j$列，剩下的就是（余因子）
|
||||||
|
>3、这一行每个数都这样算$a_{ij} \times |C_{ij}| \times (-1)^{i+j}$，最后求和
|
||||||
|
|
||||||
|
>**定理 2**
|
||||||
|
>若 $A$ 为三角阵，则 det$A$ 等于 $A$ 的主对角线上元素的乘积
|
||||||
|
|
||||||
|
### 3.2 行列式的性质
|
||||||
|
>**定理3 (行变换)**
|
||||||
|
>令 $A$ 是一个方阵.
|
||||||
|
>a. 若 $A$ 的某一行的倍数加到另一行得矩阵B ， 则det $B$ = det $A$ .
|
||||||
|
>b 若 $A$ 的两行互换得矩阵 $B$ ， 则 det $B$ = - det $A$.
|
||||||
|
>c. 若 $A$ 的某行来以 $k$ 倍得到矩阵 $B$ ， 则det $B$ = $k$ det $A$ .
|
||||||
|
>** 补充
|
||||||
|
>$$\vert A^T\vert=\vert A\vert$$
|
||||||
|
>$$\vert A^{-1}\vert=\frac{1}{\vert A\vert}$$
|
||||||
|
>
|
||||||
|
>$$|A^{*}|=|A|^{n - 1}$$
|
||||||
|
>$$\vert kA\vert=k^{n}\vert A\vert$$
|
||||||
|
|
||||||
|
>**定理4**
|
||||||
|
> 方阵 $A$ 是可逆的当且仅当 det $A \neq 0$
|
||||||
|
|
||||||
|
>**定理5**
|
||||||
|
> 若 $A$ 为一个 $n \times n$ 矩阵，则det $A^T$ = det $A$.
|
||||||
|
|
||||||
|
>**定理6 (乘法的性质)**
|
||||||
|
若 $A$ 和 $B$ 均为 $n \times n$ 矩阵，则 det $AB$ = (det $A$)( det $B$) .
|
||||||
|
|
||||||
|
* 行列式与秩的关系
|
||||||
|
>$\text{det}(A)\neq0$那么矩阵$A$是满秩的，秩$\text{rank}(A) = n$。这是因为行列式不为零意味着矩阵的列（行）向量组是线性无关的
|
||||||
|
>也就是齐次线性方程组$Ax=0$的充要条件是系数矩阵秩$\text{rank}(A) = n$
|
||||||
|
|
||||||
|
* **$r(A) = n$** $\Leftrightarrow$ **$|A| \neq 0$** $\Leftrightarrow$ **齐次线性方程组 $Ax = 0$ 只有零解 $\Leftrightarrow$ 可逆**
|
||||||
|
|
||||||
|
### 3.3 克拉默法则
|
||||||
|
>**定理7 (克拉默法则)**
|
||||||
|
设 $A$ 是一个可逆的 $n \times n$ 矩阵，对 $R$<sup>m</sup> 中任意向量 $b$ ， 方程 $Ax =b$ 的唯一解可由下式给出:
|
||||||
|
$$\displaystyle x_i=\frac{det \ \ A_i(b)}{det \ \ A},i=1,2,\cdots,,n$$
|
||||||
|
|
||||||
|
~~不太能解释~~
|
||||||
|
## 第四章
|
||||||
|
|
||||||
|
|
||||||
|
### 4.1 向量空间(vector space)与子空间(subspace)
|
||||||
|
>向量空间和向量计算法则一样
|
||||||
|
|
||||||
|
* 子空间
|
||||||
|
>定义向量空间 $V$ 的一个子空间是 $V$ 的一个满足以下三个性质的子集 $H$:
|
||||||
|
a. $V$ 中的零向量在 $H$ 中
|
||||||
|
b. $H$ 对向量加法封闭，即对 $H$ 中任意向量 $U$，$V$ ， 和 $u + v$ 仍在 $H$ 中.
|
||||||
|
c. $H$ 对标量乘法封闭， 即对 $H$ 中任意向量 $u$ 和任意标量 $C$ ，向量 $cu$ 仍在 $H$ 中.
|
||||||
|
|
||||||
|
|
||||||
|
>**定理1** 若 $v_1,v_2,\cdots,v_p$ 在向量空间 $V$ 中，则$span\{x_1,x_2,\cdots,x_i\}$是 $V$ 的一个子空间.
|
||||||
|
|
||||||
|
### 4.2 零空间、列空间和线性变换
|
||||||
|
* 矩阵的零空间(null space)
|
||||||
|
|
||||||
|
>**定义**
|
||||||
|
矩阵 $A$ 的零空间写成 $NulA$ ， 是齐次方程 $Ax = 0$ 的全体解的集合.
|
||||||
|
|
||||||
|
>**定理2** $m \times n$ 矩阵 $A$ 的零空间是$R$<sup>m</sup>的一个子空间.等价地, $m$ 个方程、$n$ 个未知数的齐次线性方程组 $Ax = 0$ 的全体解的集合是$R$<sup>m</sup>的一个子空间
|
||||||
|
|
||||||
|
* 矩阵的列空间(column space)
|
||||||
|
>**定义**
|
||||||
|
$m \times n$矩阵 $A$ 的列空间(记为 $ColA$ ) 是由 $A$ 的列的所有线性组合组成的集合.若 $A=\begin{bmatrix}
|
||||||
|
\ x_1\ x_2\ \cdots \ x_3 \ \\
|
||||||
|
\end{bmatrix}$，则 $ColA = span\{x_1,x_2,\cdots,x_i\}$.
|
||||||
|
|
||||||
|
>**定理3** $m \times n$ 矩阵 $A$ 的列空间是 $R$<sup>m</sup> 的一个子空间.
