chapter03_problems.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3/ Problem solutions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## P3.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 5 & 25\\\\2 & 1 & 32\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  5  25⎤\n",
       "⎢        ⎥\n",
       "⎣2  1  32⎦"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUG = Matrix([\n",
    "    [1, 5,   25],\n",
    "    [2, 1,   32]])\n",
    "AUG"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 15\\\\0 & 1 & 2\\end{matrix}\\right], \\  \\left( 0, \\  1\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  15⎤        ⎞\n",
       "⎜⎢        ⎥, (0, 1)⎟\n",
       "⎝⎣0  1  2 ⎦        ⎠"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUG.rref()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## P3.2\n",
    "\n",
    "In the above solution we showed how to build `AUG` matrix directly.\n",
    "This time, we'll build `AUG` by row-joining (`row_join`) a matrix of coefficients and a vector of constants."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3 & 3\\\\2 & \\frac{3}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡3   3 ⎤\n",
       "⎢      ⎥\n",
       "⎣2  3/2⎦"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([\n",
    "    [3,       3],\n",
    "    [2,  S(3)/2]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}6\\\\5\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡6⎤\n",
       "⎢ ⎥\n",
       "⎣5⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = Matrix([6,5])  # b is a column vector\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3 & 3 & 6\\\\2 & \\frac{3}{2} & 5\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡3   3   6⎤\n",
       "⎢         ⎥\n",
       "⎣2  3/2  5⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# row-join A and b to obtain the augmented matrix\n",
    "AUG = A.row_join(b)\n",
    "AUG"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### a) Alice\n",
    "\n",
    "Let's obtain the matrix `AUGA` which is the result after Alice's row operation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 1 & 2\\\\2 & \\frac{3}{2} & 5\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1   1   2⎤\n",
       "⎢         ⎥\n",
       "⎣2  3/2  5⎦"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGA = AUG.copy()  # make a copy of AUG\n",
    "AUGA[0,:] = AUGA[0,:]/3\n",
    "AUGA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 1 & 2\\\\0 & - \\frac{1}{2} & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1   1    2⎤\n",
       "⎢          ⎥\n",
       "⎣0  -1/2  1⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGA[1,:] = AUGA[1,:] - 2*AUGA[0,:]\n",
    "AUGA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 1 & 2\\\\0 & 1 & -2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  1  2 ⎤\n",
       "⎢        ⎥\n",
       "⎣0  1  -2⎦"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGA[1,:] = -2*AUGA[1,:]\n",
