{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "第2章 向量空间与矩阵"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2.1 向量与矩阵"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$n$维列向量：含有$n$个数的数组，记为  \n",
    "$$\\mathbf{a}=\\begin{bmatrix} a_1 \\\\ a_2 \\\\ \\vdots \\\\ a_n \\end{bmatrix}$$  \n",
    "$a_i$表示向量$a$的第$i$个元素。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "定义$\\mathbb{R}$为全体实数组成的集合，则由实数组成的$n$维列向量可表示为$\\mathbb{R}^n$，称为$n$维实数向量空间。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$n$维行向量记为  \n",
    "$$\\mathbf{a}=\\left[a_1, a_2, \\dots, a_n \\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "向量$\\mathbf{a}$的转置记为$\\mathbf{a}^\\top$。如果\n",
    "$$\\mathbf{a}=\\begin{bmatrix} a_1 \\\\ a_2 \\\\ \\vdots \\\\ a_n \\end{bmatrix}$$    \n",
    "则\n",
    "$$\\mathbf{a}^\\top=\\left[a_1, a_2, \\dots, a_n \\right]$$  \n",
    "相应的，可记为  \n",
    "$$\\mathbf{a}=\\left[a_1, a_2, \\dots, a_n \\right]^\\top$$  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "给定向量$\\mathbf{a}=\\left[a_1, a_2, \\dots, a_n \\right]^\\top$和向量$\\mathbf{b}=\\left[b_1, b_2, \\dots, b_n \\right]^\\top$，如果$a_i=b_i,i=1,2,\\dots,n$，则两个向量相等。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "向量加法运算：\n",
    "$$\\mathbf{a}+\\mathbf{b}=\\left[a_1+b_1, a_2+b_2, \\dots, a_n+b_n \\right]^\\top$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "向量加法运算的性质：\n",
    "1. 交换性 $\\mathbf{a}+\\mathbf{b}=\\mathbf{b}+\\mathbf{a}$\n",
    "2. 结合性 $\\left(\\mathbf{a}+\\mathbf{b}\\right)+\\mathbf{c}=\\mathbf{a}+\\left(\\mathbf{b}+\\mathbf{c}\\right)$\n",
    "3. 存在零向量  \n",
    "$$\\mathbf{0}=\\left[0,0,\\dots,0\\right]^\\top$$  \n",
    "使得\n",
    "$$\\mathbf{a}+\\mathbf{0}=\\mathbf{0}+\\mathbf{a}=\\mathbf{a}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "向量减法运算：\n",
    "$$\\mathbf{a}-\\mathbf{b}=\\left[a_1-b_1, a_2-b_2, \\dots, a_n-b_n \\right]^\\top$$\n",
    "$$\\mathbf{0}-\\mathbf{b}=-\\mathbf{b}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "向量减法运算性质：\n",
    "$$\\mathbf{b}+\\{\\mathbf{a}-\\mathbf{b}\\}=\\mathbf{a} \\\\\n",
    "-\\left(-\\mathbf{b}\\right)=\\mathbf{b} \\\\\n",
    "-\\left(\\mathbf{a}-\\mathbf{b}\\right)=\\mathbf{b}-\\mathbf{a}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "设$\\mathbf{x}=\\left[x_1,x_2,\\dots,x_n\\right]^\\top$是$\\mathbf{a}+\\mathbf{x}=\\mathbf{b}$的解，有  \n",
    "$$a_1+x_1=b_1 \\\\a_2+x_2=b_2 \\\\ \\vdots \\\\ a_n+x_n=b_n$$  \n",
    "则$$\\mathbf{x}=\\mathbf{b}-\\mathbf{a}$$\n",
    "即向量$\\mathbf{b}-\\mathbf{a}$是向量方程$\\mathbf{a}+\\mathbf{x}=\\mathbf{b}$的唯一解。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "标量向量乘法运算:\n",
    "$$\\alpha\\mathbf{a}=\\left[\\alpha a_1, \\alpha a_2, \\dots, \\alpha a_n \\right], \\quad \\alpha \\in \\mathbb{R}, \\mathbf{a} \\in \\mathbb{R}^n$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "标量向量乘法运算性质：\n",
    "1. 分配律　$$\\alpha\\left(\\mathbf{a}+\\mathbf{b}\\right)=\\alpha\\mathbf{a}+\\alpha\\mathbf{b} \\\\  \n",
