notebooks/Math/Linear Algebra/THEOREMS LINEAR DEPENDECY.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.style.use('classic')\n",
    "import seaborn as sns\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "import math\n",
    "from mpl_toolkits import mplot3d\n",
    "\n",
    "from svn_util.dplot import VectorGround\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "# Linear dependency .Definitions and theorems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Linear dependecy \n",
    "deff : let's to have set of n vectors $\\vec{r}_i\\in L^n linear space$ if exists real numbers $\\gamma_i\\in |R $ such that :\n",
    "$$\\sum_{i=n}^{n}\\gamma_i.r_i=0$$ the vecors $\\vec{r}_i$ are linear dependent <br>\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Theorem 1: <br> \n",
    "##### if the n vectors for example  $\\vec{r_1}=r_{11}.\\vec{e}_1 +r_{12}.\\vec{e}_2  + _{13}.\\vec{e}_3 $    \n",
    "##### and $\\vec{r_2}=r_{21}.\\vec{e}_1 +r_{22}.\\vec{e}_2  + _{23}.\\vec{e}_3 $   \n",
    " , $\\vec{r_3}=r_{31}.\\vec{e}_1 +r_{32}.\\vec{e}_2  + _{33}.\\vec{e}_3 $ where $\\vec{r}_{i}\\in{L^3}$ \n",
    "##### are linear dependent then   $\\det{R}=0$   \n",
    "$\\large\\det{R}=\\large{\\bigtriangleup}\n",
    "\\begin{pmatrix} \n",
    "r_{11} & r_{21} & r_{31} \\\\\n",
    "r_{12} & r_{22} & r_{32} \\\\\n",
    "r_{13} & r_{23} & r_{33}\n",
    "\\end{pmatrix} \n",
    "\\quad$ $\\large {=0}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### proof:\n",
    "  if vectors $\\vec{r_1}$ and $\\vec{r_2}$,$\\vec{r_3}$ are linear depend then by deffinition \n",
    "  we have $$\\gamma_1.\\vec{r}_1 + \\gamma_2.\\vec{r}_2 +   \\gamma_3.\\vec{r}_3  = 0 $$\n",
    "  wich can be expressed as : <br>\n",
    "   $\\gamma_1.(r_{11}\\vec{e}_1+r_{12}\\vec{e}_2+r_{13}\\vec{e}_3) $ + \n",
    "   $\\gamma_2.(r_{21}\\vec{e}_1+r_{22}\\vec{e}_2+r_{23}\\vec{e}_3) $ + \n",
    "   $ \\gamma_3.(r_{31}\\vec{e}_1+r_{32}\\vec{e}_2+r_{33}\\vec{e}_3) =  $ <br> \n",
    "   $(\\gamma_1.r_{11} + \\gamma_2.r_{21} +\\gamma_3.r_{31}).\\vec{e}_1 $ + \n",
    "   $(\\gamma_1.r_{12} + \\gamma_2.r_{22} +\\gamma_3.r_{31}).\\vec{e}_2 $\n",
    "   $(\\gamma_1.r_{13} + \\gamma_2.r_{23} +\\gamma_3.r_{33}).\\vec{e}_3 = 0 $ <br>\n",
    "   the desion are when $$ \\gamma_i.r_{ik}=0$$\n",
    "   in matrix form expressed $\\large{R*Y=0}$ <br>\n",
    "\n",
    "\n",
    "$\\begin{pmatrix} \n",
    "r_{11} & r_{21} & r_{31} \\\\\n",
    "r_{12} & r_{22} & r_{32} \\\\\n",
    "r_{13} & r_{23} & r_{33}\n",
    "\\end{pmatrix}$ *\n",
    "$\\begin{pmatrix} \n",