    "AUGA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 4\\\\0 & 1 & -2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  0  4 ⎤\n",
       "⎢        ⎥\n",
       "⎣0  1  -2⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGA[0,:] = AUGA[0,:] - AUGA[1,:]\n",
    "AUGA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### b) Bob"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & \\frac{3}{2} & 1\\\\2 & \\frac{3}{2} & 5\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  3/2  1⎤\n",
       "⎢         ⎥\n",
       "⎣2  3/2  5⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGB = AUG.copy()\n",
    "AUGB[0,:] = AUGB[0,:] - AUGB[1,:]\n",
    "AUGB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & \\frac{3}{2} & 1\\\\0 & - \\frac{3}{2} & 3\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  3/2   1⎤\n",
       "⎢          ⎥\n",
       "⎣0  -3/2  3⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGB[1,:] = AUGB[1,:] - 2*AUGB[0,:]\n",
    "AUGB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & \\frac{3}{2} & 1\\\\0 & 1 & -2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  3/2  1 ⎤\n",
       "⎢          ⎥\n",
       "⎣0   1   -2⎦"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGB[1,:] = -1*S(2)/3*AUGB[1,:]\n",
    "AUGB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 4\\\\0 & 1 & -2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  0  4 ⎤\n",
       "⎢        ⎥\n",
       "⎣0  1  -2⎦"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGB[0,:] = AUGB[0,:] - S(3)/2*AUGB[1,:]\n",
    "AUGB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### c) Charlotte"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2 & \\frac{3}{2} & 5\\\\3 & 3 & 6\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡2  3/2  5⎤\n",
       "⎢         ⎥\n",
       "⎣3   3   6⎦"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGC = AUG.copy()\n",
    "AUGC[0,:], AUGC[1,:] = AUGC[1,:], AUGC[0,:]\n",
    "AUGC"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & \\frac{3}{4} & \\frac{5}{2}\\\\3 & 3 & 6\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  3/4  5/2⎤\n",
       "⎢           ⎥\n",
       "⎣3   3    6 ⎦"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGC[0,:] = AUGC[0,:]/2\n",
    "AUGC"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & \\frac{3}{4} & \\frac{5}{2}\\\\0 & \\frac{3}{4} & - \\frac{3}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  3/4  5/2 ⎤\n",
       "⎢            ⎥\n",
       "⎣0  3/4  -3/2⎦"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGC[1,:] = AUGC[1,:] - 3*AUGC[0,:]\n",
    "AUGC"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & \\frac{3}{4} & \\frac{5}{2}\\\\0 & 1 & -2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  3/4  5/2⎤\n",
       "⎢           ⎥\n",
       "⎣0   1   -2 ⎦"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGC[1,:] = S(4)/3*AUGC[1,:]\n",
    "AUGC"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 4\\\\0 & 1 & -2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  0  4 ⎤\n",
       "⎢        ⎥\n",
