{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "73bd968b-d970-4a05-94ef-4e7abf990827",
   "metadata": {},
   "source": [
    "Chapter 02\n",
    "\n",
    "# 向量加减法\n",
    "Book_4《矩阵力量》 | 鸢尾花书：从加减乘除到机器学习 (第二版)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5727cda1-e9bd-4483-b50f-9b9ef36574ae",
   "metadata": {},
   "source": [
    "这段代码定义了两个二维列向量 $a$ 和 $b$，并计算了它们的加法与减法。具体来说，向量 $a$ 定义为：\n",
    "\n",
    "$$\n",
    "a = \\begin{bmatrix} -2 \\\\ 5 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "向量 $b$ 定义为：\n",
    "\n",
    "$$\n",
    "b = \\begin{bmatrix} 5 \\\\ -1 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "代码首先计算了 $a$ 和 $b$ 的向量加法，得到结果向量 $a + b$，其计算过程为：\n",
    "\n",
    "$$\n",
    "a + b = \\begin{bmatrix} -2 \\\\ 5 \\end{bmatrix} + \\begin{bmatrix} 5 \\\\ -1 \\end{bmatrix} = \\begin{bmatrix} 3 \\\\ 4 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "此外，代码还展示了向量减法 $a - b$ 和 $b - a$，分别为：\n",
    "\n",
    "$$\n",
    "a - b = \\begin{bmatrix} -2 \\\\ 5 \\end{bmatrix} - \\begin{bmatrix} 5 \\\\ -1 \\end{bmatrix} = \\begin{bmatrix} -7 \\\\ 6 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "$$\n",
    "b - a = \\begin{bmatrix} 5 \\\\ -1 \\end{bmatrix} - \\begin{bmatrix} -2 \\\\ 5 \\end{bmatrix} = \\begin{bmatrix} 7 \\\\ -6 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "代码中分别使用了直接运算（如 `a + b`）和 `NumPy` 函数 `np.add` 与 `np.subtract` 来完成加法和减法操作。这种处理方式展示了向量的基本线性代数操作。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "88825576-ff58-4b2e-8a6b-770f593c57bb",
   "metadata": {},
   "source": [
    "## 导入所需库"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "61e4349a-f354-4e33-a9a0-0b43ab26eb3b",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np  # 导入NumPy库，用于数值计算"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ea419dfa-5bf7-4c35-b465-8ff0416ce120",
   "metadata": {},
   "source": [
    "## 定义两个列向量"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "ee8dadac-372a-43a9-8a21-f2e54d27b4c6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-2],\n",
       "       [ 5]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.array([[-2], [5]])  # 定义向量a，值为[-2, 5]\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "1749cae5-c1ac-491b-ba00-0e93066130ac",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 5],\n",
       "       [-1]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = np.array([[5], [-1]])  # 定义向量b，值为[5, -1]\n",
    "b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68440eab-d48e-450b-ad71-43adcc4c30b3",
   "metadata": {},
   "source": [
    "## 计算向量加法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "7c6885fd-59eb-457b-967f-fbeebee26a96",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3],\n",
       "       [4]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_plus_b = a + b  # 使用直接加法计算a和b的和\n",
    "a_plus_b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "2bbd2e21-91c4-43b5-b18d-597011c73214",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3],\n",
       "       [4]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_plus_b_2 = np.add(a, b)  # 使用np.add函数计算a和b的和\n",
    "a_plus_b_2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "54f2167f-edeb-48f4-81e4-89e27e459fd8",
   "metadata": {},
   "source": [
    "## 计算向量减法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "f6e97112-bfc4-44e9-9a9d-d168d4ba030e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-7],\n",
       "       [ 6]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_minus_b = a - b  # 计算a减去b的差\n",
    "a_minus_b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "a1b483cd-6bfc-450c-a883-d86a89ea897b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-7],\n",
       "       [ 6]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_minus_b_2 = np.subtract(a, b)  # 使用np.subtract函数计算a减去b的差\n",
    "a_minus_b_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "b1b4beab-4cab-48b1-b90d-065ee5971e9e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 7],\n",
       "       [-6]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b_minus_a = b - a  # 计算b减去a的差\n",
    "b_minus_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "2a010763-e0f1-457f-acf9-e8e0182d271d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 7],\n",
       "       [-6]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b_minus_a_2 = np.subtract(b, a)  # 使用np.subtract函数计算b减去a的差\n",
    "b_minus_a_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "85a80909-2aac-49ed-bb7a-f8cc6b80ee7d",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ecd322f4-f919-4be2-adc3-69d28ef25e69",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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