|
||||||
|
|
||||||
|
* 线性变换的核与值域
|
||||||
|
>线性变换 见1.8
|
||||||
|
* 核(零空间 $Nul A$)
|
||||||
|
>线性变换 $T$ 的核(或零空间)是 $V$ 中所有满足 $T(u) = 0$ 的向量 $u$ 的集合
|
||||||
|
|
||||||
|
### 4.3 线性无关集(linearly independent set)和基(basis)
|
||||||
|
* 线性无关 见1.7
|
||||||
|
|
||||||
|
>**定理5 (生成集定理)**
|
||||||
|
令$S = \{v_1,v_2,\cdots,v_p\}$是$V$中的向量集，$H = span\{v_1,v_2,\cdots,v_p\}$.
|
||||||
|
a.若 $S$ 中某一个向量(比如说 $v_k$ ) 是 $S$ 中其余向量的线性组合，则 $S$ 中去掉$v_k$ 后形成的集合仍然可以生成 $H$.
|
||||||
|
b. 若$H \neq \{0\}$ ，则 $S$ 的某一子集是 $H$ 的一个基.
|
||||||
|
|
||||||
|
* NulA 和ColA 的基
|
||||||
|
>**定理6**
|
||||||
|
矩阵 $A$ 的主元列构成 $ColA$ 的一个基.
|
||||||
|
|
||||||
|
|
||||||
|
### 4.5 向量空间的维数(dimension)
|
||||||
|
>**定理9**
|
||||||
|
若向量空间 $V$ 具有一组基(n个基向量)， 则 $V$ 中任意包含多于 $n$ 个向量的集合一
|
||||||
|
定线性相关.
|
||||||
|
|
||||||
|
~~这是期中考证明题，没做出来~~
|
||||||
|
|
||||||
|
>**定理10** 若向量空间 $V$ 有一组基含有 $n$ 个向量，则 $V$ 的每一组基一定恰好含有 $n$ 个向量.
|
||||||
|
|
||||||
|
* $NulA$ 的维数是方程 $Ax=0$ 中自由变量的个数，$ColA$ 的维数是 $A$ 中主元列的个数.
|
||||||
|
|
||||||
|
### 4.6 秩(rank)
|
||||||
|
|
||||||
|
* $ColA^T = Row A$.
|
||||||
|
>**定理13** 若两个矩阵 $A$ 和 $B$ 行等价，则它们的行空间相同.若 $B$ 是阶梯形矩阵，则 $B$ 的非零行构成 $A$ 的行空间的一个基同时也是 $B$ 的行空间的一个基
|
||||||
|
|
||||||
|
~~?看不太懂~~
|
||||||
|
|
||||||
|
*以下比较重要*
|
||||||
|
|
||||||
|
>**定义**
|
||||||
|
$A$ 的秩即 $A$ 的列空间的维数
|
||||||
|
|
||||||
|
>**定理14 (秩定理)**
|
||||||
|
$m \times n$ 矩阵 $A$ 的列空间和行空间的维数相等，这个公共的维数(即 $A$ 的秩)还等于 $A$ 的主元位置的个数且,满足方程
|
||||||
|
$$rank\ A+dim\ \ Nul \ A = n$$
|
||||||
|
|
||||||
|
>**定理 (可逆矩阵定理(续))**
|
||||||
|
令 $A$ 是一个 $n \times n$ 矩阵，则下列命题中的每一个均等价于 $A$ 是可逆矩阵:
|
||||||
|
a. $A$ 的列构成$R$<sup>n</sup>的一个基.
|
||||||
|
b. $ColA=$$R$<sup>n</sup>.
|
||||||
|
c. $dim \ ColA = n$.
|
||||||
|
d. $rank A = n$.
|
||||||
|
e. $Nul A = \{0\}$.
|
||||||
|
f. $dim \ NulA=0$.
|
||||||
|
|
||||||
|
|
||||||
|
### 4.7 基的变换
|
||||||
|
~~先欠着~~
|
||||||
|
|
||||||
|
## 第五章
|
||||||
|
|
||||||
|
|
||||||
|
### 5.1 特征向量(eigenvector)与特征值(eigenvalue)
|
||||||
|
>定义 $A$ 为 $n \times n$ 矩阵，$x$ 为非零向量， 若存在数 $λ$ 使 $Ax=λx$ 有非平凡解 $x$， 则称 $λ$ 为 $A$的特征值,$x$ 称为对应于 $λ$ 的特征向量
|
||||||
|
也可写作$(A-λI)x=0$
|
||||||
|
|
||||||
|
>**定理1**
|
||||||
|
三角矩阵的主对角线的元素是其特征值.
|
||||||
|
|
||||||
|
>**定理2**
|
||||||
|
$λ_1,\cdots,λ_r$ 是 $n \times n$ 矩阵 $A$ 相异的特征值，$v_1,\cdots,v_r$是与$λ_1,\cdots,λ_r$对应的特征向量，那么向量集合{$v_1,\cdots,v_r$}线性无关.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
* 一、逆矩阵的特征值
|
||||||
|
若矩阵$A$可逆，$\lambda$是$A$的特征值，则$A^{-1}$的特征值是$\displaystyle \frac{1}{\lambda}$，特征向量不变。
|
||||||
|
|
||||||
|
* 二、转置矩阵的特征值
|
||||||
|
矩阵$A$与其转置矩阵$A^T$具有相同的特征值。
|
||||||
|
|
||||||
|
* 三、伴随矩阵的特征值
|
||||||
|
若$A$可逆，$A$的特征值为$\lambda_i$（$i = 1,2,\cdots,n$，$\lambda_i\neq0$），则伴随矩阵$A^*$的特征值为$\displaystyle \frac{\vert A\vert}{\lambda_i}$，特征向量不变。
|
||||||
|
|
||||||
|
### 5.2 特征方程(eigen equation)
|
||||||
|
>**定理(可逆矩阵定理(续))**
|
||||||
|
设 $A$ 是 $n \times n$ 矩阵，则 $A$ 是可逆的当且仅当
|
||||||
|
a.0不是 $A$ 的特征值.
|
||||||
|
b.$A$ 的行列式不等于零.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
>**定理3 (行列式的性质)**
|
||||||
|
设 $A$ 和 $B$ 是 $n \times n$ 矩阵.
