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‎LA_00_正文前_V2.pdf

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‎LA_01_01_什么是向量.pdf

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‎LA_01_02_坐标系.pdf

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‎LA_01_03_向量加减法.pdf

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‎LA_01_04_向量标量乘法.pdf

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‎LA_01_05_01.ipynb

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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "f04dd603-a2d1-48ce-8c17-9f1dba8de1ee",
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"metadata": {},
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"source": [
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"Chapter 01\n",
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"\n",
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"# 单位向量\n",
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"《线性代数》 | 鸢尾花书：数学不难"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ccafb456-2453-4c82-8a65-b1963a370cb2",
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"metadata": {},
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"source": [
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"这段代码的核心目标是计算向量 $a$ 的 **L2 范数（即欧几里得范数）**，然后基于此计算单位向量，并最终验证单位向量与原始长度的关系。数学上，该过程涉及向量的归一化和数乘运算，具体分析如下：\n",
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"\n",
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"### 1. 计算向量 $a$ 的 L2 范数\n",
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"\n",
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"首先，给定二维向量：\n",
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"\n",
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"$$\n",
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"a = \\begin{bmatrix} 3 \\\\ 4 \\end{bmatrix}\n",
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"$$\n",
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"\n",
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"L2 范数（或欧几里得范数）定义如下：\n",
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"\n",
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"$$\n",
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"\\|a\\|_2 = \\sqrt{3^2 + 4^2} = \\sqrt{9 + 16} = \\sqrt{25} = 5\n",
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"$$\n",
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"\n",
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"代码中的 `np.linalg.norm(a)` 计算了该范数，并将其存储在变量 `norm_a` 中。\n",
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"\n",
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"### 2. 计算单位向量\n",
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"\n",
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"单位向量的定义是 **原始向量除以其范数**，即：\n",
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"\n",
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"$$\n",
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"\\hat{a} = \\frac{a}{\\|a\\|_2} = \\frac{1}{5} \\begin{bmatrix} 3 \\\\ 4 \\end{bmatrix} = \\begin{bmatrix} 0.6 \\\\ 0.8 \\end{bmatrix}\n",
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"$$\n",
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"\n",
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"代码中 `unit_a = a / norm_a` 实现了这一计算，得到向量 $\\hat{a}$，它的特点是方向与 $a$ 相同，但长度为 1：\n",
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"\n",
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"$$\n",
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"\\|\\hat{a}\\|_2 = \\sqrt{0.6^2 + 0.8^2} = \\sqrt{0.36 + 0.64} = \\sqrt{1} = 1\n",
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"$$\n",
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"\n",
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"### 3. 长度 $\\times$ 方向向量\n",
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"\n",
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"在向量运算中，如果将单位向量 $\\hat{a}$ 乘以原始向量的范数 $\\|a\\|_2$，理论上应该得到原向量：\n",
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"\n",
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"$$\n",
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"\\|a\\|_2 \\cdot \\hat{a} = 5 \\cdot \\begin{bmatrix} 0.6 \\\\ 0.8 \\end{bmatrix} = \\begin{bmatrix} 3 \\\\ 4 \\end{bmatrix} = a\n",
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"$$\n",
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"\n",
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"代码中 `norm_a * unit_a` 计算了这个结果，并验证了单位向量的正确性。\n",
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"\n",
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"### 结论\n",
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"\n",
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"1. `np.linalg.norm(a)` 计算向量 $a$ 的模长，结果为 $5$。\n",
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"2. `a / norm_a` 计算单位向量 $\\hat{a}$，即 **保持方向不变但长度为 1**。\n",
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"3. `norm_a * unit_a` 验证单位向量乘以原范数是否能复原向量 $a$，结果证明计算正确。\n",
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"\n",
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"这段代码展示了 **向量归一化** 的基本数学原理，在机器学习、计算机视觉和物理建模等领域都有重要应用，例如方向向量的计算、余弦相似度、梯度归一化等。"
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]
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},
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{
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"cell_type": "markdown",
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"id": "9e5f832e-6b83-4973-88a7-52f63fe92b0e",
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"metadata": {},
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"source": [
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"## 初始化"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "2889d4f3-5ebd-46aa-885b-99bbf6e744aa",
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np"
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]
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},
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{
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"cell_type": "markdown",
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"id": "fd7c49d7-03ab-4b61-9326-535ba2eae184",
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"metadata": {},
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"source": [
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"## 定义向量 a"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "b20fbb44-18d3-43f5-ae82-a44b37cf266b",
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"metadata": {},
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"outputs": [],
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"source": [
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"a = np.array([3, 4])\n",
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"a"
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]
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},
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{
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"cell_type": "markdown",
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"id": "9633ae00-50e4-4d75-8cd4-7c20150ca580",
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"metadata": {},
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"source": [
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"## 计算向量的长度"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"id": "78a0127a-255d-43b6-8931-68cb32bfaf76",
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"metadata": {},
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"outputs": [],
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"source": [
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"norm_a = np.linalg.norm(a)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "535e230f-6131-41d1-867e-2ade3ee63eb4",
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"metadata": {},
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"source": [
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"## 计算单位向量"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"id": "6bc32359-2ea2-4312-a760-89c370e868f0",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([0.6, 0.8])"
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]
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},
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"execution_count": 10,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"unit_a = a / norm_a\n",
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"unit_a"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ce0a2a1d-8fe1-40f0-9ef0-716fdced62fd",
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"metadata": {},
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"source": [
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"## 长度 $\\times$ 方向向量"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"id": "954b7525-dcef-47cb-af13-40c741ca3891",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([3., 4.])"
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]
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},
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"execution_count": 12,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"norm_a * unit_a"
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]
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},
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{
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"cell_type": "markdown",
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"id": "f48c7a73-872b-403c-9d99-86e7ef6635bd",
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"metadata": {},
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"source": [
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"作者\t**生姜DrGinger** \n",
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"脚本\t**生姜DrGinger** \n",
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"视频\t**崔崔CuiCui** \n",
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"开源资源\t[**GitHub**](https://github.com/Visualize-ML) \n",
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"平台\t[**油管**](https://www.youtube.com/@DrGinger_Jiang)\t\t\n",
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"\t\t[**iris小课堂**](https://space.bilibili.com/3546865719052873)\t\t\n",
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"\t\t[**生姜DrGinger**](https://space.bilibili.com/513194466) "
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python [conda env:base] *",
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"language": "python",
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"name": "conda-base-py"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.12.7"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}

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