    "\\gamma_{11} \\\\\n",
    "\\gamma_{12} \\\\\n",
    "\\gamma_{13}\n",
    "\\end{pmatrix}$ =\n",
    "$\\begin{pmatrix} \n",
    "0 \\\\\n",
    "0 \\\\\n",
    "0\n",
    "\\end{pmatrix}\n",
    "\\quad$\n",
    "\n",
    "<br> From theory of linear algebra we know that the above matrix eqation has only desion when $\\det{R}=0$\n",
    "\n",
    "and $\\det{R}=\\det{R^T}$ is true as well\n",
    "\n",
    "R^T = transpose matrix of R wich  is <br>\n",
    "\n",
    "\n",
    "$\\large{R^T}=$ $\\begin{pmatrix} \n",
    "r_{11} & r_{21} & r_{31} \\\\\n",
    "r_{12} & r_{22} & r_{32} \\\\ r_{13} & r_{23} & r_{33}\\end{pmatrix}\\quad$ <br>\n",
    "\n",
    "from above discussion $\\Rightarrow$  $\\large{det{R^T}=0}$ and threorem is proven"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Example : <br>\n",
    "let's to have $r_1=2.\\vec{e}_1+3.\\vec{e}_2$   and    $r_1=4.\\vec{e}_1+6.\\vec{e}_2$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {},
   "outputs": [
    {
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rZ+Kvv+77OSIitD8KDh2qWW0UmJRqrvn5+XC5XEhPT0ePHj0wffp0REZGIjExEQUFBaLLAwDk5OSILsE0zCqX4uJivPjii4iOjkZGRhYaNvwEAPD119rQcOfO2u3699ca1ZVnb2VlwHvvaatxfTl6VGtSlfn0UyAqquoh46rUq1f124LKv5/WPefqPq4dO2qXly3zff+gICA8/OrDx1amwuvYKEo119LSUgDAH//4Rzz22GNo164dEhMTERQUhB07dgiuTpORkSG6BNMwq1xOnDiBqVOn4uLFi6hfPxt/+EMYfv0V2L376u8xXbNGW4CUkqIN7fpSUKA1qCvl5wNLlxq/Ori8hg0vnxG3bKl9dx9X91vhnU5taNmXa6/1/XadQKDC69goSjXX8PBwBAcHo1WrVp7rQkND0bx5c5w8ebLS+8TFxcFms8Fms2HTpk2w2WxwuVyen6enp8PhcHguO51OJCUleT1GamoqsrKyPJdzcnJgt9u9bmO325GTk4NJkyYBALKysiqszEtKSvLafszhcHh9BJTL5fJ7fW5G1Ddp0iRL1wcY9/tzX2/V+oz4/UVGRnr+XVhYiBtvvBGvv74dZWWXm6uv+tq0ycLs2dqWiPn5+T7rCwnR5kevrO/zz4Ft237GPfcU4777tOfr1Ok0XnjhHABtIdW995aiT5/lun5/bdtq102bNhsAPP+9zpx5ud5rrvH9+zt9usyzXWOgHV+Xy4UTJ05Yuj49vz+HwwGbzYb4+Hj0dg9N6KDcDk0jRoxAdHQ0JkyYAECbh42Li8PIkSPRs2dPz+24QxNRzaSlpWHYsGFYuHAhWrVqhQcffBBjxoRi2TJtc/6wsKs/Rny89n358sp/fscdwK23Vv/9oi+/rL0tx6gdmsaOBebP1+Znyy96On0aaNoUuOEG4PDhyu9bWKjNNa9dq239SNbFT8Wphf79++Ozzz7Dp59+itzcXCxYsAAAcOeddwqujChwFRQUYNiwYQCAkSNHok+fPggNDUVmJtC9e/Uaa3XceKP3wiIjnT9/9cceNkx7m88333hf77787LO+7+t+C0779rWvkQKHcs31T3/6E5555hksXboUw4cPx6FDhzB79uxa/3VitMqGE2XFrPJo+Ntqn4KCAk9WpxPIyTFuT19AW6z000/Vv/2vv2rfT52q+nbnz2vbEjZvDnz3ne/bdewITJig7RX8yy/acS0sBKZN0xZt/TYgVqmdO7W52VtvrX79ViP769hIyr3PFdDOXvv37y+6jEr17dtXdAmmYVY5pP02Rrt8+XLUr1/fk/XECW3XoiefNO657r1Xe6tNXp7vVcOAdpvVqy9vCPGXv2jN8403gMoORWio9vaasrKrf4Td9OnagqZHHgGA0diwQdsxatKkqt8CtG2bdrvgAD6lkfl1bDTl5lyri3OuRFdXUFDgOWstu/Lz1mrhanOupaXA9ddrK4P79NH9dKYpLdWa+4IFV9+6kcTjnCsRCVV+ONgMwcHavOf775vydIZZt07b6P+hh0RXQmZhc7WY8kvWZcesge3K4WA3f