|
||||||
|
a. $A$ 可逆的元要条件是 det$A \neq 0$.
|
||||||
|
b. det $AB =$ (det $A$) (det$B$).
|
||||||
|
c. det $A^T$ = det $A$.
|
||||||
|
d. 若 $A$ 是三角形矩阵，那么det $A$ 是 $A$ 主对角线元素的乘积.
|
||||||
|
e. 对 $A$ 作行替换不改变其行列式值.作一次行交换，行列式值符号改变一次数来一行后，
|
||||||
|
行列式值等于用此数来原来的行列式值.
|
||||||
|
|
||||||
|
>**定理4**
|
||||||
|
若 $n \times n$ 矩阵 $A$ 和 $B$ 是相似的，那么它们有相同的特征多项式，从而有相同的特征值(和相同的重数).
|
||||||
|
|
||||||
|
### 5.3 对角化(diagonalize)
|
||||||
|
>**定理5 (对角化定理)**
|
||||||
|
$n \times n$ 矩阵 $A$ 可对角化的充分必要条件是 $A$ 有 $n$ 个线性无关的特征向量.
|
||||||
|
事实上， $A=PDP^{-1}$ , $D$ 为对角矩阵的充分必要条件是 $P$ 的列向量是 $A$ 的 $n$ 个线性无关的特征向量.此时，$D$ 的主对角线上的元素分别是 $A$ 的对应于 $P$ 中特征向量的特征值.
|
||||||
|
|
||||||
|
>**定理6**
|
||||||
|
有 $n$ 个相异特征值的$n \times n$ 矩阵可对角化.
|
||||||
|
|
||||||
|
>**定理7**
|
||||||
|
~~似乎不重要，因为我也读不懂~~
|
||||||
|
|
||||||
|
>**定理8 (对角矩阵表示)**
|
||||||
|
设 $A=PDP^{-1}$ ， 其中 $D$ 为 $n \times n$ 对角矩阵，若 $R$<sup>n</sup> 的基$\beta$由 $P$ 的列向量组成，那么 $D$ 是变换 $x$ → $Ax$的$\beta$-矩阵.
|
||||||
|
|
||||||
|
## 第六章
|
||||||
|
|
||||||
|
### 6.1 内积、长度和正交性
|
||||||
|
|
||||||
|
* 内积
|
||||||
|
内积的英文是 “inner product” 或 “dot product”
|
||||||
|
> **定理1**
|
||||||
|
> 设 $v$，$u$ 和 $w$ 是 $R$<sup>n</sup> 中的向量, $c$ 是一个数,那么
|
||||||
|
|
||||||
|
|
||||||
|
$a. \ \ \ u \cdot v = v \cdot u$
|
||||||
|
|
||||||
|
$b.\ \ \ (u +v) \cdot w = u \cdot w +v \cdot w$
|
||||||
|
|
||||||
|
$c. \ \ \ (cu) \cdot v=c(u \cdot v)=u \cdot (cv)$
|
||||||
|
|
||||||
|
$d. \ \ \ u \cdot u \geq 0，并且u \cdot u=0 成立的充分必要条件是u=0$
|
||||||
|
|
||||||
|
|
||||||
|
* 向量的长度
|
||||||
|
|
||||||
|
$$||v|| ^2 = v \cdot v$$
|
||||||
|
|
||||||
|
$$dist(u,v)=||u-v||$$
|
||||||
|
|
||||||
|
* 正交向量
|
||||||
|
正交向量的英文是 “orthogonal vectors” 或 “perpendicular vectors”
|
||||||
|
|
||||||
|
>定义如果 $u \cdot v = 0$ ，如 $R$<sup>n</sup> 中的两个向量 $u$ 和 $v$ 是(相互) 正交的.
|
||||||
|
|
||||||
|
>对于一个方阵$A$，Col$A$中的向量与Nul$A$中的向量正交。
|
||||||
|
|
||||||
|
>**定理2 (毕达哥拉斯(勾股)定理)**
|
||||||
|
|
||||||
|
$$||u+v||^2=||u||^2+||v||^2$$
|
||||||
|
|
||||||
|
* 正交补
|
||||||
|
正交补的英文是 “orthogonal complement”
|
||||||
|
|
||||||
|
>1.向量 $x$ 属于 $W$<sup>⊥</sup> 的充分必要条件是向量 $x$ 与生成空间 $W$ 的任一向量都正交.
|
||||||
|
>2. $W$<sup>⊥</sup> 止是 $R$<sup>n</sup> 的一个子空间.
|
||||||
|
|
||||||
|
>**定理3**
|
||||||
|
$( Row A )$<sup>⊥</sup> = $Nul A$ 且 $( ColA )$<sup>⊥</sup> = $Nul A$<sup>T</sup>
|
||||||
|
|
||||||
|
### 6.2 正交集
|
||||||
|
* 正交集的英文是 “orthogonal set” 或 “orthonormal set”
|
||||||
|
>**定理4**
|
||||||
|
如果 $S=\{x_1,x_2,\cdots,x_i\}$ 是由 $R$<sup>n</sup> 中非零向量构成的正交集，那么 $S$ 是线性无关集，因此构成 $S$ 所生成的子空间的一组基.
|
||||||
|
|
||||||
|
>**定理5**
|
||||||
|
假设$\{x_1,x_2,\cdots,x_i\}$是 $R$<sup>n</sup> 中于空间 $W$ 的正文基,对 $W$ 中的每个向量y,线性组合 $y=x_1c_1+\cdots+x_ic_i$ 中的权可以由 $c_j=(y \cdot u_j)/(u_j \cdot u_j)$计算
|
||||||
|
|
||||||
|
#### 正交投影 **先欠着** ~~懒得写~~
|
||||||
|
|
||||||
|
|
||||||
|
>**定理6**
|
||||||
|
一个 $m \times n$ 矩阵 U 具有单位正交列向量的充分必要条件是 $U$<sup>T</sup> $U$ = $I$.