       "⎣0  1  -2⎦"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGC[0,:] = AUGC[0,:] - S(3)/4*AUGC[1,:]\n",
    "AUGC"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## P3.3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "# define agmented matrices for three systems of eqns. with unique sol'ns\n",
    "AUGA = Matrix([\n",
    "        [ -1, -2, -2],\n",
    "        [  3, 3, 0]])\n",
    "\n",
    "AUGB = Matrix([\n",
    "        [ 1, -1, -2,  1],\n",
    "        [-2,  3,  3, -1],\n",
    "        [-1,  0,  1,  2]])\n",
    "\n",
    "AUGC = Matrix([\n",
    "        [ 2, -2,  3, 2],\n",
    "        [ 1, -2, -1, 0],\n",
    "        [-2,  2,  2, 1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-1 & -2 & -2\\\\3 & 3 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-1  -2  -2⎤\n",
       "⎢          ⎥\n",
       "⎣3   3   0 ⎦"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAM4AAAAzCAYAAADfEzcDAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAKTUlEQVR4Ae2d63XVOBDHQ04KyCYdQAdhqYDQQYAKFjpgD9/4xoEOYCvYhQ5gKwDSwaYDQjpg/z+hcWRZ9vVDvvGNpXMUy3qMRiPNQ9L45s7Pnz/3poRXr14dptor/yqVX/LyUaDQPh8th0I66NNAE/RB9d7p+TlR/5vy7kb57/X+PMorr/kpwLycRmCZo0dRXnntQQEviKDpY6U7Bf+dTRrHM80nPWGGRlD+f8p8rmeKqRr1S8Z8FNAcPBN0Jr0wzkgyi3YnavqX4kOlW5lnvwu+GsJ9F3ommaarbSkrFNhFCmitnwvv14pYUq2h1VQTAKTXiZ6PW1sPKBAcOBlGvK90KycPANmrqvp64yt+1/Oe4hvlXfRqfMOVhCcm8J8ejd/1vORd+UxuCTNRQPT9qPhI8YNicv0nNY4qM2HvFJON+uIrOIe+c2Ch/oC7taC+kRpf9GSxvVWaRYjZuVU8xgzY4wiTYwYT7wsODPNN6XhfM6aL0qaDAtBcxSgOFEgjJBlHtT4pvlWjSZJN7a8UsblB4u9G7zNmqE8GDON+tG6URtPxDiMvPaAp/wiRFP4wPmNAc5cwPwWgN8LrMO6qwTiqxII78pMU19+ld7RlivG/KP80RYyFDQ6t8iOBJ4cwCITFa82F0XMwOqIxQhaz3sz9CkaDcXyl23AYwMK7rEZ6nbD91dLNHRiEgxnD93oEv1INKRhXKO9ZKIDWeRYLqhrjqPCFKjEhnCrsbNA4+iyqoyUPUGPAxOUwIw4csuypLKVN47rlfSIFRGcEGLSGgapQYxzlshfhRKFNylUNF54wpkiNw7RQH+Za1DA1LzANJlptEheF5O1EBkWC1qnWTMU4waTstLYZMG/HA+oupSqHAu81V5wQlrA9CtjlfnXCdhD0/VJpTsFugwlgWiUYXpU0bcS9zmzBS6d/1UElpXp0hnmWpL/yOQn8rCdWQQlbpIBoDl/APNDeCa2QcdgsG2dtEa38XfmBAji1aC1v1ktQcFD/3L1MDoLlJJ2ehWkmU3M0ALQ9/pp3FS+cqaYEtjMLivub2xIQAuwH4mAaZyeEhObmTAO4p2fFNEozeamxxWMt7/ko8I8H5U5jbY/z1GfuxGLqSQskBG4qcXA38Fp4aIRFB+GIQHugZ3wYADN1maM3Nq4lM/QU3NSW9UJ0DrTGOEwQdtys5ss2Z1Nj4S7qUk8WmQtKo1WfKNZu5H+VLuuvcEWjwPxcdmIiWCQPF5zFMb5w4jqDtbTUgKYGx7Hhqxq68dkeB8lMZvYgRN3kC7BJf3yt2ADjQzb36RDa5aX6MbPmgd5xF09uwFW2pADdwLs6yQmQWxz+oikC6jg1p8qzm/fZHW3VFwsb2jWciVXG4Qr+Z2eKeAUMDdDdmWoHAoAUJs6ibQR/kqPo0JGF9dU3Ujk2c8Iqi00L9ywHC9sYoF9DCKgGzsrD0fa1nm6h+roIT7yPs6w5DxMn4ktFBLQJSiXrQXXxwaR/mGio1ubbsz21O93X0zpxmfVuyluhQC8KoFEajrNaYGhLTM1KuivNYuW9UV95owIwFYc4E9O3acEhfX71lU9CxsnC/UOwKHVvDQWeaOGm/BsX6WjrcQVnLK0hwXjkHoxz5Fta5hBApe7KKaDFh83ftnYow3yKg5lIbr8QF27pHZw5KOodNFbD+wjGMa6zzN6ASsVCAVEArdK4xugpzU1o3wQhwXnsbzMcwjjHHuuUZLiJAZU+d4sCbMZT+2NjipRAtrVmQvsmRgzOd0d0zHhqGmcEjNKkUMBZLMYIQ8lhQntouxz1wXkM47p21R4nsN9yIFVgrIcCaJYurZKihGmjWR1tUx0HeexxxmgcQDiNE8AqyUKBPBQIBHFKqlte26FCHiS6obQxfHcrX4rGcWq252auF9BSaVUUYPEbI8QDZwOekuqmcRqHCjGAGd/BeayJeQnjXHnkbDAz4lpAL4kCmYQliy/FHAwV1xdzteLdAh4G54FW2suEi8Hv82S9j9F4TlMdqLHZmW1Sow8SnXVEFLulnd1XKYWI+j9RftJ/KVV/W3nCiwVnLkEsMBbhVn5w0C9UfkWHBdxwlRlAg3PVxQewEQSXr1UZT+UbpjTrjPuTh9bA5+XAxUD2eTLmsZ/RXMA4pnH6dDa4jogyu69SCik/Gb38l1Lt584TfjANv9lV+fIpjYAxP65ZzRj1Zd7wuMQQx64DFl+X+wwLtNPRNgcugoFghClNw21yJuby1QS6kr0DfVzCOGbnZTfVNJikr5LyzVdp7AXUxlEyGarkFqXSS3R3Z9JqnzcIT6QzNGMR/KY4a1BffCB3NqUTtcdZck8Rr2O0Ty0oj3kwrVorC19UbxIual8JoBBuKq26zrTUc5CppvowDeFqX3+ssQPmsvP9YTANYirvi+JpgEi+HncHEhJvCT84yIdyLO4pASFQfaE6AVAOXPp0DyOP0TbGI9/3RTRb2FPs3DZkWRym0cI6NlGUrzVgit3oDw56wWV73NHzIDg4eE76nDsXLpsGoX5Y/OCackrd1NwYx+1xqIzWMdtwU+Ne5Z4Qm+pmNw83dbiUctGnzbTgIGNP5SbQ5kQZD+FcHxMyHkzMseZ3Tly6aMZ+rI32Xe0oM8Y53/c1mSTL9FmTH8YUpl1CgKaFzGYMy1ab1iKGaZiHjXuCHERSf2OkbrJrwWKe+aQ79cVqsk2YmROXEG6Y9rhN+ez8AfAEp9I4nIxwZDjldCXEsW/6uG/FldRDYnOEm0sLbJVsLCjw32qnAzoTflNxY2vhTjtN49iXbTn3HKZVUkMzbTTZvk4B38U8TSomRPnBwYVOnuYHSwALyd39OMZRJqYaqvapYpYgmMAj0FkcLM9O9OLyVb2LVs680TPHydSqaLfFwZpS4SplzzQOaX5wzQp5zxFQa3BqHEzjzHrJF3e6xHcxC/co5QcHlzg5dZw49OAU1An7kHGwr9nj5GQeYKZO6zj6rvkq6X11QbTmMGCnfnBwdZN0PWAEXOUhUTGOJtHdK6hw7FHddRc+JZhsxnb2RwEbA8qYIdqgiU1Y2Y8N8iRvyslPRiwLKCigObGTwupw4SAiDbepTB6uH7ZHiaoMft3oqzQYYs8GGoNbmKpuWm+T/1JPyFmqgRvMY5MSAmXPWcJyKMD1QO3/RtUYRwuNo1CY56VilrsEz4BZYA2lo/rOpj2H9r2pvnCbw1NjU7elfCAFNE9sXRBwtYvdylQL4LHIX6iBnXwFRSVZKLA6Cpi2qZ0ANxhHDIMdh6mA1imhUGC1FPDaBo1T82KHIA3G8VSi4jM1REWVUCiwVgqwD+WndRv7/STjqCIa57UiDUsoFFgdBcQDrH08OdyFZ0yAJONQSQ3wl+LCZ8x3C3E/5b1QYGcooDXPnQ2fHrQeLtVO1eKR0VDxkyJmW3WGHdfzndQOE1S/od4S7UrWBAqIxjWaC1T8PgH6OpuKplxKs7+vfhMhRYlWjWOVBYhjOBiITVJb4Eb1RxCLlmqjVN58zIlC90w09YKI36ngn491Cv7/AaWneFpPXiMKAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & -2\\\\0 & 1 & 2\\end{matrix}\\right], \\  \\left( 0, \\  1\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  -2⎤        ⎞\n",
       "⎜⎢        ⎥, (0, 1)⎟\n",
       "⎝⎣0  1  2 ⎦        ⎠"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGA.rref()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & -1 & -2 & 1\\\\-2 & 3 & 3 & -1\\\\-1 & 0 & 1 & 2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1   -1  -2  1 ⎤\n",
       "⎢              ⎥\n",
       "⎢-2  3   3   -1⎥\n",
       "⎢              ⎥\n",
       "⎣-1  0   1   2 ⎦"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 0 & -4\\\\0 & 1 & 0 & -1\\\\0 & 0 & 1 & -2\\end{matrix}\\right], \\  \\left( 0, \\  1, \\  2\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  0  -4⎤           ⎞\n",
       "⎜⎢           ⎥           ⎟\n",
       "⎜⎢0  1  0  -1⎥, (0, 1, 2)⎟\n",
       "⎜⎢           ⎥           ⎟\n",
       "⎝⎣0  0  1  -2⎦           ⎠"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGB.rref()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2 & -2 & 3 & 2\\\\1 & -2 & -1 & 0\\\\-2 & 2 & 2 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡2   -2  3   2⎤\n",
       "⎢             ⎥\n",
       "⎢1   -2  -1  0⎥\n",
       "⎢             ⎥\n",
       "⎣-2  2   2   1⎦"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGC"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 0 & - \\frac{2}{5}\\\\0 & 1 & 0 & - \\frac{1}{2}\\\\0 & 0 & 1 & \\frac{3}{5}\\end{matrix}\\right], \\  \\left( 0, \\  1, \\  2\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  0  -2/5⎤           ⎞\n",