2cdOxbYurVmc6/+Ut2s06dr7+ENCvJzQX4m4+vYX9hcLSY7O1t0CaZh1sBVfnXwkCFDvH5Wm6wbN2r7D2dkXP73xo2V3zYyEpg6VVtUJFp1sk6bpn3cnIU+H6TWZHsd+xPnXH3gnCuRb0G/nYIVFBQIW2k/ZIjWsK7o7ZbicGi7T6Wl+f6wAbIezrkSkel8DQebXweQlaVt5m9VzZtrnwzExqoeJd+KQ0S1U9VwsNlCQ7UPL7ey2n6IAAU+nrlazJX7XsqMWQNPdVYHy5K1OlTKCqiXVw82V4sZO3as6BJMw6yBpbrDwTJkrS6VsgLq5dWDC5p84IImosuM3iyCyMq4oImITGH2ZhFEgY7N1WLKf/ag7Jg1MNR0dXAgZ60plbIC6uXVg83VYvLy8kSXYBpmtb7arA4O1Ky1oVJWQL28enDO1QfOuRJZY7MIIrNxzpWI/MYqm0UQBSI2V4txuVyiSzANs1qXns0iAi2rHiplBdTLqwebq8UkJyeLLsE0zGpdelYHB1pWPVTKCqiXVw/Oufogas7V5XIhLCzMtOcTiVmtKS0tDcOGDcPy5ctrtcVhIGXVS6WsgDp5OecqIRVeuG7Maj1G7B0cKFmNoFJWQL28erC5EpEHN4sgMgabq8Wkp6eLLsE0zGotRq0ODoSsRlEpK6BeXj3YXC0mIiJCdAmmYVbrMPKj5Kye1UgqZQXUy6sHFzT5wE0kSCXcLILoMi5oIiLduFkEkfHYXC3G6XSKLsE0zCqekcPBblbN6g8qZQXUy6sHm6vFpKSkiC7BNMwqnj9WB1s1qz+olBVQL68enHP1gXOuJDu9m0UQyYpzrkRUK/4YDiaiy9hciRTEzSKI/IvN1WJSU1NFl2AaZhXD36uDrZTV31TKCqiXVw82V4uJjY0VXYJpmNV8ZgwHWyWrGVTKCqiXVw8uaPKBC5pIRtwsgujquKCJiKqNm0UQmYfN1WJycnJEl2AaZjWPmauDRWc1k0pZAfXy6sHmajEZGRmiSzANs5rHzNXBorOaSaWsgHp59eCcqw+ccyVZcLMIoprhnCsRVYmbRRCJweZKJDFuFkEkBpurxdjtdtElmIZZ/UvU6mAeV3mpllcPNleL6du3r+gSTMOs/iNyOJjHVV6q5dWDzdViYmJiRJdgGmb1H5HDwTyu8lItrx5srkSS4WYRROKxuVpMVlaW6BJMw6zGs8LqYB5XeamWVw82V4vJzs4WXYJpmNV4VlgdzOMqL9Xy6sFNJHzgJhIUaLhZBJExuIkEEQGwxnAwEV3G5kokASsMBxPRZWyuFpOUlCS6BNMwqzGstjqYx1VequXVQ9k51ylTpmD79u2YNWsWunTpUuHnouZcnU4noqKiTHs+kZhVv4KCAs9Za1mZNf5T5nGVlyp5OedaS1u2bMGFCxdEl1EpFV64bsyqnxWHg3lc5aVaXj2Ua65OpxPLly/HxIkTRZdCpIvVhoOJ6DKlmmtpaSmmT5+O+Ph4REZGii6nUg6HQ3QJpmHW2rPy6mAeV3mpllcPpZrrunXrULduXTz00EOiS/EpLy9PdAmmYdbas+JwsBuPq7xUy6tHqOgCzHL48GGsXbsWixcvFl1KlQYMGCC6BNMwa+1YfTiYx1VequXVQ5kz17179+L06dMYMGAAHnjgATzwwAMAgIkTJ2Lq1Kk+7xcXFwebzQabzYZNmzbBZrPB5XJ5fp6enu41VOJ0OissV09NTfXakzMnJ6fC5yLa7Xbk5OR4LmdlZSE1NdXrNklJSXA6nZ7LDocD6enpnssul4v1SV7fqFGjMGzYMAQFBWHIkCGWq8/qvz/Wx/p81edwOGCz2RAfH4/evXtDL2XeilNQUIBffvnF67qEhASMHz8e3bp1qzAHK+qtOC6XC2FhYaY9n0jMWnNBQUEAtNezFc9aAR5XmamSl2/FqYEGDRqgTZs2Xl+AtrTcSoubkpOTRZdgGmatGasPB7vxuMpLtbx6KHPmWpn777/fcptIqPKXIcCsNeHeLCIoKAilpaUGVmY8Hld5qZLXiDPXqy5oys/PR3h4eK0e3OoyMzNFl1CBCi9cN2atPvfq4HPnzhlRjl/xuMpLtbx6XHVYOD4+Hl988YUZtRBRJQJlOJiILrtqc83Pz8frr7+OKVOm4NSpU2bUpLTyq99kx6xX594swr06OBDwuMpLtbx6VHtB086dOzF06FBs3brVn/UoLyIiQnQJpmHWqwuk4WA3Hld5qZZXj6s216effhohISEoKytDQUEBZs6ciUmTJuHkyZNm1KccI95fFSiYtWqBOhzM4yov1fLqcdXmmpCQgEWLFqFdu3ae6/71r38hISEBH374oV+LI1JVIA4HE9Fl1RoWjo6OxsKFCzF8+HDUqVMHZWVlOH/+PFJSUmCz2XDs2DF/16mM8juQyI5ZfQvE4WA3Hld5qZZXj2rPuQYHB2PAgAFYunQpbrvtNs/12dnZ+Otf/4p169b5pUDVpKSkiC7BNMxauUAdDnbjcZWXann1qPUmEps2bcLbb7+NwsJC7YGCgtChQwdMnDgRN9xwg6FFiiBqEwlSWyBtFkEkK1M2kfDl0UcfxZ133om5c+dix