    "\\left(\\alpha+\\beta\\right)\\mathbf{a}=\\alpha\\mathbf{a}+\\beta\\mathbf{a}$$  \n",
    "2. 结合性　$\\alpha\\left(\\beta\\mathbf{a}\\right)=\\left(\\alpha\\beta\\right)\\mathbf{a}$\n",
    "3. 标量１满足　$1\\mathbf{a}=\\mathbf{a}$\n",
    "4. 任意标量$\\alpha$满足　$\\alpha\\mathbf{0}=\\mathbf{0}$\n",
    "5. 标量０满足　$0\\mathbf{a}=\\mathbf{0}$\n",
    "6. 标量-1满足　$\\left(-1\\right)\\mathbf{a}=-\\mathbf{a}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果方程$$\\alpha_1\\mathbf{a}_1+\\alpha_2\\mathbf{a}_2+\\dots+\\alpha_k\\mathbf{a}_k=\\mathbf{0}$$中所有的系数$\\alpha_i\\left(i=1,2,\\dots,k\\right)$都等于零，则称向量集合$\\{\\mathbf{a_1},\\mathbf{a_2},\\dots,\\mathbf{a_k}\\}$是线性无关的，否则称为线性相关的．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果向量集合中只包含一个$\\mathbf{0}$向量元素，由于对于任意$\\alpha\\neq0$，都有$\\alpha\\mathbf{0}=\\mathbf{0}$，因此该集合是线性相关的．所有包含$\\mathbf{0}$向量的集合都是线性相关的．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果集合中只包括单个非零向量$\\mathbf{a}\\neq\\mathbf{0}$，只有$\\alpha=0$时，才有$\\alpha\\mathbf{0}=\\mathbf{0}$成立，因此该集合是线性无关的．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "给定向量$\\mathbf{a}$，如果存在标量$\\alpha_1,\\alpha_2,\\dots,\\alpha_k$，使得\n",
    "$$\\mathbf{a}=\\alpha_1\\mathbf{a}_1+\\alpha_2\\mathbf{a}_2+\\dots+\\alpha_k\\mathbf{a}_k$$\n",
    "则称向量$\\mathbf{a}$为$\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_k$的线性组合．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "命题２.1 向量结合$\\{\\mathbf{a_1},\\mathbf{a_2},\\dots,\\mathbf{a_k}\\}$是线性相关的，当且仅当集合中的一个向量可以表示为其他向量的线性组合．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "令$\\mathcal{V}\\subset\\mathbb{R}^n$，如果对于$\\mathbf{a},\\mathbf{b}\\in\\mathcal{V}$，都有$\\mathbf{a}+\\mathbf{b}\\in\\mathcal{V},\\alpha\\mathbf{a}\\in\\mathcal{V}$（$\\alpha$为任意标量），即$\\mathcal{V}$在向量加法运算和标量向量乘法运算下是封闭的，则称$\\mathcal{V}$为$\\mathbb{R}$的子空间．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "令$\\mathbf{a}\\in\\mathcal{V}$，因为$\\left(-1\\right)\\mathbf{a}=-\\mathbf{a}$，所以$-\\mathbf{a}\\in\\mathcal{V}$；因为$\\mathbf{a}+\\left(-\\mathbf{a}\\right)=\\mathbf{0}$，所以$\\mathbf{0}\\in\\mathcal{V}$，即每个子空间都包含$\\mathbf{0}$向量．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "设$\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_k\\in\\mathbb{R}$，它们所有线性组合的集合记为\n",
    "$$span\\left[\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_k\\right]=\\left\\{\\sum_{i=1}^{k}\\alpha_i\\mathbf{a}_i:\\alpha_1,\\alpha_2,\\dots,\\alpha_k\\in\\mathbb{R}\\right\\}$$  \n",
    "称为$\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_k$张成的子空间．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "对于向量$\\mathbf{a}$，子空间$span\\left[\\mathbf{a}\\right]$由向量$\\alpha\\mathbf{a}$组成，其中$\\alpha\\in\\mathbb{R}$．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果向量$\\mathbf{a}$可表示为$\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_k$的线性组合，则\n",