       "⎜⎢             ⎥           ⎟\n",
       "⎜⎢0  1  0  -1/2⎥, (0, 1, 2)⎟\n",
       "⎜⎢             ⎥           ⎟\n",
       "⎝⎣0  0  1  3/5 ⎦           ⎠"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGC.rref()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## P3.4\n",
    "\n",
    "These three systems of equations have infinitely many solutions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### P3.4 a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-1 & -2 & -2\\\\3 & 6 & 6\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-1  -2  -2⎤\n",
       "⎢          ⎥\n",
       "⎣3   6   6 ⎦"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGA = Matrix([\n",
    "    [ -1, -2,  -2],\n",
    "    [  3,  6,   6]])\n",
    "AUGA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 2 & 2\\\\0 & 0 & 0\\end{matrix}\\right], \\  \\left( 0,\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  2  2⎤      ⎞\n",
       "⎜⎢       ⎥, (0,)⎟\n",
       "⎝⎣0  0  0⎦      ⎠"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGA.rref()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left[\\begin{matrix}-2\\\\1\\end{matrix}\\right]\\right]$"
      ],
      "text/plain": [
       "⎡⎡-2⎤⎤\n",
       "⎢⎢  ⎥⎥\n",
       "⎣⎣1 ⎦⎦"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGA[0:2,0:2].nullspace()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "# the solutions to the sytem of equations represented by AUGA\n",
    "# is of the form    point + nullspace\n",
    "point = AUGA.rref()[0][:,2]\n",
    "nullspace = AUGA[0:2,0:2].nullspace()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡2⎤\n",
       "⎢ ⎥\n",
       "⎣0⎦"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# the point is also called he particular solution\n",
    "point"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-2\\\\6\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-2⎤\n",
       "⎢  ⎥\n",
       "⎣6 ⎦"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# if the augmented matrix AUGA is [A|b], then the point satisfies A*point = b\n",
    "print( AUGA[0:2,0:2]*point == AUGA[:,2] )\n",
    "AUGA[0:2,0:2]*point"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Finding the null space"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-2\\\\1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-2⎤\n",
       "⎢  ⎥\n",
       "⎣1 ⎦"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# the nullspace of A is one dimensional and spanned by\n",
    "n = nullspace[0]\n",
    "n\n",
    "# every vector n in the nullspace of A satisfies  A*n=0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Matrix([[-2], [6]]) True\n",
      "Matrix([[-2], [6]]) True\n",
      "Matrix([[-2], [6]]) True\n",
      "Matrix([[-2], [6]]) True\n",
      "Matrix([[-2], [6]]) True\n",
      "Matrix([[-2], [6]]) True\n",
      "Matrix([[-2], [6]]) True\n",
      "Matrix([[-2], [6]]) True\n",