|
||||||
|
|
||||||
|
>**定理7**
|
||||||
|
假设 $U$ 是一个具有单位正交列的 $m \times n$ 矩阵，且 $x$ 和 $y$ 是 $R$<sup>n</sup> 中的向量，那么
|
||||||
|
a. $||Ux|| = ||x|| .$
|
||||||
|
b. $(Ux) \cdot (Uy) =x \cdot y$
|
||||||
|
c. $(Ux) \cdot (Uy) = 0$ 的充分必要条件是 $x \cdot y = 0$
|
||||||
|
|
||||||
|
>**定理9 (最佳逼近定理)**
|
||||||
|
假设 $W$ 是 $R$<sup>n</sup> 的一个子空间，$y$ 是 $R$<sup>n</sup> 中的任意向量， $\widehat{y}$ 是 $y$ 在 $W$ 上的正支投影，那么 $\widehat{y}$ 是 $W$ 中最接近 $y$ 的点，也就是
|
||||||
|
$$||y-\widehat{y}||<||y-v||$$
|
||||||
|
>对所有属于 $W$ 又异于 $\widehat{y}$ 的 $v$ 成立.
|
||||||
|
|
||||||
|
### 6.4 格拉姆-施密特方法
|
||||||
|
|
||||||
|
#### 格拉姆 - 施密特方法
|
||||||
|
|
||||||
|
设$\left\{\boldsymbol{v}_{1},\boldsymbol{v}_{2},\cdots,\boldsymbol{v}_{n}\right\}$是内积空间$V$中的一组线性无关向量。
|
||||||
|
首先$\boldsymbol{u}_{1}=\boldsymbol{v}_{1}$；对于$k = 2,3,\cdots,n$，
|
||||||
|
$$\boldsymbol{u}_{k}=\boldsymbol{v}_{k}-\sum_{j = 1}^{k - 1}\frac{\left\langle\boldsymbol{v}_{k},\boldsymbol{u}_{j}\right\rangle}{\left\langle\boldsymbol{u}_{j},\boldsymbol{u}_{j}\right\rangle}\boldsymbol{u}_{j}$$
|
||||||
|
即从$\boldsymbol{v}_{k}$中减去它在已构造正交向量$\boldsymbol{u}_{1},\boldsymbol{u}_{2},\cdots,\boldsymbol{u}_{k - 1}$上的投影，得到新正交向量$\boldsymbol{u}_{k}$。
|
||||||
|
|
||||||
|
|
||||||
|
### 6.5 最小二乘问题
|
||||||
|
* 最小二乘的英文是 “least squares” 或 “least square method”；
|
||||||
|
最小二乘解的英文是 “least squares solution”。
|
||||||
|
|
||||||
|
* **定义**
|
||||||
|
$$||b-A\widehat{x}||\leq||b-Ax||$$
|
||||||
|
|
||||||
|
>**定理13**
|
||||||
|
方程 $Ax=b$ 的最小二乘解集和法方程 $A$<sup>T</sup> $Ax = A$<sup>T</sup> $b$ 的非空解集一致.
|
||||||
|
|
||||||
|
>**定理14**
|
||||||
|
设 $A$ 是 $m \times n$ 矩阵. 下面的条件是逻辑等价的:
|
||||||
|
a.对于 $R$<sup>n</sup> 中的每个 $b$ ， 方程 $Ax =b$ 有唯一最小二乘解.
|
||||||
|
b.$A$ 的列是线性无关的.
|
||||||
|
c.矩阵 $A$<sup>T</sup> $A$是可逆的.
|
||||||
|
当这些条件成立时，最小二乘解￡有下面的表示:
|
||||||
|
$$\widehat{x}=( A^T A)^{-1}A^Tb$$
|
||||||
|
|
||||||
|
|
||||||
|
### 6.7 内积空间
|
||||||
|
|
||||||
|
>**定义**
|
||||||
|
向量空间 $V$ 上的内积是一个函数，对每一对属于$V$的向量 $u$ 和 $v$，存在一个实数$\langle u,v \rangle$满足下面公理，其中 $u$，$v$，$w$ 属于$V$，$C$ 为所有数.
|
||||||
|
1.$\langle u,v\rangle= \langle v,u \rangle$
|
||||||
|
2.$\langle u +v, w\rangle =\langle u, w\rangle +\langle v,w\rangle$
|
||||||
|
3.$\langle$c$u,v\rangle=$c$\langle u, v\rangle$
|
||||||
|
4.$\langle u,u\rangle \geq 0$且$\langle u,u\rangle =0$ 的充分必要条件是 $u=0$
|
||||||
|
一个赋予上面内积的向量空间称为**内积空间**
|
||||||
|
|
||||||
|
* 内积空间的英文是 “inner product space” 或 “pre-Hilbert space”
|
||||||
|
|
||||||
|
|
||||||
|
>**定理16 (柯西-施瓦茨不等式)**
|
||||||
|
对 $V$ 中任意向量 $u$ 和 $v$，有
|
||||||
|
$$|\langle u,v \rangle| \leq ||u||\ \ ||v||$$
|
||||||
|
|
||||||
|
|
||||||
|
>**定理17 (三角不等式)**
|
||||||
|
对属于$V$ 的所有向量$u$,$v$，有
|
||||||
|
$$||u-v||\leq||u||+||v||$$
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
## 第七章
|
||||||
|
|
||||||
|
### 7.1 对称矩阵的对角化
|
||||||
|
|
||||||
|
就是$A^T=A$
|
||||||
|
|
||||||
|
>**定理1** 如果 $A$ 是对称矩阵，那么不同特征空间的任意两个特征向量是正交的.
|
||||||
|
|
||||||
|
>**定理2** 一个$n \times n$ 矩阵 $A$ 可正交对角化的充分必要条件是 $A$ 是对称矩阵.
|
||||||
|
|
||||||
|
### 7.2 二次型
|
||||||
|
* 二次型是一个定义在 $R$<sup>n</sup> 上的函数， 它在向量 $x$ 处的值可由表达式$Q(x) = x^T Ax$ 计算，其中 $A$ 是一个 $n \times n$ 对称矩阵.矩阵 $A$ 称为关于二次型的矩阵.
