01-方程组的几何解释.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing(use_latex='mathjax')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. 例子定义"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x, y = symbols('x y')\n",
    "a, b, c, d = symbols('a, b, c, d')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left [ 2 x - y = 0, \\quad - x + 2 y = 3\\right ]$$"
      ],
      "text/plain": [
       "[2⋅x - y = 0, -x + 2⋅y = 3]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = Eq(2 * x - y, 0)\n",
    "eq2 = Eq(-x + 2 * y, 3)\n",
    "[eq1,eq2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. Row Picture"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & -1\\\\-1 & 2\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2   -1⎤\n",
       "⎢      ⎥\n",
       "⎣-1  2 ⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[2, -1], [-1, 2]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}x\\\\y\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡x⎤\n",
       "⎢ ⎥\n",
       "⎣y⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = Matrix([[x], [y]])\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}0\\\\3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0⎤\n",
       "⎢ ⎥\n",
       "⎣3⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = Matrix([[0], [3]])\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right], \\quad \\left[\\begin{matrix}x\\\\y\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\3\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "⎛⎡1  2⎤, ⎡x⎤, ⎡0⎤⎞\n",
       "⎜⎢    ⎥  ⎢ ⎥  ⎢ ⎥⎟\n",
       "⎝⎣3  4⎦  ⎣y⎦  ⎣3⎦⎠"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A, X , b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}x + 2 y\\\\3 x + 4 y\\end{matrix}\\right] = \\left[\\begin{matrix}0\\\\3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ x + 2⋅y ⎤ = ⎡0⎤\n",
       "⎢         ⎥   ⎢ ⎥\n",
       "⎣3⋅x + 4⋅y⎦   ⎣3⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(A * X, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. Column Picture"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}2\\\\-1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2 ⎤\n",
       "⎢  ⎥\n",
       "⎣-1⎦"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "col1 = Matrix([[2], [-1]])\n",
    "col1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}-1\\\\2\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-1⎤\n",
       "⎢  ⎥\n",
       "⎣2 ⎦"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "col2 = Matrix([[-1], [2]])\n",
    "col2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}0\\\\3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0⎤\n",
       "⎢ ⎥\n",
       "⎣3⎦"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = Matrix([[0], [3]])\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( x, \\quad \\left[\\begin{matrix}2\\\\-1\\end{matrix}\\right], \\quad y, \\quad \\left[\\begin{matrix}-1\\\\2\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\3\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "⎛x, ⎡2 ⎤, y, ⎡-1⎤, ⎡0⎤⎞\n",
       "⎜   ⎢  ⎥     ⎢  ⎥  ⎢ ⎥⎟\n",
       "⎝   ⎣-1⎦     ⎣2 ⎦  ⎣3⎦⎠"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x, col1, y, col2, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}2 x - y\\\\- x + 2 y\\end{matrix}\\right] = \\left[\\begin{matrix}0\\\\3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2⋅x - y ⎤ = ⎡0⎤\n",
       "⎢        ⎥   ⎢ ⎥\n",
       "⎣-x + 2⋅y⎦   ⎣3⎦"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(x * col1 + y * col2, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4. 矩阵乘法规则"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}a & b\\\\c & d\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a  b⎤\n",
       "⎢    ⎥\n",
       "⎣c  d⎦"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[a, b], [c, d]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}x\\\\y\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡x⎤\n",
       "⎢ ⎥\n",
       "⎣y⎦"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = Matrix([[x], [y]])\n",
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}a x + b y\\\\c x + d y\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a⋅x + b⋅y⎤\n",
       "⎢         ⎥\n",
       "⎣c⋅x + d⋅y⎦"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * B"
   ]
  }
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