      "Matrix([[-2], [6]]) True\n",
      "Matrix([[-2], [6]]) True\n",
      "Matrix([[-2], [6]]) True\n"
     ]
    }
   ],
   "source": [
    "# so solution to A*x=b is any (point+s*n) where s is any real number\n",
    "# since  A*(point+s*n) = A*point + sA*n = A*point + 0 = b.\n",
    "# Let's verify claim for values of s in the range -5,-4,-3,-2,-1,0,1,2,3,4,5\n",
    "for s in range(-5,6):\n",
    "    print( AUGA[0:2,0:2]*(point + s*n), \n",
    "           AUGA[0:2,0:2]*(point + s*n) == AUGA[:,2] )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### P3.4 b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & -1 & -2 & 1\\\\-2 & 3 & 3 & -1\\\\-1 & 2 & 1 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1   -1  -2  1 ⎤\n",
       "⎢              ⎥\n",
       "⎢-2  3   3   -1⎥\n",
       "⎢              ⎥\n",
       "⎣-1  2   1   0 ⎦"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGB = Matrix([\n",
    "        [ 1, -1, -2,   1],\n",
    "        [-2,  3,  3,  -1],\n",
    "        [-1,  2,  1,   0]])\n",
    "AUGB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & -3 & 2\\\\0 & 1 & -1 & 1\\\\0 & 0 & 0 & 0\\end{matrix}\\right], \\  \\left( 0, \\  1\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  -3  2⎤        ⎞\n",
       "⎜⎢           ⎥        ⎟\n",
       "⎜⎢0  1  -1  1⎥, (0, 1)⎟\n",
       "⎜⎢           ⎥        ⎟\n",
       "⎝⎣0  0  0   0⎦        ⎠"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGB.rref()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3 s + 2\\\\s + 1\\\\s\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡3⋅s + 2⎤\n",
       "⎢       ⎥\n",
       "⎢ s + 1 ⎥\n",
       "⎢       ⎥\n",
       "⎣   s   ⎦"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "point_B = AUGB.rref()[0][:,3]\n",
    "nullspace_B = AUGB[0:3,0:3].nullspace()[0]\n",
    "s = symbols('s')\n",
    "point_B + s*nullspace_B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### P3.4 c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2 & -2 & 3 & 2\\\\0 & 0 & 5 & 3\\\\-2 & 2 & 2 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡2   -2  3  2⎤\n",
       "⎢            ⎥\n",
       "⎢0   0   5  3⎥\n",
       "⎢            ⎥\n",
       "⎣-2  2   2  1⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGC = Matrix([\n",
    "        [ 2, -2, 3,  2],\n",
    "        [ 0,  0, 5,  3],\n",
    "        [-2,  2, 2,  1]])\n",
    "AUGC"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQEAAABMCAYAAABtXQL7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAPbUlEQVR4Ae2dS44cxxGGR4NZG2MSELz16AZ8nECjG5DUCSzdgAJ33BHSDSgewJBEQAegdAJaXHhPHsAAacJrA/T/ZVeUs6qzujOr65E9HQFUV1Y+I//MiIx8VPVnnz59OnNyBMYi8PTp00ulfaTrW7nvjs3H062HwEVu0WrgPxT3O91/S6WR/x35EydFLxX+MBXgfseLQNPm100NUAZZpHRvFfEqFVlhn6X83S8PAeEHrq903ZX7Y06qLCWgzJ4rs3e6JxVAr6Af9Ewjx/QufnD3zUBA/eGNavJG9weFNfo+Ef8r+ZXmk8jmtL3UFsjpS6HwQlfWwLtXCTQNjLn310x4n8NIZlyPdoIIqH/82K+2/PByJdAHZsSzsMRi/7eub3RtYd3P8rzvET8rA0w8NAqZZpkWcXp3OwKOwGoIYAUwICenXTFX+ywBFADmxV5tEmea41aerCH8oit77pKT7xJx1uJd5ZoZ/V71/ELX9/Jzq2uJRj