|
||||||
|
### 7.4 SVD
|
||||||
|
SVD是奇异值分解（Singular Value Decomposition）的英文缩写。它是一种重要的矩阵分解方法。对于任意一个实矩阵$A_{m\times n}$（$m$行$n$列），都可以分解为
|
||||||
|
$$A = U\Sigma V^{T}$$
|
||||||
|
的形式。其中$U$是$m\times m$的正交矩阵，$V$是$n\times n$的正交矩阵，$\Sigma$是$m\times n$的对角矩阵，其对角线上的元素$\sigma_{ii}$（$i = 1,2,\cdots,\min(m,n)$）称为奇异值，并且$\sigma_{ii}\geq0$，这些奇异值按照从大到小的顺序排列在$\Sigma$的对角线上。
|
||||||
|
|
||||||
|
|
||||||
|
|||||||
File diff suppressed because it is too large
Load Diff
528
package-lock.json
generated
528
package-lock.json
generated
@@ -15,10 +15,11 @@
|
|||||||
"vue": "3.5.12"
|
"vue": "3.5.12"
|
||||||
},
|
},
|
||||||
"devDependencies": {
|
"devDependencies": {
|
||||||
|
"markdown-it-mathjax3": "^4.3.2",
|
||||||
"pagefind": "^1.3.0",
|
"pagefind": "^1.3.0",
|
||||||
"sass": "^1.76.0",
|
"sass": "^1.76.0",
|
||||||
"typescript": "^5.4.5",
|
"typescript": "^5.4.5",
|
||||||
"vitepress": "1.6.3"
|
"vitepress": "1.6.4"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
"node_modules/@algolia/abtesting": {
|
"node_modules/@algolia/abtesting": {
|
||||||
@@ -2307,6 +2308,16 @@
|
|||||||
}
|
}
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
|
"node_modules/@xmldom/xmldom": {
|
||||||
|
"version": "0.9.8",
|
||||||
|
"resolved": "https://registry.npmjs.org/@xmldom/xmldom/-/xmldom-0.9.8.tgz",
|
||||||
|
"integrity": "sha512-p96FSY54r+WJ50FIOsCOjyj/wavs8921hG5+kVMmZgKcvIKxMXHTrjNJvRgWa/zuX3B6t2lijLNFaOyuxUH+2A==",
|
||||||
|
"devOptional": true,
|
||||||
|
"license": "MIT",
|
||||||
|
"engines": {
|
||||||
|
"node": ">=14.6"
|
||||||
|
}
|
||||||
|
},
|
||||||
"node_modules/acorn": {
|
"node_modules/acorn": {
|
||||||
"version": "8.15.0",
|
"version": "8.15.0",
|
||||||
"resolved": "https://registry.npmjs.org/acorn/-/acorn-8.15.0.tgz",
|
"resolved": "https://registry.npmjs.org/acorn/-/acorn-8.15.0.tgz",
|
||||||
@@ -2344,6 +2355,16 @@
|
|||||||
"node": ">= 14.0.0"
|
"node": ">= 14.0.0"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
|
"node_modules/ansi-colors": {
|
||||||
|
"version": "4.1.3",
|
||||||
|
"resolved": "https://registry.npmjs.org/ansi-colors/-/ansi-colors-4.1.3.tgz",
|
||||||
|
"integrity": "sha512-/6w/C21Pm1A7aZitlI5Ni/2J6FFQN8i1Cvz3kHABAAbw93v/NlvKdVOqz7CCWz/3iv/JplRSEEZ83XION15ovw==",
|
||||||
|
"devOptional": true,
|
||||||
|
"license": "MIT",
|
||||||
|
"engines": {
|
||||||
|
"node": ">=6"
|
||||||
|
}
|
||||||
|
},
|
||||||
"node_modules/argparse": {
|
"node_modules/argparse": {
|
||||||
"version": "1.0.10",
|
"version": "1.0.10",
|
||||||
"resolved": "https://registry.npmjs.org/argparse/-/argparse-1.0.10.tgz",
|
"resolved": "https://registry.npmjs.org/argparse/-/argparse-1.0.10.tgz",
|
||||||
@@ -2368,6 +2389,13 @@
|
|||||||
"url": "https://github.com/sponsors/antfu"
|
"url": "https://github.com/sponsors/antfu"
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}
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}
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"resolved": "https://registry.npmjs.org/braces/-/braces-3.0.3.tgz",
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||||||
@@ -2421,6 +2449,45 @@
|
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"url": "https://github.com/sponsors/wooorm"
|
"url": "https://github.com/sponsors/wooorm"
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}
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}
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"devOptional": true,
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"license": "MIT",
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"cheerio-select": "^1.5.0",
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"htmlparser2": "^6.1.0",
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"parse5": "^6.0.1",
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"parse5-htmlparser2-tree-adapter": "^6.0.1",
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"funding": {
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"url": "https://github.com/cheeriojs/cheerio?sponsor=1"
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"devOptional": true,
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"css-what": "^6.0.1",
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"domelementtype": "^2.2.0",
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"domhandler": "^4.3.1",
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"domutils": "^2.8.0"
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"funding": {
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"url": "https://github.com/sponsors/fb55"
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@@ -2500,6 +2567,36 @@
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"node": ">= 8"
|
"node": ">= 8"
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}
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}
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"devOptional": true,
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"css-what": "^6.0.1",
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"domutils": "^2.8.0",
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"funding": {
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"url": "https://github.com/sponsors/fb55"
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"version": "6.2.2",
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"devOptional": true,
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"license": "BSD-2-Clause",