44dKCsrw549e/DMM8/glltuqdFjzZkzp7ZlEEklkIeDiegyXR8517RpU0ydOhWZmZmYNWsWXC4XLl68iOzs7Grdv6yszLMROZHqAn04mIgu071x/7lz57Bz504UFRUB0IaHy8rKqvVFFV35MUsyY9bLZFodzOMqL9Xy6qHrzPWf//wn5s+fj7y8PE9TbdmypeezUqnmYmNjRZdgGma9TKbhYB5XeamWV49aNdczZ85g3rx5+PrrrwFow7vBwcGIi4tDQkIC6tSpY2iRKunevbvoEkzDrJryw8EFBfXx6adAv35mVWY8Hld5qZZXjxo3V4fDgYULF6KgoMAztNu6dWtMmjQJMTExhhdIJLMrh4Pz8oCkJOC++4Da7jR38iRQUgI0b25oqYaZORPo0AHo00d0JUT+U+05119++QUvvPACZsyY4WmsISEhePrpp7FkyRI2VoPk5OSILsE0qmbt2RNo2VL7atgwD0Aumje/hJYtgVtuAX78ERgwoHbPc+gQ8NxzQIMGhpRdK1c7rjYbsGgRIMM25Sq9hgH18upRreb64YcfYujQofj22289i5FuvvlmLFq0CAkJCQgN1TV1S+VkZGSILsE0qmb99FPg55+BV15JA9AKy5d/jqNHg/Hzz0BcHNC2LfDaazV/jnPngKefBubOBX6bwgUAnDoFjB4N9OoFdO4MdOoEzJgBFBfrz+WWmws884x2xv3ww/XQpQuQnAz8+mvF24aEAO++CyQmAidOGFeDCCq9hgH18upx1U0knn/+efz73/8GoM2thoaGYvDgwRg4cCBCQkJMKVIEbiJB/lTZZhEFBUDv3kBGBtCsWc0f85lngG7dtO9up04Bjz0GpKQA/+//addt2qQ18dhY4IsvAL3v+jl5EnjwQWDZMsC9eduPPwK33w5cfz3w9dfA735X8X6vvabdbuVKfc9PZDQjNpG46pmr+z2rZWVliImJwZIlSzyflENEtVPZ6uAGDYDt22vXWHftAjIzgaFDva+fOhUYN+5yYwWARx8FRo0C/vd/gVdeqU313t5/X3v+F1+8fF3bttrw948/AqtXV36/UaO0Rn/ggP4aiKymWsPCderUwYgRI7BgwQK0bt3a3zURSc0fm0XMmKE11itnaD77DEhI0L6X9+c/a9/XrtX/3J06aYuvWrb0vv78ee27r4iNGwM9egC//TqIpHLV5tqxY0csXboUTz75JIKDde85QVdht9tFl2AaFbP6Y7OIs2eBjRuBhx+u+LP27bXh5jNnvK+PjNS+G/EhJ/fdpz3+okXaZbvdjuJi4P/+D2jaFHjiCd/3vfde4IMP9NcgikqvYUC9vHpctVumpKSgRYsWZtRCAPr27Su6BNOomLVhw7sArET37hexebPWlMaNA0aO1M4mv/++5o/92WfaIqFOnSr+LD0dOHZMm2Mt7/Bh7fvNN/t+3EuXgHvuAVq3Bg4erH49ffr0xYQJ2ln0hx8CTZr4vu299wL79gXuwiaVXsOAenn14DJfi1HpLU2qZX3rrbcAjEVaWhmOHQtBQoLWXFJSgP37tcVMLVsCb75Zs8fesQPo2lVrsFcKCan8/a6rVmnfR4/2/bjFxVpT/eUXbdVvdHTVdfzwg/Z4TmcMLl3S5lPLz/VW5sYbte/Z2dpq5kCj0msYUC+vHhznJfKzwsJCTJo0CaNGzQFwHAkJg3HsGHD6tPZ2lRYttFW9DRpoi41qau9e7TGq68cftebaqxcwfLjv29Wrpy02On4cuOOOqz9ux47Al18COTnAvHnAXXcBkyYBVW0jHhGh/QFw6FD16ycKBGyuFpOVlSW6BNOokPXIkSNo3bo1ZsyYAeA6TJnSAYD29pS779bedwoA/ftrzfbKs7eyMuC994C//c33cxw9Wv3dnC5eBJ56CujSpXqLmerVq/nq5aysLPTpAwwcqC20mjnT922DgoDwcCA/v2bPYRUqvIbLUy2vHmyuFlMrUcL9AAAgAElEQVTdj+uTgQpZQ0JCcOrUqd/+/b/o2bMZfv0V2L1bWwhUlTVrgPHjtWFj98rbyhQUaA2qOkaO1BrxZ59V/z415T6u7j8U7Hagqs99v/Za4MIF/9Tibyq8hstTLa8ebK4WM7yqcTrJqJC1/PvBGzdujHbt2iEzUzsjvVpzHTgQmD1b2xKx6ueo3m5LL78MuFzARx9d3h6xqqZdHaWlWhNt2VIbngYuH1f3Ge/p05Xv1OR2+rTY7Rr1UOE1XJ5qefVgcyXyo+a/rSY6ceIETpw4gaioKGRmamdrRn3ASESE1qCqsmyZtmp45UrgmmsuX3/lphM1deqUtp3j0aPAli3eP3PXFBamvSWnMoWF2lkr35BAsuFqYSI/iY+PBwAMGzYMvyu3/19mptZYw8KMeZ4bb6y6uW7dCvznP0BqqjbH6XbxIlBUVPVjnz+vne36ejtNZCTQqpXWHJ980vtn27Zp3wcPrnwlM3D5LTjt21ddB1Gg4ZmrxSQlJYkuwTQyZ3U6nVixYgUAYMmSJZ6sTqe2mvZqQ8I10akT8NNPlf/s//5Pa3offaR9zFtMzOWvm24CoqJ8P+7589r7YJs3B777zvftFi3ShoAvXdIuJyUl4eBBbdvD22+vekHTzp1Ao0bArbdePacVyfwaroxqefXgmavFjB07VnQJppE5a/nhYOBy1hMngOuuq3iWp8e99wIvvQTk5VVcNTx4sPZpOfv2VX7fqs4YQ0O1s9KysqrnRB95RLvd5Mna8PDFiy/D4dAWY40fX/UZ+rZt2s5Sgbr5m8yv4cqollcPNleLiarqVEIysmatbDjYnTU21pgtB8u76y5tePbrryt+APnu3bV/3Dp1gG++qd5tO3W6vDEFcG217lNaqs3XLlhQq/IsQdbXsC+q5dUjQP9eJLKmK4eDzRAcDAwbpn06TSBZt07b1P+hh0RXQmQ8NleLcTgcokswjYxZrxwOdvN31rFjtYVLvuZezVTdrNOna+/hLb/IKtDI+Bquimp59WBztZi8vDzRJZhGtqy+VgcDtcu6caP2cXEZGZf/vXFj5beNjNQ+u/W112r8NIarTtZp07SPm+vRw4SC/Ei21/DVqJZXj6DMzMwqdv5UV2FhIfr06YP8/Hw0atRIdDlkcU6n03PWWlbVZrp+NmSI1rAM+jQ7v3A4tN2n0tJ8v0WHSKSzZ88iPDwcmzdvrvVnLvPMlcgAvoaDzZaWBmRlXd4tyYqaNwfeeYeNleTG5moxLpdLdAmmkSVrVcPBbmZlDQ3V3nfaoYMpT1epq2Xt1Cmw51nLk+U1XF2q5dWDzdVikpOTRZdgGhmyVnd1sAxZq4tZ5aVaXj045+qDqDlXl8uFMKP2xbM4GbIG/XYKduLECZ9nrYAcWauLWeWlSl7OuUpIhReuW6Bnrc5wsFugZ60JZpWXann1YHMlqgURm0UQUeBgc7WY9PR00SWYJpCz1nR1cCBnrSlmlZdqefVgc7WYiCt3XpdYoGatyXCwW6BmrQ1mlZdqefXggiYfuIkEVcYqm0UQkf9wQRORyayyWQQRWRubq8U4jf48MgsLtKy1GQ52C7SsejCrvFTLqwebq8WkpKSILsE0gZRV7+rgQMqqF7PKS7W8enDO1QfOuVJ51d0sgogCnxFzrqEG12Rp7733HrZt24bc3FzUq1cP3bp1w/Dhw7kCjqqkZziYiNSk1LDw7t27ERcXh9TUVEydOhWHDh3Cq6++KrossjBuFkFEtaFUc50+fTp69uyJG264AR06dMCoUaOwa9cuFBQUiC7NIzU1VXQJpgmErEatDg6ErEZhVnmpllcPpZrrlfLz81GnTh3UrVtXdCkesbGxokswjdWzGjkcbPWsRmJWeamWVw9lFzQVFxdjzJgxaNeuHWw2W4Wfc0GT2rhZBJG6uIlELZWUlOCNN94AAIwcOVJwNWRF3CyCiPRQrrmWlpbCbrfjyJEjmDFjxlWHhOPi4mCz2WCz2bBp0ybYbDa4XC7Pz9PT0+FwODyXnU4nkpKSvB4jNTUVWVlZnss5OTmw2+1et7Hb7cjJyUFOTg4AICsrq8L8RlJSktebuB0Oh9dG2i6Xy+/1uRlRX05OjiXrGzRoEABtODgzM9OQ+hITEw3//QHWPL7Z2dmWrs/I35+7JqvW52ZUffHx8ZauT8/vz+FwwGazIT4+Hr1794ZeSg0Ll5WVYcaMGfjhhx8wf/58NGnSxOdtRQ0L2+12TJo0ybTnE8mKWf01HGzFrP7CrPJSJa8Rw8JKNdfZs2fjq6++wrRp03Ddddd5rg8PD0dISIjXbTnnqiZuFkFE3ESihjZv3gwA+Nvf/uZ1/Zo1axAVFSWiJLIQbhZBREZRqrlmZmaKLoEsiptFEJGRlFvQZHVXTvTLzEpZ/b062EpZ/Y1Z5aVaXj3YXC2mb9++okswjVWymjEcbJWsZmBWeamWVw+lFjTVBBc0qYGbRRDRlbiJBJFO3CyCiPyBzdViyr/ZWnais5q5Olh0VjMxq7xUy6sHm6vFlN/dRnYis5q9OpjHVU4qZQXUy6sH51x94Jyr3LhZBBH5wjlXolrgZhFE5G9srqQUbhZBRGZgc7WYKz+xQWYisopaHczjKieVsgLq5dWDzdVixo4dK7oE/yko8LpodlaRw8FSH9crMKu8VMurB5urxUj9AQIpKcAXX3gumplV9HCw1Mf1CswqL9Xy6qHUxv1kktdfB776quL1x48Db74J5OYC11xjakncLIKIzMQzV4txOByiS9Bv8mRg61bvr2XLgNBQ4KOPPI3VrKxWWB0sxXGtJmaVl2p59eCZq8Xk5eWJLsE/wsKADRuA1q09V5mRVfRwsJu0x7USzCov1fLqwU0kfOAmEnLgZhFEVFNGbCLBM1fyD5cLmDkTOH0aiIzU5lmnTwfCw00rwQrDwUSkJs65WozL5RJdgn7nzgF33w2UlABz5wLJycCf/uS1Uhjwb1arDAe7SXFcq4lZ5aVaXj3YXC0mOTlZdAn6JSZq72n9+9+1y8ePAx9+CHTvfvk277+PT3r1AgYPBhYsMLwEq60OluK4VhOzyku1vHpwztUHUXOuLpcLYWFhpj2f4fLzgeuuA+67D7j/fuDSJaBBA+Cpp4BmzbTbHDwIbNoE14gRCAOAmBhg1SrgrrsMKSE+Ph4rVqzAsGHDLHHWCkhwXGuAWeWlSl7OuUoo4F+4+/cDFy