+yMtQvftPFtICpPFOtQRpUAsoAIcU825nBYM6JAOVplsUHBd/TtVdLJbJZxWtt3lU+i67PdKdhzxp+/tD9K12uCFbpFdUX+p04fKv+8UBX6DcpjndNB7ACWPTJWQxM5b3lp7w+6nqo61sF/rQVoWKPNXlX2d8Imkvd24aEH/mZpq8YOWdtLQTURxgckF+zIJOsJJWAEmMBYAmgSZzWR4D5HSvxfXotj2u1FxaWkyOQQgAZvlIfeZwKxC+pBOT/RFfuliD5OM2LwLWyZwrVJ1usJXxxUseiczHK0F9wsxA12NkWZ9ALPFN7MHhgDdBGSdpaE1AiLAC3ApJwLe+p9sgZ5W8tz1lYl8DcdGtxDfDLymRxEIsxuTaQsgRMY0y+I1DGt8duEDABt1E/BsasgxxFEadz9wkhIOG3tSST7U7tO0qgGXVYD2B7IdXpOon9oRoEblfDiTNSKwIogjuSa6z8DvWnA5wMhDAfbhQ1Cu53Vapk1GQnI7UgtyQ2NtqnyjQrgXMDs5JwOPhNM+Xh7wXM2ko7M0emGeC/1tXp030lYGeNJ9sW3MnWgoHqgFg2R/eWG3zrAqmU8jK/2c8JuADTBMdLaj+seyrAdnNnHaedDigCHepaF7sCCIxTPQiglK8S7JglcOOUdqKu7nU4AvQTzpt0pgSxJWBTAVtEOLxIz2EqBDhenTrwgWXDga7FlHYzWNBXkt8PULjxyRTltp47o478nNZD4JWKZqDvTAlaS0ABdjyYiE4VISBBYqfmg+7M6QLJjeWGMP5t4zP/r8pkBAmnF3Wn/A4pHAXwXndeOOOV8le6e3/qoLTqg1mMKIKWYksgBKjRLGIbaUqH8mdUowPda/Ll/DsLFa91p+NUSyvzzqj/RDzYtOC+nr9ssFsEs6asXd8PQEF8acwoPvNQFAEHiWZft7By/Z5GQG1A2xG4PR1QAELJNXtDqSxbfISZo6I1eVfZHwVWtaa1+EM50YfgMyae6XSz9624UHcPIsCAy1Yhh4fCgG/TATMPvKEGsfOAPQigAKD+libPt0KI/9SAwD8aJlprwJSArQegJZwcgSkRQAGYgpgyX89rHAK8kg4xnQxkSsDm57yV5uQIjEHApgH9UR8F4BbmGETnSWOWgFn/7VuEV0153ljzAH/jc9X8kr6DIkiN+m5h1tMDTMbbdjJLwDwsQj0sOyfHhABbme0Iw+KTnlmR9n5VSSuqLVDUgeQOg/+FOfCNI2yi+a8j8H8Emr7CV6EQbrb9/q47A8ivzZ1O9Rf5h/MCcvMdxHbLUG6nOhBAKdNWXO8u9GOrhK6tBYbTMAISbvpIu02pZ858/FP38Nq57o/1zFmGPw/n4iEVIGDWwMYSEEPBobsFVMCjs3AkCDzr8ckrzT6Y9ECp8JE2YvDHUgsLgzQc1N/f3fj6ryMwgIBGfOb78aIfx5qP/jV01ckGxoGar+c9EW8m66Ge56qOVdgtgfXa9qhLVsd8rIv9Z74xeNRfpKIuqodNkWtsF9Zi4PEQMlm/JBPWBIJDd9MO+M9KqkT8phkmSTV/olEzbzSK+KODMhev5k9bxNMPuhD+X3Q/41nuoyPxjSXDm49b/Mtv9j6rMhiQbc2FszvIJC9jtdaW3LyPwbHf5PcCFb+EbhEZSyA4SlIeElfMM2LwspC9aUalw0smh+Q7RdpaeRNfvAOOgGFq838QZr1NUe1J8hBvjC60JQq95pE0WV/xzGDIoqYJYRtPfrP3WZVBm4Idr2jba9oIPy/YsRvTkp5RUvBqA3gblul438QL6VECRjTirCSmq/0Tjcp54+tCVf1pCx1QF396GQu89SE7gTprf5o4c0b6rfUM1W+pPkv5ndfCVTYKCUyx/PoEr2ad9MP2PVs7hXgogct9KSYM5w3C1rSJ8uW48rUqvSQvUfHBWTNvfV5Xf1Zb0ZEwV+PdAEYs/H/WdWz0SHVKrWcs1S/ADqXalwH7GhCWQksNr/Dcj9/GyXC00wGLayaCPc9xp6J0nD7RcaCO2bPxWuy3Zt4WA6GwIAQEs5SFQRareBGtmrWK3LqId9o+VmZx0qX6BcK+69N+KWGH50cxs5luk8GQ54US2ZqACWJmPmXRMjWW8VKW+YGxa+btwKrNmly4YdVxHTuhzLY+prNkv1BZ8JCiMN1qsO6HwzOKN2XB9OMOPi85HTABTymbjmYa5Ha+gJp5m6/WnrMhwBrGW3uI7qv2Cwk+CoBpwNZiZcMjPHemCY3/vltHBlECRiaI9rzG3Q4urVH2vjJr5m0f7x6+GwHM4rH9f85+wYLgj1IGW1uWTXXgGd5Lqa2r8r5kOmD0X3OMvP+pSWf3fjZtwf0APZvGXWJdIlH8zg6wNm8pflf3U+fZ+2ckilPyZyOfr1gp2rgzOja8rNZnhR2r/5wJ4IWtIWJNYIwl0JHRWAnE7qFCd/n/pwm0eyeuKsM2F34pzWV+Q4sznbymfqiZt6nrOlV+wqxEwHOK/VdOpCXjrNUvVC7bkme671IARBlSXoTtolZGqSPTgZQG3JXBIWEsZKQ0F5WBthZnNt6L/NbM2yIAnHAhDD42EPVhWLRfSCg5tfhFrADk5qhwSm7geZe10q+LPZu8hWdGfzIhs05ACJ3+hzlO6oDDXfnzMsqSCqlfu5p56/NaxXPTMRmt7L8F6Khscy2mzFUWh5YO7TfIQErIwDm7XxzKi9LfUXn3de8vBKIYUjsAyOzB1vO5MlmMVDkq8kF3KhVIbhQQe52d01Kb0OV+a+ZtORSKS0JwMF1RAsxhGcEWVQAqkwM2HOs9hN4o8f1UBrn9QvHox6N5UXqwROGg1HgRyy78OEacUnQMnqaA5RxHWAKpzMfllpcKxjlgYpoX8Bf9E40dbFbLm/AKHUS8s50F1fKnLau1HYKhK5jyuh9iEZgS2yC7/bu3X0zAC+1rSrXPAUoqRdfyTFnWqbixHwoLCrJv0wE8LAD3bARYyrxv7sxWXknGlfM2dJikpIqzxBVudEb6D6vZoWPNUlAiU5WH9dFalokoe72UHr7PdPF23pbAyS+rzyreaF6UFkWTTYofBlHdx0wHOlP/86jU25HbnY5ALgJPFJGOyDTghTrlQQKZW2gvHvPoQ5UPI+q+1fhescnHKXhJZtzzZCAdYwWQDQobYi3kDEvAwLMA/J0cgb0ISPAQ/HYNQM/P9Mz8fOrtw0FeVCb99uDzJcqHQzkH/W/iVLwMVrYJUDlYAewYjFVaNuAH2ccSCNpA946J0JTnN0dgEAF1wv6oH0xT+TM9WIp4k27oRF0pD0y5WOAcS1PysosHeDxkemgDfpB9lICdmbaAXYV7mCMQI8CHTtjWMrKBZMw81fIouqv81NZZUR4WWXkxMrISHw7rmH/ufUpehspseBvaLRhK1vfvtBPTAWswC+gn8GdHYAgBPisWL6RhGbyUn/WpoXTV+je8T6ZYpq6o+JuCNxvwgwEQKwHmGU6OQAkCz9QpWZxiTh7mmXo+xEwtKdvjjkfABvygrGMlYNphfNae8qQQkMBjPle53XtSDVFeWRvwgxI4bxoyZCO3BZZn6ykcAUegegQk4wz2NuBvlEDDtc3hXAlU34zOoCNwEAIm45y2xJIL5wS4s7hDoEXAbzZS4XbIgbmk/+9AJtI14lYjT5lwnmq0e03FbeAP/zuA3+smoOjoYpOm6KZOw2ES/9+BItTCu+XV4VZzWxbCe0rRw/8PqsLtIa/zpvbmYVpiFlDUaZb6hnsx/85bGWQ141VWk5OLfaepsQ38G0tADWp7vRZhLmTYPrKy4jJg6Fp82IJFHLaU23krQ7pmvMpqclqxbcpvA387HQCGMEeQIFqkOaDhOOmHRMZhgUL+hK9FzlsZ8jXjVVaTE4ndDLLId7soSNXPo/q/bNyzCGLmKG+HGCK25nc6b2UY14xXWU1OLrbJdmsFgECsBOwLJfyZwRxkAm6jflyGWQdrTQect7g19rtrxms/96cb4+um6j/FELRKQNod7YCAmraI4y3ltlcclyqvpBznrQSt5hhxWRKPPTMCQbYl62b1h+JaJdAUzh9J8pmmOzMwY6N9KmsbWTg3sAY5b2Wo14xXWU1OJHYj01jaHQVA9ftKgO+cQWY2bJ4m+BUTNg1Imfzm1x5gmKDI7Cyct2yoQsSa8SqryUnFNgu/MxUAgY4SUOPalGDU+9QZkJL/VSKeWQKdBYtEvDm9nLcydGvGq6wmpxGbgZ1dgb2WAHDwiSimBKY58JuKsDRSB5I4qVjD/w44b/ktXXNb5tfiBGJKlpnecyW/RdCxBMBDCexTTWO/X0Y2SVLeMOH/O5BEZ9izRtxq5GkYwZMP4WOwEAP8Fl1s+Ww8UASP1dBYBDaXH4ha7M2o7/87UAzbWY241chTObI3OAUyrOo90MUXn5KyPKQE0BiPdaFBJv1oRMPIpHmKx0nIeSuDsWa8ympyo2Pb+t6gzG1NB4AjatxgDdxoiLxyjsDNRoCBnM+pD+68JZUAmCgRUwLMB5tP4O3kCDgCR4KAZBhrnunAoBVAVQaVAIEi/iQUa+AqPPmPI+AIHAUCklmEn4/37P08+U4loIzYU+R6rsvJEXAEjgeBF2KV/1hMbgvG1dipBIioTHhvnL88sgWGOL27HQFHoDIEJKuc8eFCdvfS0O5APyGZ/a7Mf9aV3GaIErxVnOgxONmeyGKon9Cfbx4C6gv86YVPMedrWiz3hxmyGjjIUgLKjNN8CDHHe4eUAKuPQweMBlcmAxf+c2oI2IdmT63es9dXcspaAOsA2Ufw/we3mGLQkNnANQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & -1 & 0 & \\frac{1}{10}\\\\0 & 0 & 1 & \\frac{3}{5}\\\\0 & 0 & 0 & 0\\end{matrix}\\right], \\  \\left( 0, \\  2\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  -1  0  1/10⎤        ⎞\n",
       "⎜⎢              ⎥        ⎟\n",
       "⎜⎢0  0   1  3/5 ⎥, (0, 2)⎟\n",
       "⎜⎢              ⎥        ⎟\n",
       "⎝⎣0  0   0   0  ⎦        ⎠"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AUGC.rref()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}s + \\frac{1}{10}\\\\s\\\\\\frac{3}{5}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡s + 1/10⎤\n",
       "⎢        ⎥\n",
       "⎢   s    ⎥\n",
       "⎢        ⎥\n",
       "⎣  3/5   ⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "constants = AUGC.rref()[0][:,3]\n",
    "\n",
    "# construct point_C by placing the constants into the location of the pivots\n",
    "pivots = AUGC.rref()[1]\n",
    "point_C = zeros(3,1)\n",
    "for idx, pivot in enumerate(pivots):\n",
    "    point_C[pivot] = constants[idx]\n",
    "\n",
    "nullspace_C = AUGC[0:3,0:3].nullspace()[0]\n",
    "s = symbols('s')\n",
    "point_C + s*nullspace_C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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