    "$$\\mathrm{span}\\left[\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_k,\\mathbf{a}\\right]=\\mathrm{span}\\left[\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_k\\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "给定子空间$\\mathcal{V}$，如果存在线性无关的向量集合$\\{\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_k\\}\\subset\\mathcal{V}$使得$\\mathcal{V}=\\mathrm{span}\\left[\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_k\\right]$，则称$\\{\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_k\\}$是子空间$\\mathcal{V}$的一组基．子空间$\\mathcal{V}$中的所有基都包含相同数量的向量，这一数量称为$\\mathcal{V}$的维数，记为$\\mathrm{dim}\\mathcal{V}$．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "命题２.2 如果$\\{\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_k\\}$是子空间$\\mathcal{V}$的一组基，则$\\mathcal{V}$中的任意向量$\\mathbf{a}$都可以唯一的表示为\n",
    "$$\\mathbf{a}=\\alpha_1\\mathbf{a}_1+\\alpha_2\\mathbf{a}_2+\\dots+\\alpha_k\\mathbf{a}_k$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "给定$\\mathcal{V}$的一组基$\\{\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_k\\}$和向量$\\mathbf{a}\\in\\mathcal{V}$，如果\n",
    "$$\\mathbf{a}=\\alpha_1\\mathbf{a}_1+\\alpha_2\\mathbf{a}_2+\\dots+\\alpha_k\\mathbf{a}_k$$  \n",
    "则系数$\\alpha_i,i=1,2,\\dots,k$称为向量$\\mathbf{a}$对应于基$\\{\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_k\\}$的坐标．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbb{R}^n$的标准基定义为  \n",
    "$$\\mathbf{e}_1=\\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{bmatrix},\\mathbf{e}_2=\\begin{bmatrix} 0 \\\\ 1 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{bmatrix},\\dots,\\mathbf{e}_n=\\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ \\vdots \\\\ 1 \\end{bmatrix}$$  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在标准基下，向量$\\mathbf{x}$可表示为　\n",
    "$$\\mathbf{x}=\\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{bmatrix}=x_1\\mathbf{e}_1+x_2\\mathbf{e}_2+\\dots+x_n\\mathbf{e}_n$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "令$\\mathbb{C}$表示复数集合，$\\mathbb{C}^n$表示$n$维复数向量集合．集合$\\mathbb{C}^n$具有与$\\mathbb{R}^n$类似的属性，其中标量可以取复数．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "矩阵：行列数组，$m$行$n$列矩阵称为$m\\times n$矩阵，记为\n",
    "$$\\mathbf{A}=\\begin{bmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\end{bmatrix}$$  \n",
    "位于矩阵第$i$行第$j$列的实数$a_{ij}$称为矩阵的第$\\left(i,j\\right)$个元素．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果认为矩阵$\\mathbf{A}$是由$n$个列向量组成的，则每列都是$\\mathbb{R}^m$空间的一个列向量.  \n",
    "如果认为矩阵$\\mathbf{A}$是由$m$个行向量组成的，则每行都是$\\mathbb{R}^n$空间的一个列向量．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$m \\times n$矩阵$\\mathbf{A}$的转置矩阵$\\mathbf{A}^\\top$是一个$n\\times m$矩阵　　\n",
    "$$\\mathbf{A}^\\top=\\begin{bmatrix} a_{11} & a_{21} & \\cdots & a_{m1} \\\\ a_{12} & a_{22} & \\cdots & a_{m2} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{1n} & a_{2n} & \\cdots & a_{mn} \\end{bmatrix}$$  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "符号$\\mathbb{R}^{m\\times n}$表示由$m\\times n$矩阵组成的集合，矩阵中每个元素都是实数．  \n",