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"engines": {
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"node": ">= 6"
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},
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"funding": {
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"url": "https://github.com/sponsors/fb55"
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@@ -3094,12 +3191,81 @@
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"node": ">=0.3.1"
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"node": ">=0.3.1"
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"entities": "^2.0.0"
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"funding": {
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"url": "https://github.com/cheeriojs/dom-serializer?sponsor=1"
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"devOptional": true,
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"funding": {
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"url": "https://github.com/fb55/entities?sponsor=1"
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{
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"license": "BSD-2-Clause"
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"url": "https://github.com/fb55/domhandler?sponsor=1"
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"devOptional": true,
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"license": "BSD-2-Clause",
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"dependencies": {
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"domelementtype": "^2.2.0",
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"domhandler": "^4.2.0"
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"funding": {
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"url": "https://github.com/fb55/domutils?sponsor=1"
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@@ -3188,12 +3354,35 @@
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"@esbuild/win32-x64": "0.21.5"
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},
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"funding": {
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"url": "https://github.com/sponsors/sindresorhus"
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"devOptional": true,
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@@ -3279,15 +3468,6 @@
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"node": "^8.16.0 || ^10.6.0 || >=11.0.0"
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"node": "^8.16.0 || ^10.6.0 || >=11.0.0"
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"license": "Apache-2.0",
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"engines": {
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}
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@@ -3382,6 +3562,36 @@
|
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"url": "https://github.com/sponsors/wooorm"
|
"url": "https://github.com/sponsors/wooorm"
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"devOptional": true,
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"funding": [
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{
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"type": "github",
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"url": "https://github.com/sponsors/fb55"
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}
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],
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"license": "MIT",
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"dependencies": {
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"domelementtype": "^2.0.1",
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}
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"integrity": "sha512-p92if5Nz619I0w+akJrLZH0MX0Pb5DX39XOwQTtXSdQQOaYH03S1uIQp4mhOZtAXrxq4ViO67YTiLBo2638o9A==",
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"devOptional": true,
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"license": "BSD-2-Clause",
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"funding": {
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"url": "https://github.com/fb55/entities?sponsor=1"
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},
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"node_modules/iconv-lite": {
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@@ -3489,6 +3699,36 @@
|
|||||||
"js-yaml": "bin/js-yaml.js"
|
"js-yaml": "bin/js-yaml.js"
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"node_modules/juice": {
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@@ -3660,6 +3900,17 @@
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"markdown-it": "bin/markdown-it.js"
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}
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@@ -3672,6 +3923,19 @@
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"devOptional": true,
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"speech-rule-engine": "^4.0.6"
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@@ -3840,6 +4104,13 @@
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"devOptional": true,
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"node_modules/mermaid": {
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@@ -3868,6 +4139,13 @@
