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‎LA_01_02_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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"# 使用numpy.meshgrid() 创建二维网格数组\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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"这段代码的数学核心是创建一个二维网格，其中 $x_1$ 和 $x_2$ 分别表示平面上的两个变量。我们可以从数学角度详细描述其作用如下：\n",
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"\n",
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"首先，$x_1$ 和 $x_2$ 是从 $-4$ 到 $4$ 之间等间距取值的数列，步长为 $1$：\n",
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"\\[\n",
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"x_1 = x_2 = \\{-4, -3, -2, -1, 0, 1, 2, 3, 4\\}\n",
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"\\]\n",
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"这两个数列定义了一系列离散的坐标点，它们将用于构造一个网格。\n",
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"\n",
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"**网格生成（Meshgrid）**\n",
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"\n",
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"使用 $\\text{np.meshgrid}(x_1, x_2)$，代码生成了两个矩阵 $xx_1$ 和 $xx_2$，它们的形状均为 $(9, 9)$。这些矩阵的作用是形成一个二维坐标网格，其中：\n",
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"- $xx_1$ 的每一行都与 $x_1$ 相同，表示 $x_1$ 方向上的坐标重复扩展。\n",
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"- $xx_2$ 的每一列都与 $x_2$ 相同，表示 $x_2$ 方向上的坐标重复扩展。\n",
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"\n",
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"用数学表示，网格上的每个点 $(x_1, x_2)$ 由：\n",
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"\\[\n",
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"xx_1(i, j) = x_1(j), \\quad xx_2(i, j) = x_2(i)\n",
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"\\]\n",
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"其中 $i, j$ 分别是矩阵的行索引和列索引。\n",
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"\n",
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"最终，$xx_1$ 和 $xx_2$ 可以被看作是二维平面上 $x_1$ 和 $x_2$ 轴上的坐标网格，它们常用于绘制等高线图、三维曲面图或者计算某些二维函数值。\n",
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"\n",
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"**示例网格点**\n",
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"例如，$(x_1, x_2)$ 的部分取值：\n",
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"\\[\n",
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"\\begin{aligned}\n",
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"xx_1 &= \n",
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"\\begin{bmatrix}\n",
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"-4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\\\\n",
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"-4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\\\\n",
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"\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n",
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"-4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4\n",
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"\\end{bmatrix} \\\\\n",
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"xx_2 &= \n",
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"\\begin{bmatrix}\n",
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"-4 & -4 & -4 & -4 & -4 & -4 & -4 & -4 & -4 \\\\\n",
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"-3 & -3 & -3 & -3 & -3 & -3 & -3 & -3 & -3 \\\\\n",
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"\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n",
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"4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4\n",
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"\\end{bmatrix}\n",
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"\\end{aligned}\n",
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"\\]\n",
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"这些矩阵的每个元素对应于二维平面上的一个坐标点 $(x_1, x_2)$，用于后续的数学计算或可视化。\n",
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"\n",
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"总结来说，这段代码的作用是构建一个二维网格，它定义了在 $[-4,4] \\times [-4,4]$ 这个离散点集上的坐标点分布，为进一步计算函数值或绘图提供基础。"
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]
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},
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{
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"cell_type": "markdown",
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"id": "398d0a9c-4884-4750-a4eb-f234428eb21f",
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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": 1,
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"id": "592d9f86-794b-4b36-89c3-099726910e65",
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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": "e3b97692-7fe2-4f0d-badf-48d46849d7d6",
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"metadata": {},
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"source": [
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"## 一维数组"
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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": 2,
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"id": "ba06a502-610c-49bf-9200-02bb7c7736f1",
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"metadata": {},
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"outputs": [],
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"source": [
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"# 生成 x1 和 x2 的一维数组，范围 [-4, 4]，步长为 1\n",
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"x1 = np.arange(-4, 5, 1)\n",
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"x2 = np.arange(-4, 5, 1)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "63975350-6c23-4a8e-981a-a8bd775c0710",
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"metadata": {},
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"source": [
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"## 二维网格数据"
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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": 3,
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"id": "e87412bc-0d80-43d5-8998-133654a75faf",
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"metadata": {},
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"outputs": [],
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"source": [
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"# 生成网格\n",
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"xx1, xx2 = np.meshgrid(x1, x2)"
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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": "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([[-4, -3, -2, -1, 0, 1, 2, 3, 4],\n",
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" [-4, -3, -2, -1, 0, 1, 2, 3, 4],\n",
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" [-4, -3, -2, -1, 0, 1, 2, 3, 4],\n",
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" [-4, -3, -2, -1, 0, 1, 2, 3, 4],\n",
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" [-4, -3, -2, -1, 0, 1, 2, 3, 4],\n",
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" [-4, -3, -2, -1, 0, 1, 2, 3, 4],\n",
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" [-4, -3, -2, -1, 0, 1, 2, 3, 4],\n",
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" [-4, -3, -2, -1, 0, 1, 2, 3, 4],\n",
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" [-4, -3, -2, -1, 0, 1, 2, 3, 4]])"
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]
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},
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"execution_count": 4,
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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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"xx1"
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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": 5,
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"id": "9a34e318-63f0-416c-b364-c35b1ed64a71",
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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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"(9, 9)"
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]
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},
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"execution_count": 5,
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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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"xx1.shape"
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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": "eed8c293-8ede-4c9b-a715-049f9f24ff43",
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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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"2"
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]
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},
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"execution_count": 6,
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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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"xx1.ndim"
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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": 7,
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"id": "7eb025ba-bf5e-424b-8332-4d7682dd8f52",
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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([[-4, -4, -4, -4, -4, -4, -4, -4, -4],\n",
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" [-3, -3, -3, -3, -3, -3, -3, -3, -3],\n",
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" [-2, -2, -2, -2, -2, -2, -2, -2, -2],\n",
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" [-1, -1, -1, -1, -1, -1, -1, -1, -1],\n",
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" [ 0, 0, 0, 0, 0, 0, 0, 0, 0],\n",
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" [ 1, 1, 1, 1, 1, 1, 1, 1, 1],\n",
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" [ 2, 2, 2, 2, 2, 2, 2, 2, 2],\n",
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" [ 3, 3, 3, 3, 3, 3, 3, 3, 3],\n",
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" [ 4, 4, 4, 4, 4, 4, 4, 4, 4]])"
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]
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},
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"execution_count": 7,
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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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"xx2"
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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": "69d33c20-60a6-4217-998a-2b5dff3e4f41",
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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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"(9, 9)"
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]
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},
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"execution_count": 8,
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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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"xx2.shape"
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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": 9,
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"id": "3857e1c9-ae1b-410f-9bcf-e925b3432eb3",
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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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"2"
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]
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},
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"execution_count": 9,
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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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"xx2.ndim"
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]
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},
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{
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"cell_type": "markdown",
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"id": "3fbd2e7e-62b5-4f02-bba1-93c0081a2318",
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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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