    "$\\mathbb{R}^n$中的列向量可视为$\\mathbb{R}^{n\\times 1}$中的元素．$n$维行向量视为$\\mathbb{R}^{1\\times n}$中的元素．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "2.2 矩阵的秩"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$m\\times n$矩阵\n",
    "$$\\mathbf{A}=\\begin{bmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\end{bmatrix}$$  \n",
    "的第$k$列用$\\mathbf{a}_k$表示  \n",
    "$$\\mathbf{a}_k=\\begin{bmatrix} a_1k \\\\ a_2k \\\\ \\vdots \\\\ a_mk \\end{bmatrix}$$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "矩阵$\\mathbf{A}$中线性无关列的最大数据称为矩阵$\\mathbf{A}$的秩，记为$\\mathrm{rank}\\mathbf{A}$．　　\n",
    "$$\\mathrm{rank}\\mathbf{A}=\\mathrm{dim}\\thinspace\\mathrm{span}\\left[a_1,a_2,\\dots,a_n\\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "命题２.3 一下运算中，矩阵$\\mathbf{A}$的秩保持不变：\n",
    "1. 矩阵$\\mathbf{A}$的某些列乘以非零标量；\n",
    "2. 矩阵内部交换次序；\n",
    "3. 在矩阵中加入列，该列是其他列的线性组合．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果矩阵$\\mathbf{A}$的行数等于列数，则该矩阵为方阵．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "行列式是与方阵$\\mathbf{A}$对应的一个标量，记为$\\mathrm{det}\\mathbf{A}$或$|\\mathbf{A}|$．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "方阵的行列式是各列的函数，具有一下性质：\n",
    "1. 方阵$\\mathbf{A}＝\\left[a_1,a_2,\\dots,a_n\\right]$的行列式是各列的线性函数，即对与任意$\\alpha,\\beta\\in\\mathbb{R}$和$\\mathbf{a}_k^{\\left(1\\right)},\\mathbf{b}_k^{\\left(2\\right)}\\in\\mathbb{R}^n$，有　　\n",
    "$$\\mathrm{det}=\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\alpha\\mathbf{a}_k^{\\left(1\\right)}+\\beta\\mathbf{a}_k^{\\left(2\\right)},\\mathbf{a}_{k+1},\\dots,\\mathbf{a}_n\\right] \\\\\n",
    "=\\alpha\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{a}_k^{\\left(1\\right)},\\mathbf{a}_{k+1},\\dots,\\mathbf{a}_n\\right] \\\\ \n",
    "+\\beta\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{a}_k^{\\left(2\\right)},\\mathbf{a}_{k+1},\\dots,\\mathbf{a}_n\\right]$$  \n",
    "2. 如果对于某个$k$，有$\\mathbf{a}_k=\\mathbf{a}_{k+1}$，则  \n",
    "$$\\mathrm{det}\\mathbf{A}=\\mathrm{det}\\left[a_1,\\dots,a_k,a_{k+1},\\dots,a_n\\right]=\\mathrm{det}\\left[a_1,\\dots,a_k,a_k,\\dots,a_n\\right]=0$$  \n",
    "3. 令  \n",
    "$$\\mathbf{I}_n=\\left[\\mathbf{e}_1,\\mathbf{e}_2,\\dots,\\mathbf{e}_n\\right]=\\begin{bmatrix} 1 & 0 & \\cdots & 0   \\\\ 0 & 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & 1 \\end{bmatrix}$$  \n",
    "其中$\\{\\mathbf{e}_1,\\mathbf{e}_2,\\dots,\\mathbf{e}_n\\}$是$\\mathbb{R}_n$的标准基，则  \n",
    "$$\\mathrm{det}\\mathbf{I}_n=1$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果性质１中$\\alpha=\\beta=0$，则  \n",
    "$$\\mathrm{det}\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{0},\\mathbf{a}_{k+1},\\mathbf{a}_n\\right]=0$$  \n",
    "即如果方阵中一列为$\\mathbf{0}$，则该方阵的行列式等于0．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果在方阵中的一列中加上另外一列与某个标量的乘积，行列式的值不会发生变化．  \n",
    "$$\\mathrm{det}\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{a}_k+\\alpha\\mathbf{a}_j,\\mathbf{a}_{k+1},\\dots,\\mathbf{a}_j,\\dots,\\mathbf{a}_n\\right]  \\\\\n",