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"devOptional": true,
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"license": "Apache-2.0"
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@@ -4324,6 +4602,19 @@
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"node": ">=8.6"
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"devOptional": true,
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"mime": "cli.js"
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"engines": {
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@@ -4336,6 +4627,13 @@
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@@ -4405,6 +4703,27 @@
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@@ -4417,6 +4736,19 @@
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"url": "https://github.com/fb55/nth-check?sponsor=1"
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@@ -4456,6 +4788,23 @@
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@@ -4768,6 +5117,16 @@
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"devOptional": true,
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@@ -4796,6 +5155,31 @@
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|
||||||
@@ -4887,6 +5271,13 @@
|
|||||||
"node": ">=8.0"
|
"node": ">=8.0"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
|
"node_modules/tr46": {
|
||||||
|
"version": "0.0.3",
|
||||||
|
"resolved": "https://registry.npmjs.org/tr46/-/tr46-0.0.3.tgz",
|
||||||
|
"integrity": "sha512-N3WMsuqV66lT30CrXNbEjx4GEwlow3v6rr4mCcv6prnfwhS01rkgyFdjPNBYd9br7LpXV1+Emh01fHnq2Gdgrw==",
|
||||||
|
"devOptional": true,
|
||||||
|
"license": "MIT"
|
||||||
|
},
|
||||||
"node_modules/trim-lines": {
|
"node_modules/trim-lines": {
|
||||||
"version": "3.0.1",
|
"version": "3.0.1",
|
||||||
"resolved": "https://registry.npmjs.org/trim-lines/-/trim-lines-3.0.1.tgz",
|
"resolved": "https://registry.npmjs.org/trim-lines/-/trim-lines-3.0.1.tgz",
|
||||||
@@ -4906,6 +5297,13 @@
|
|||||||
"node": ">=6.10"
|
"node": ">=6.10"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
|
"node_modules/tslib": {
|
||||||
|
"version": "2.8.1",
|
||||||
|
"resolved": "https://registry.npmjs.org/tslib/-/tslib-2.8.1.tgz",
|
||||||
|
"integrity": "sha512-oJFu94HQb+KVduSUQL7wnpmqnfmLsOA/nAh6b6EH0wCEoK0/mPeXU6c3wKDV83MkOuHPRHtSXKKU99IBazS/2w==",
|
||||||
|
"devOptional": true,
|
||||||
|
"license": "0BSD"
|
||||||
|
},
|
||||||
"node_modules/typescript": {
|
"node_modules/typescript": {
|
||||||
"version": "5.9.2",
|
"version": "5.9.2",
|
||||||
"resolved": "https://registry.npmjs.org/typescript/-/typescript-5.9.2.tgz",
|
"resolved": "https://registry.npmjs.org/typescript/-/typescript-5.9.2.tgz",
|
||||||
@@ -5055,6 +5453,16 @@
|
|||||||
"node": ">=8"
|
"node": ">=8"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
|
"node_modules/valid-data-url": {
|
||||||
|
"version": "3.0.1",
|
||||||
|
"resolved": "https://registry.npmjs.org/valid-data-url/-/valid-data-url-3.0.1.tgz",
|
||||||
|
"integrity": "sha512-jOWVmzVceKlVVdwjNSenT4PbGghU0SBIizAev8ofZVgivk/TVHXSbNL8LP6M3spZvkR9/QolkyJavGSX5Cs0UA==",
|
||||||
|
"devOptional": true,
|
||||||
|
"license": "MIT",
|
||||||
|
"engines": {
|
||||||
|
"node": ">=10"
|
||||||
|
}
|
||||||
|
},
|
||||||
"node_modules/vfile": {
|
"node_modules/vfile": {
|
||||||
"version": "6.0.3",
|
"version": "6.0.3",
|
||||||
"resolved": "https://registry.npmjs.org/vfile/-/vfile-6.0.3.tgz",
|
"resolved": "https://registry.npmjs.org/vfile/-/vfile-6.0.3.tgz",
|
||||||
@@ -5168,9 +5576,9 @@
|
|||||||
}
|
}
|
||||||
},
|
},
|
||||||
"node_modules/vitepress": {
|
"node_modules/vitepress": {
|
||||||
"version": "1.6.3",
|
"version": "1.6.4",
|
||||||
"resolved": "https://registry.npmjs.org/vitepress/-/vitepress-1.6.3.tgz",
|
"resolved": "https://registry.npmjs.org/vitepress/-/vitepress-1.6.4.tgz",
|
||||||
"integrity": "sha512-fCkfdOk8yRZT8GD9BFqusW3+GggWYZ/rYncOfmgcDtP3ualNHCAg+Robxp2/6xfH1WwPHtGpPwv7mbA3qomtBw==",
|
"integrity": "sha512-+2ym1/+0VVrbhNyRoFFesVvBvHAVMZMK0rw60E3X/5349M1GuVdKeazuksqopEdvkKwKGs21Q729jX81/bkBJg==",
|
||||||
"license": "MIT",
|
"license": "MIT",
|
||||||
"dependencies": {
|
"dependencies": {
|
||||||
"@docsearch/css": "3.8.2",
|
"@docsearch/css": "3.8.2",
|
||||||
@@ -5524,6 +5932,15 @@
|
|||||||
"nanoid": "^4.0.0"
|
"nanoid": "^4.0.0"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
|
"node_modules/vue-command-palette/node_modules/fuse.js": {
|
||||||
|
"version": "6.6.2",
|
||||||
|
"resolved": "https://registry.npmjs.org/fuse.js/-/fuse.js-6.6.2.tgz",
|
||||||
|
"integrity": "sha512-cJaJkxCCxC8qIIcPBF9yGxY0W/tVZS3uEISDxhYIdtk8OL93pe+6Zj7LjCqVV4dzbqcriOZ+kQ/NE4RXZHsIGA==",
|
||||||
|
"license": "Apache-2.0",
|
||||||
|
"engines": {
|
||||||
|
"node": ">=10"
|
||||||
|
}
|
||||||
|
},
|
||||||
"node_modules/vue-command-palette/node_modules/nanoid": {
|
"node_modules/vue-command-palette/node_modules/nanoid": {
|
||||||
"version": "4.0.2",
|
"version": "4.0.2",
|
||||||
"resolved": "https://registry.npmjs.org/nanoid/-/nanoid-4.0.2.tgz",
|
"resolved": "https://registry.npmjs.org/nanoid/-/nanoid-4.0.2.tgz",
|
||||||
@@ -5548,12 +5965,90 @@
|
|||||||
"integrity": "sha512-L2RPSAwUFbgZH20etwrXyVyCBu9OxRSi8T/38QsvnkJyvq2LufW2lDCOzm7t/U9C1mkhJGWYfCuFBCmIuNivrg==",
|
"integrity": "sha512-L2RPSAwUFbgZH20etwrXyVyCBu9OxRSi8T/38QsvnkJyvq2LufW2lDCOzm7t/U9C1mkhJGWYfCuFBCmIuNivrg==",
|
||||||
"license": "MIT"
|
"license": "MIT"
|
||||||
},
|
},
|
||||||
|