    "=\\mathrm{det}\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{a}_k,\\mathbf{a}_{k+1},\\dots,\\mathbf{a}_j,\\dots,\\mathbf{a}_n\\right] \\\\\n",
    "+\\alpha\\mathrm{det}\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{a}_j,\\mathbf{a}_{k+1},\\dots,\\mathbf{a}_j,\\dots,\\mathbf{a}_n\\right] \\\\\n",
    "=\\mathrm{det}\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{a}_k,\\mathbf{a}_{k+1},\\dots,\\mathbf{a}_j,\\dots,\\mathbf{a}_n\\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果交换方阵中的列次序，则行列式的符号将发生改变．\n",
    "$$\\mathrm{det}\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{a}_k,\\mathbf{a}_{k+1},\\dots,\\mathbf{a}_n\\right] \\\\\n",
    "=\\mathrm{det}\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{a}_k+\\mathbf{a}_{k+1},\\mathbf{a}_{k+1},\\dots,\\mathbf{a}_n\\right] \\\\\n",
    "=\\mathrm{det}\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{a}_k+\\mathbf{a}_{k+1},\\mathbf{a}_{k+1}-\\left(\\mathbf{a}_k+\\mathbf{a}_{k+1}\\right),\\dots,\\mathbf{a}_n\\right] \\\\\n",
    "=\\mathrm{det}\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{a}_k+\\mathbf{a}_{k+1},-\\mathbf{a}_k,\\dots,\\mathbf{a}_n\\right] \\\\\n",
    "=-\\mathrm{det}\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{a}_k+\\mathbf{a}_{k+1},\\mathbf{a}_k,\\dots,\\mathbf{a}_n\\right] \\\\\n",
    "=-\\left(\\mathrm{det}\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{a}_k,\\mathbf{a}_k,\\dots,\\mathbf{a}_n\\right]+ \\\\ \\mathrm{det}\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{a}_{k+1},\\mathbf{a}_k,\\dots,\\mathbf{a}_n\\right]\\right) \\\\\n",
    "=-\\mathrm{det}\\left[\\mathbf{a}_1,\\dots,\\mathbf{a}_{k-1},\\mathbf{a}_{k+1},\\mathbf{a}_k,\\dots,\\mathbf{a}_n\\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "给定$m\\times n$矩阵$\\mathbf{A}$，其中$p$阶子式是一个$p\\times p$矩阵的行列式，该$p\\times p$行列式由矩阵$\\mathbf{A}$去掉$m-p$行和$n-p$列获得，其中$p\\leqslant\\min{\\{m,n\\}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "命题2.4 如果一个$m\\times n\\left(m\\geqslant n\\right)$矩阵$\\mathbf{A}$具有非零的$n$阶子式，则$\\mathbf{A}$的各列是线性无关的，即$\\mathrm{rank}\\mathbf{A}=n$．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果矩阵存在一个非零子式，则与非零子式相对应的列都是线性无关的．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果矩阵$\\mathbf{A}$具有$r$阶子式$|\\mathbf{M}|$，有以下性质1.$|\\mathbf{M}|\\neq 0$；2.从$\\mathbf{A}$中抽取出一行和一列，增加到$\\mathbf{M}$中，由此得到的新子式为零，则\n",
    "$$\\mathrm{rank}\\mathbf{A}=r$$即矩阵$\\mathbf{A}$的秩等于它非零子式的最高阶数．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "一个非奇异（可逆）的矩阵是一个行列式非零的方阵．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "设$\\mathbf{A}$是$n\\times n$方阵，$\\mathbf{A}$是非奇异的，当且仅当存在$n\\times n$方阵$\\mathbf{B}$，使得\n",
    "$$\\mathbf{A}\\mathbf{B}=\\mathbf{B}\\mathbf{A}=\\mathbf{I}_n$$\n",
    "其中，$\\mathbf{I}_n$表示$n\\times n$单位矩阵：\n",
    "$$\\mathbf{I}_n=\\begin{bmatrix} 1 & 0 & \\cdots & 0   \\\\ 0 & 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & 1 \\end{bmatrix}$$  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "矩阵$\\mathbf{B}$称为矩阵$\\mathbf{A}$的逆矩阵，记为$\\mathbf{B}=\\mathbf{A}^{-1}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2.3 线性方程组"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "包含$n$个标量的$m$个方程可表示为向量等式\n",
    "$$x_1\\mathbf{a}_1+x_2\\mathbf{a}_2+\\dots+x_n\\mathbf{a}_n=\\mathbf{b}$$\n",
    "其中，\n",
    "$$\\mathbf{a}_j=\\begin{bmatrix} a_1j \\\\ a_2j \\\\ \\vdots \\\\ a_mj \\end{bmatrix},\\mathbf{b}=\\begin{bmatrix} b_1 \\\\ b_2 \\\\ \\vdots \\\\ b_m \\end{bmatrix}$$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "该方程可表示为矩阵形式\n",
    "$$\\mathbf{A}\\mathbf{x}=\\mathbf{b}$$\n",
    "其中，$\\mathbf{A}$为系数矩阵\n",
    "$$\\mathbf{A}=\\left[\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_n\\right]$$ \n",
    "$\\mathbf{x}$为未知数向量\n",
    "$$\\mathbf{x}=\\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{bmatrix}$$\n",
    "增广矩阵定义为\n",
    "$$\\left[\\mathbf{A},\\mathbf{b}\\right]=\\left[\\mathbf{a}_1,\\mathbf{a}_2,\\dots,\\mathbf{a}_n,\\mathbf{b}\\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "定理2.1 方程组$\\mathbf{A}\\mathbf{x}=\\mathbf{b}$有解，当且仅当\n",
    "$$\\mathrm{rank}\\mathbf{A}=\\mathrm{rank}\\left[\\mathbf{A},\\mathbf{b}\\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "定理2.2 方程$\\mathbf{A}\\mathbf{x}=\\mathbf{b}$中$\\mathbf{A}\\in\\mathbb{R}^{n\\times n}$且$\\mathrm{rank}\\mathbf{A}=m$．可以通过为$n-m$个未知数赋予任意值并求解其他未知数来获得$\\mathbf{A}\\mathbf{x}=\\mathbf{b}$的解．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2.4 内积和范数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "实数$a$的绝对值记为$|a|$，定义为\n",
    "$$ |a|=\\left\\{\n",
    "\\begin{aligned}\n",
    "a,\\quad a\\geqslant 0 \\\\\n",
    "-a,\\quad a < 0\n",
    "\\end{aligned}\n",
    "\\right.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "实数绝对值的性质：\n",
    "1. $|a|=|-a|$  \n",
    "2. $-|a|\\leqslant a\\leqslant|a|$\n",
    "3. $|a+b|\\leqslant |a|+|b|$\n",
    "4. $||a|-|b||\\leqslant |a-b|\\leqslant|a|+|b|$\n",
    "5. $|ab|=|a||b|$\n",
    "6. 如果$|a|\\leqslant c$且$|b|\\leqslant d$，则$|a+b|\\leqslant c+d$\n",
    "7. $|a|\\leqslant b \\Leftrightarrow -b\\leqslant a\\leqslant b$\n",
    "8. $|a|\\geqslant b \\Leftrightarrow \\left(a\\geqslant b\\lor -a\\geqslant b\\right)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "对于$\\mathbf{x},\\mathbf{y}\\in\\mathbb{R}^n$，定义欧式内积为\n",
    "$$\\langle\\mathbf{x},\\mathbf{y}\\rangle=\\sum_{i=1}^nx_i y_i=\\mathbf{x}^\\top\\mathbf{y}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "内积是一个实值函数$\\langle\\cdot\\thinspace,\\cdot\\rangle:\\mathbb{R}^n\\times\\mathbb{R}^n\\to\\mathbb{R}$，具有如下性质：\n",
    "1. 非负性：$\\langle\\mathbf{x},\\mathbf{y}\\rangle\\geqslant0$，当且仅当$\\mathbf{x}=\\mathbf{0}$时，$\\langle\\mathbf{x},\\mathbf{x}\\rangle=0$．  \n",
    "2. 对称性：$\\langle\\mathbf{x},\\mathbf{y}\\rangle=\\langle\\mathbf{y},\\mathbf{x}\\rangle$  \n",