"node_modules/web-resource-inliner": {
|
||||||
|
"version": "6.0.1",
|
||||||
|
"resolved": "https://registry.npmjs.org/web-resource-inliner/-/web-resource-inliner-6.0.1.tgz",
|
||||||
|
"integrity": "sha512-kfqDxt5dTB1JhqsCUQVFDj0rmY+4HLwGQIsLPbyrsN9y9WV/1oFDSx3BQ4GfCv9X+jVeQ7rouTqwK53rA/7t8A==",
|
||||||
|
"devOptional": true,
|
||||||
|
"license": "MIT",
|
||||||
|
"dependencies": {
|
||||||
|
"ansi-colors": "^4.1.1",
|
||||||
|
"escape-goat": "^3.0.0",
|
||||||
|
"htmlparser2": "^5.0.0",
|
||||||
|
"mime": "^2.4.6",
|
||||||
|
"node-fetch": "^2.6.0",
|
||||||
|
"valid-data-url": "^3.0.0"
|
||||||
|
},
|
||||||
|
"engines": {
|
||||||
|
"node": ">=10.0.0"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"node_modules/web-resource-inliner/node_modules/domhandler": {
|
||||||
|
"version": "3.3.0",
|
||||||
|
"resolved": "https://registry.npmjs.org/domhandler/-/domhandler-3.3.0.tgz",
|
||||||
|
"integrity": "sha512-J1C5rIANUbuYK+FuFL98650rihynUOEzRLxW+90bKZRWB6A1X1Tf82GxR1qAWLyfNPRvjqfip3Q5tdYlmAa9lA==",
|
||||||
|
"devOptional": true,
|
||||||
|
"license": "BSD-2-Clause",
|
||||||
|
"dependencies": {
|
||||||
|
"domelementtype": "^2.0.1"
|
||||||
|
},
|
||||||
|
"engines": {
|
||||||
|
"node": ">= 4"
|
||||||
|
},
|
||||||
|
"funding": {
|
||||||
|
"url": "https://github.com/fb55/domhandler?sponsor=1"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"node_modules/web-resource-inliner/node_modules/entities": {
|
||||||
|
"version": "2.2.0",
|
||||||
|
"resolved": "https://registry.npmjs.org/entities/-/entities-2.2.0.tgz",
|
||||||
|
"integrity": "sha512-p92if5Nz619I0w+akJrLZH0MX0Pb5DX39XOwQTtXSdQQOaYH03S1uIQp4mhOZtAXrxq4ViO67YTiLBo2638o9A==",
|
||||||
|
"devOptional": true,
|
||||||
|
"license": "BSD-2-Clause",
|
||||||
|
"funding": {
|
||||||
|
"url": "https://github.com/fb55/entities?sponsor=1"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"node_modules/web-resource-inliner/node_modules/htmlparser2": {
|
||||||
|
"version": "5.0.1",
|
||||||
|
"resolved": "https://registry.npmjs.org/htmlparser2/-/htmlparser2-5.0.1.tgz",
|
||||||
|
"integrity": "sha512-vKZZra6CSe9qsJzh0BjBGXo8dvzNsq/oGvsjfRdOrrryfeD9UOBEEQdeoqCRmKZchF5h2zOBMQ6YuQ0uRUmdbQ==",
|
||||||
|
"devOptional": true,
|
||||||
|
"license": "MIT",
|
||||||
|
"dependencies": {
|
||||||
|
"domelementtype": "^2.0.1",
|
||||||
|
"domhandler": "^3.3.0",
|
||||||
|
"domutils": "^2.4.2",
|
||||||
|
"entities": "^2.0.0"
|
||||||
|
},
|
||||||
|
"funding": {
|
||||||
|
"url": "https://github.com/fb55/htmlparser2?sponsor=1"
|
||||||
|
}
|
||||||
|
},
|
||||||
"node_modules/web-worker": {
|
"node_modules/web-worker": {
|
||||||
"version": "1.5.0",
|
"version": "1.5.0",
|
||||||
"resolved": "https://registry.npmjs.org/web-worker/-/web-worker-1.5.0.tgz",
|
"resolved": "https://registry.npmjs.org/web-worker/-/web-worker-1.5.0.tgz",
|
||||||
"integrity": "sha512-RiMReJrTAiA+mBjGONMnjVDP2u3p9R1vkcGz6gDIrOMT3oGuYwX2WRMYI9ipkphSuE5XKEhydbhNEJh4NY9mlw==",
|
"integrity": "sha512-RiMReJrTAiA+mBjGONMnjVDP2u3p9R1vkcGz6gDIrOMT3oGuYwX2WRMYI9ipkphSuE5XKEhydbhNEJh4NY9mlw==",
|
||||||
"license": "Apache-2.0"
|
"license": "Apache-2.0"
|
||||||
},
|
},
|
||||||
|
"node_modules/webidl-conversions": {
|
||||||
|
"version": "3.0.1",
|
||||||
|
"resolved": "https://registry.npmjs.org/webidl-conversions/-/webidl-conversions-3.0.1.tgz",
|
||||||
|
"integrity": "sha512-2JAn3z8AR6rjK8Sm8orRC0h/bcl/DqL7tRPdGZ4I1CjdF+EaMLmYxBHyXuKL849eucPFhvBoxMsflfOb8kxaeQ==",
|
||||||
|
"devOptional": true,
|
||||||
|
"license": "BSD-2-Clause"
|
||||||
|
},
|
||||||
|
"node_modules/whatwg-url": {
|
||||||
|
"version": "5.0.0",
|
||||||
|
"resolved": "https://registry.npmjs.org/whatwg-url/-/whatwg-url-5.0.0.tgz",
|
||||||
|
"integrity": "sha512-saE57nupxk6v3HY35+jzBwYa0rKSy0XR8JSxZPwgLr7ys0IBzhGviA1/TUGJLmSVqs8pb9AnvICXEuOHLprYTw==",
|
||||||
|
"devOptional": true,
|
||||||
|
"license": "MIT",
|
||||||
|
"dependencies": {
|
||||||
|
"tr46": "~0.0.3",
|
||||||
|
"webidl-conversions": "^3.0.0"
|
||||||
|
}
|
||||||
|
},
|
||||||
"node_modules/which": {
|
"node_modules/which": {
|
||||||
"version": "2.0.2",
|
"version": "2.0.2",
|
||||||
"resolved": "https://registry.npmjs.org/which/-/which-2.0.2.tgz",
|
"resolved": "https://registry.npmjs.org/which/-/which-2.0.2.tgz",
|
||||||
@@ -5569,6 +6064,13 @@
|
|||||||
"node": ">= 8"
|
"node": ">= 8"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
|
"node_modules/wicked-good-xpath": {
|
||||||
|
"version": "1.3.0",
|
||||||
|
"resolved": "https://registry.npmjs.org/wicked-good-xpath/-/wicked-good-xpath-1.3.0.tgz",
|
||||||
|
"integrity": "sha512-Gd9+TUn5nXdwj/hFsPVx5cuHHiF5Bwuc30jZ4+ronF1qHK5O7HD0sgmXWSEgwKquT3ClLoKPVbO6qGwVwLzvAw==",
|
||||||
|
"devOptional": true,
|
||||||
|
"license": "MIT"
|
||||||
|
},
|
||||||
"node_modules/xml-js": {
|
"node_modules/xml-js": {
|
||||||
"version": "1.6.11",
|
"version": "1.6.11",
|
||||||
"resolved": "https://registry.npmjs.org/xml-js/-/xml-js-1.6.11.tgz",
|
"resolved": "https://registry.npmjs.org/xml-js/-/xml-js-1.6.11.tgz",
|
||||||
|
|||||||
@@ -19,6 +19,7 @@
|
|||||||
"doc": "docs"
|
"doc": "docs"
|
||||||
},
|
},
|
||||||
"devDependencies": {
|
"devDependencies": {
|
||||||
|
"markdown-it-mathjax3": "^4.3.2",
|
||||||
"pagefind": "^1.3.0",
|
"pagefind": "^1.3.0",
|
||||||
"sass": "^1.76.0",
|
"sass": "^1.76.0",
|
||||||
"typescript": "^5.4.5",
|
"typescript": "^5.4.5",
|
||||||
|
|||||||
Reference in New Issue
Block a user