    "3. 可加行：$\\langle\\mathbf{x}+\\mathbf{y},\\mathbf{z}\\rangle=\\langle\\mathbf{x},\\mathbf{z}\\rangle+\\langle\\mathbf{y},\\mathbf{z}\\rangle$  \n",
    "4. 齐次性：对于任意$r\\in\\mathbb{R}$，总有$\\langle r\\mathbf{x},\\mathbf{y}\\rangle=r\\langle\\mathbf{x},\\mathbf{y}\\rangle$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "给定向量$\\mathbf{x}$和$\\mathbf{y}$，如果$\\langle\\mathbf{x},\\mathbf{y}\\rangle=0$，则称$\\mathbf{x}$和$\\mathbf{y}$是正交的．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "向量$\\mathbf{x}$的欧式范数定义为\n",
    "$$\\|\\mathbf{x}\\|=\\sqrt{\\langle\\mathbf{x},\\mathbf{x}\\rangle}=\\sqrt{\\mathbf{x}^\\top\\mathbf{x}}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "定理2.3 柯西－施瓦茨不等式　　对于$\\mathbb{R}^n$中任意两个向量$\\mathbf{x}$和$\\mathbf{y}$，有\n",
    "$$|\\langle\\mathbf{x},\\mathbf{y}\\rangle|\\leqslant\\|\\mathbf{x}\\|\\|\\mathbf{y}\\|$$\n",
    "成立．进一步，当且仅当对于某个$\\alpha\\in\\mathbb{R}$有$\\mathbf{x}=\\alpha\\mathbf{y}$时，该不等式的等号成立．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "向量$\\mathbf{x}$的欧式范数$\\|\\mathbf{x}\\|$具有如下性质：\n",
    "1. 非负性：$\\|\\mathbf{x}\\|\\geqslant 0$，当且仅当$\\mathbf{x}\\geqslant\\mathbf{0}$时，$\\|\\mathbf{x}\\|= 0$； \n",
    "2. 齐次性：$\\|r\\mathbf{x}\\|=|r|\\|\\mathbf{x}\\|,r\\in\\mathbb{R}$；\n",
    "3. 三角不等式：$\\|\\mathbf{x}＋\\mathbf{y}\\|\\leqslant\\|\\mathbf{x}\\|+\\|\\mathbf{y}\\|$．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$p$范数定义为\n",
    "$$\\|\\mathbf{x}\\|_{p}=\\left\\{\n",
    "\\begin{aligned}\n",
    "\\left(|x_1|^p+|x_2|^p+\\dots+|x_n|^p\\right)^{1/p}, 1\\leqslant p<\\infty \\\\\n",
    "\\max{\\{|x_1|,|x_2|,\\dots,|x_n|\\}}, \\qquad p=\\infty\n",
    "\\end{aligned}\n",
    "\\right.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果对于所有$\\varepsilon>0$，都存在一个$\\delta>0$，使得$\\|\\mathbf{y}-\\mathbf{x}\\|<\\delta\\Rightarrow\\|\\mathbf{f}\\left(\\mathbf{y}\\right)-\\mathbf{f}\\left(\\mathbf{x}\\right)\\|<\\varepsilon$，则函数$\\mathbf{f}:\\mathbb{R}^n\\to\\mathbb{R}^m$在点$\\mathbf{x}$是连续的．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果函数$\\mathbf{f}$在$\\mathbb{R}^n$中任意点都是连续的，称该函数在$\\mathbb{R}^n$中是连续的．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "对于复数空间$\\mathbb{C}^n$，内积$\\langle\\mathbf{x},\\mathbf{y}\\rangle$定义为$\\sum_{i=1}^n x_i\\overline{y}_i$，上划线表示共轭．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "复数空间$\\mathbb{C}^n$上的内积是一个复值函数，具有如下性质：\n",
    "1. $\\langle\\mathbf{x},\\mathbf{x}\\rangle\\geqslant0$，当且仅当$\\mathbf{x}=\\mathbf{0}$时，$\\langle\\mathbf{x},\\mathbf{x}\\rangle=0$．  \n",
    "2. $\\langle\\mathbf{x},\\mathbf{y}\\rangle=\\overline{\\langle\\mathbf{y},\\mathbf{x}\\rangle}$  \n",
    "3. $\\langle\\mathbf{x}+\\mathbf{y},\\mathbf{z}\\rangle=\\langle\\mathbf{x},\\mathbf{z}\\rangle+\\langle\\mathbf{y},\\mathbf{z}\\rangle$  \n",
    "4. 对于任意$r\\in\\mathbb{C}$，总有$\\langle r\\mathbf{x},\\mathbf{y}\\rangle=r\\langle\\mathbf{x},\\mathbf{y}\\rangle$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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