4ASUnAH/9Y+W1++AGYORNh48Zpl7t2BbZvN6S5Wm042C3gj2sNMKu8VMurB4eFyVgNGmjffxuW9bJ7t/b9oYdwbMkSfPjhZu3y0aPa2asBrDYcTERqYnO1mPT0dNEl6NOhA9Cp0+VGCgBFRcALLwAXL2qXr70WCfPfRt++j+LAunVAWRnQp4/up7by6uCAP641wKzyUi2vHhwWtpiIiAjRJei3YYM2LPzvfwMhIUBpKTB6NNCiBQBg165d+PzzT1Av6E4cH2vDzbv+BQTr+zvPqsPBblIc12piVnmpllcPLmjygZtI+M/QocOxfPnbeBnAYgDZJ07gd4WFQJs2tX5MbhZBREYxYkETh4XJdElJiZjYqBE+BrB48WL8rqAA2Lmz1o9n5eFgIlITm6vFOJ1O0SX4XbuTJzHt3Dl8A6DviBFAdDTQqlWtHsvqw8FuKhxXN2aVl2p59WBztZiUlBTRJfjf3Xdj6KBBCAK0xUxlZcA999TqoQJldbASx/U3zCov1fLqweZqMdOmTRNdQsAIpOFglY4rs8pLtbx6sLlSQAqU4WAiUhObKwWkQBkOJiI1sblaTGpqqugSLC+QhoPdVDquzCov1fLqweZqMbGxsaJLsLRAHQ5W6bgyq7xUy6sHm6vFdC//sWxUQaAOB6t0XJlVXqrl1YPNlQJGIA4HE5Ga2FwtJicnR3QJlhSow8FuKh1XZpWXann1YHO1mIyMDNElWFKgDge7qXRcmVVequXVg83VYiZNmiS6BMuRYThYpePKrPJSLa8ebK5kaYE+HExEamJzJUsL9OFgIlITm6vF2O120SVYhgzDwW4qHVdmlZdqefVgc7WYvn37ii7BEmQbDlbpuDKrvFTLqwebq8XExMSILsESZBsOVum4Mqu8VMurB5srWY5Mw8FEpCY2V4vJysoSXYJQsg0Hu6l0XJlVXqrl1YPN1WKys7NFlyCUbMPBbiodV2aVl2p59WBztZjhw4eLLkEYmYeDVTquzCov1fLqweZKliDrcDARqSlUdAEirF69Ghs2bEBBQQG6dOmCxMRENGnSRHRZSrPUcHBBAdCggegqiCiAKXfmumXLFrz77rsYM2YMFixYgMLCQrzyyiuiy/JISkoSXYLpLDccnJICfPGFoQ+p0nFlVnmpllePoMzMzDLRRZjp2WefRbdu3TBs2DAAwLFjx/DUU09hyZIluPnmmz23KywsRJ8+fZCfn49GjRqZVp/T6URUVJRpzyfKkCFDsHLlShw/ftxz1lpWZvJL8fXXga++qnj98ePAiRNAbi5wzTWGPJUqxxVgVpmpkvfs2bMIDw/H5s2bUb9+/Vo9hlLDwsXFxTh48KDXpPz111+PqKgo7Nmzx6u5iqLCC7c8ocPBkydXvO7YMeDRR4GPPjKssQJqHVdmlZdqefVQalj47NmzKC0tRePGjb2uj4iIQF5enqCqyMzh4E8++RyrVq1CSUlJ5TcICwM2bAC6dDGlHiKSk1JnrqYPO9aCw+FA7969RZfhd+fPn/f8+4EHHsD7779vyvMmJAzDhQslSE5+DW+88SKeeuop7xs0aaJ9GUyV4wowq8xUy6uHUmeu4eHhCA4OxpkzZ7yuz8vLQ0RERKX3iYuLg81mg81mw6ZNm2Cz2eByuTw/T09Ph8Ph8Fx2Op0VJv1TU1O9djbJycmp8OkSdrsdOTk5njPorKwspKamet0mKSkJTqfTc9nhcCA9Pd1z2eVy+b0+N731tWvXznP9wIEDTfsqKipEaelg5OYWYciQofjHu+8Cr70GPP88Lr76KrbHxsJVbojaqN/fihUrDP39GV2fkcf35MmTlq7PyN+f+79Xq9bnZlR98+fPt3R9en5/DocDNpsN8fHxhvwBoeSCpjvuuAN//etfAQDHjx/HX/7yF8ssaCL/uv32u5GdnYWBAwfh1QljET1sGNCnD/Dyy9oN1q8HgoOBP/9ZaJ1EJI4RC5qUOnMFgH79+mH9+vX46quvcODAAcycOROdOnWyxGIm8r+PP/4H9u/PwapV7yB64ULtPa1//7v2w+PHgQ8/BLp3977Tf/8L/PZ2ISKi6lBqzhUAHn74YZw5cwbz5s3zbCIxfvx40WV5uFwuhIWFiS7DFCKyulcnIz8fWLkSuO8+YMYM4NIlbeOI2bOBZs0u3+Gtt4BvvwUOHdL1vDyuclIpK6BeXj2UO3MFgKeeegrr16+Hw+HAG2+8YandmZKTk0WXYBqhWffvBy5cAJKSgEmTtLfljB3r3VgB4LnnDDlr5XGVk0pZAfXy6qHcnGt1iZpzVekvQ6FZ9+4Ffv97YN8+oNziKgDA7t3Arbdevvzll9qc7Jdf1vrpeFzlpFJWQJ28nHOVkAovXDehWTt0ADp10hqpW1ER8MILwMWLhj8dj6ucVMoKqJdXD+XmXIk8NmzQhoX//W8gJAQoLQVGjwZatBBdGREFOJ65Wkz5923JTnjW6Ghg7VptyPfFF7VVw35qrMKzmohZ5aVaXj3YXC3G12YWMmJWOTGrvFTLqwcXNPnATSQIALBsmTZ8vHMnMHw4MHQo0Lat6KqIyI/4qThE/paQoH0REdUAh4UtpvzembJjVjkxq7xUy6sHm6vFpKSkiC7BNMwqJ2aVl2p59eCcqw+ccyUiUhM3kSAiIrIgNlciIiKDsblazJUfECwzZpUTs8pLtbx6sLlaTGxsrOgSTMOscmJWeamWVw8uaPKBC5qIiNTEBU1EREQWxOZqMTk5OaJLMA2zyolZ5aVaXj3YXC0mIyNDdAmmYVY5Mau8VMurB+dcfeCcKxGRmjjnSkREZEFsrkRERAZjc7UYu90uugTTMKucmFVequXVg83VYvr27Su6BNMwq5yYVV6q5dWDzdViYmJiRJdgGmaVE7PKS7W8erC5EhERGYzN1WKysrJEl2AaZpUTs8pLtbx6sLlaTHZ2tugSTMOscmJWeamWVw9uIuEDN5EgIlITN5EgIiKyIDZXIiIig7G5WkxSUpLoEkzDrHJiVnmpllcPzrn6IGrO1el0IioqyrTnE4lZ5cSs8lIlL+dcJaTCC9eNWeXErPJSLa8ebK5EREQGY3O1GIfDIboE0zCrnJhVXqrl1YPN1WLy8vJEl2AaZpUTs8pLtbx6cEGTD9xEgohITVzQREREZEFsrhbjcrlEl2AaZpUTs8pLtbx6sLlaTHJysugSTMOscmJWeamWVw/Oufogas7V5XIhLCzMtOcTiVnlxKzyUiUv51wlpMIL141Z5cSs8lItrx5srkRERAZjc7WY9PR00SWYhlnlxKzyUi2vHmyuFhMRESG6BNMwq5yYVV6q5dWDC5p84CYSRERq4oImIiIiC2JztRin0ym6BNMwq5yYVV6q5dVDmea6efNmjBo1Cn369EG/fv3w0ksv4dixY6LLqiAlJUV0CaZhVjkxq7xUy6uHMnOur7/+Ojp27IhbbrkFALBkyRIcO3YMy5YtQ2hoaIXbc86ViEhNRsy5Vuwqkpo8ebLX5QkTJqB///44fPgwoqOjBVVFREQyUmZY+Er5+fkAwLNSIiIynJLNtaysDGlpaejatSsiIyNFl+MlNTVVdAmmYVY5Mau8VMurR8APC8+ZMwebNm3y+fPY2FjMmzfP67qFCxfip59+wptvvunv8mosNjZWdAmmYVY5Mau8VMurR8CfuT777LNYu3atz6+XX37Z6/ZLlizBl19+idmzZ6Np06ZXffy4uDjYbDbYbDZs2rQJNpvN6zMN09PT4XA4PJedTieSkpK8HiM1NRVZWVmeyzk5ObDb7V63sdvtyMnJQffu3QEAWVlZFf5KTEpK8loK73A4vLYjc7lcfq/PzYj6unfvbun6AON+f//85z8tXZ+Rv7/bbrvN0vUZ+ftz//dq1frcjKpv7dq1lq5Pz+/P4XDAZrMhPj4evXv3hl7KrBYGgBUrVmDjxo2YN28eWrduXeVtuVqYiEhN3KGpBlavXo01a9YgKSkJDRs2xOnTp3H69GlcvHhRdGleyv/1JjtmlROzyku1vHoo01w//PBDXLhwAZMmTcITTzzh+frPf/4jujQvGRkZokswDbPKiVnlpVpePZQaFq4JDgsTEamJw8JEREQWxOZKRERkMDZXi7lyibrMmFVOzCov1fLqweZqMX379hVdgmmYVU7MKi/V8urB5moxMTExokswDbPKiVnlpVpePdhciYiIDMbmajHltwmTHbPKiVnlpVpePdhcLSY7O1t0CaZhVjkxq7xUy6sHN5HwgZtIEBGpiZtIEBERWRCbKxERkcHYXC3mys8alBmzyolZ5aVaXj045+qDqDlXp9OJqKgo055PJGaVE7PKS5W8nHOVkAovXDdmlROzyku1vHqwuRIRERmMzdViHA6H6BJMw6xyYlZ5qZZXDzZXi8nLyxNdgmmYVU7MKi/V8urBBU0+cBMJIiI1cUETERGRBbG5WozL5RJdgmmYVU7MKi/V8urB5moxycnJokswDbPKiVnlpVpePdhcLeb+++8XXYJpmFVOzCov1fLqweZqMZmZmaJLMA2zyolZ5aVaXj3YXImIiAwWKroAqyor096hdPbsWVOf9z//+Y/pzykKs8qJWeWlSl53RncfqA02Vx+KiooAAK1atTL9ucPDw01/TlGYVU7MKi+V8hYVFaFBgwa1ui83kfChtLQUp06dQt26dREUFCS6HCIiMklZWRmKiorQtGlTBAfXbvaUZ64+BAcHIzIyUnQZREQkQG3PWN24oImIiMhgbK5EREQG47CwRW3evBlbt27FoUOHEBoaik6dOmHEiBG4/vrrRZdmiNWrV2PDhg0oKChAly5dkJiYiCZNmoguy1Dvvfcetm3bhtzcXNSrVw/dunXD8OHDERERIbo0v5syZQq2b9+OWbNmoUuXLqLL8Yv9+/dj8eLF2LNnD6655hp06dIFL7/8suiyDFdQUICFCxciKysLRUVFiI6OxjPPPIPY2FjRpem2bds2bNy4Efv370dhYSE+++wzhISEeH6em5uLOXPmYM+ePWjcuDEGDx6Mhx9+uFqPzTNXi8rOzkavXr2QkpKC2bNno7i4GC+88AIuXbokujTdtmzZgnfffRdjxozBggULUFhYiFdeeUV0WYbbvXs34uLikJqaiqlTp+LQoUN49dVXRZfld1u2bMGFCxdEl+FXhw8fhs1mQ8eOHbFo0SIsWLAAPXr0EF2WX7z11lvYt28fpk6diqVLlyImJgbJyck4d+6c6NJ0u3DhAjp37oyBAwdW+NmlS5eQlJSE8PBwLF68GIMGDcKcOXPw3XffVeuxeeZqUZMnT/a6PGHCBPTv3x+HDx9GdHS0oKqM8cEHH+CJJ57AvffeCwCYOHEinnrqKRw4cAA333yz4OqMM336dK/Lo0aNwqhRo1BQUKB7sYRVOZ1OLF++HAsWLMCTTz4puhy/SUtLwz333IOhQ4d6rmvdurXAivxn79696NOnD37/+98DABISErB+/Xrk5uZ6rgtUPXv2BAB8//33FX72zTff4OTJk3j77bdRr149tGnTBtnZ2fjggw+qNRrDM9cAkZ+fDwAB/9myxcXFOHjwIG6//XbPdddffz2ioqKwZ88egZX5X35+PurUqYO6deuKLsUvSktLMX36dMTHx0u90r6kpATffvstoqKiMG7cODz++OMYP348Dh48KLo0v7jllluwfft25Ofno6SkBB9//DGaNWuGNm3aiC7Nr3JychATE4N69ep5ruvcuTP27t1brfuzuQaAsrIypKWloWvXrgH/P62zZ8+itLQUjRs39ro+IiICeXl5gqryv+LiYqxcuRK9e/f2mtORybp161C3bl089NBDokvxq/z8fLhcLqSnp6NHjx6YPn06IiMjkZiYiIKCAtHlGW7MmDEIDw9Hv3790KtXL6xevRrTpk2T9o9EtzNnzlRYH1GT/09xWNhkc+bMwaZNm3z+PDY2FvPmzfO6buHChfjpp5/w5ptv+rs8v9OznVigKikpwRtvvAEAGDlypOBq/OPw4cNYu3YtFi9eLLoUvystLQUA/PGPf8Rjjz0GAEhMTERcXBx27NiBXr16iSzPcOvXr8fPP/+MWbNmoVGjRvjkk08wefJkvP3220rt1lRTbK4me/bZZzFo0CCfP7/mmmu8Li9ZsgRffvkl5s+fj6ZNm/q7PL8LDw9HcHAwzpw543V9Xl6elKtoS0tLYbfbceTIEcybN0/av/b37t2L06dPY8CAAV7XT5w4Effffz+mTJkiqDLjuV/D5bdGDQ0NRfPmzXHy5EmBlRnvwoULeOeddzBr1izP6uC2bdsiKysLn3/+OR5//HHBFfpP48aNceTIEa/ravL/KTZXkzVo0KDai1lWrFiBjz/+GPPmzUPz5s39XJk56tSpg+joaHz//feeRQHHjx+H0+kM+MURVyorK8PMmTOxZ88ezJ8/P+Dny6ty9913o3379l7XJSQkwGazoVu3boKq8o9rrrkGbdu2xdGjRz3XlZSUwOl04rrrrhNYmfEuXbqES5cuVdgCMCgoyHMGL6uYmBisXbsWRUVFnj+Kd+3ahQ4dOlTr/myuFrV69WqsWbMGr776Kho2bIjTp08DABo2bFjh7DbQ9OvXDwsWLEC7du3QvHlzLFy4EJ06dZJqpTCgTQHs3LkT06ZNAwDPMQwPD5du3tXXH41RUVEBv06gMv3798fMmTNx2223ISYmBhs2bAAA3HnnnYIrM1b9+vVx6623YuHChRg9ejTCw8OxdetWOJ1OdO3aVXR5up09exYnT570/KF04MABhISEoEWLFujWrRuaNWsGu92OIUOGYO/evfjiiy8qvAvAF27cb1H/8z//gxMnTlS4fu7cubjtttsEVGSsVatWeW0iMX78eOk2kbj//vsrvX7NmjWIiooyuRrz3X///VJvIrFu3Tr84x//wLlz59C+fXuMGTNGyhW0v/zyCxYvXoxdu3ahqKgIrVu3Rnx8PLp37y66NN22bt0Ku91e4Xr3/2ePHDni2USiSZMmGDRoEB555JFqPTabKxERkcH4VhwiIiKDsbkSEREZjM2ViIjIYGyuREREBmNzJSIiMhibKxERkcHYXImIiAzG5kpERGQwNlciIiKDcW9hIsXMmzcPGRkZAIDIyEi88847qF+/frXum5KSgo0bNwIAmjVrhnfeeafaH0RBpBKeuRIp5tlnn/V8eot739jq+OGHHzxNGQCef/55NlYiH9hciRRTr1492Gw2z+WPPvoI33//fZX3KS4uxsyZMz0fdt+jRw/pPgGGyEhsrkQK6tatG3r27Ang8ufOulwun7dftmwZcnNzAWgfmTd69GhT6iQKVGyuRIoaNWoUGjduDAA4duwYli1bVunt9u3bh3/84x+ey6NHj0ZERIQpNRIFKjZXIkU1atQIY8aM8Vxev3499u7d63WbS5cuYcaMGSgtLQWgfRj4Aw88YGqdRIGIzZVIYffddx/uueceAEBpaSlmzJiBixcven7+3nvv4b///S8AoH79+hg3bpyQOokCDZsrkeLGjRvnWfV76NAhrFq1CgDw3//+1/NvABgxYgQiIyOF1EgUaNhciRTXpEkTjBw50nN51apVOHDgAGbMmIFLly4BAG6//XY88sgjokokCjhBmZmZZaKLICLxJkyYgH/9618AtCHgwsJCAEBYWBjS0tJw/fXXiyyPKKDwzJWIAACJiYkICwsDAE9jBYCEhAQ2VqIaYnMlIgBAVFQU4uLivK7r0MCfwdwAAAJXSURBVKEDnnjiCUEVEQUuNlciAqDtwvTll196XRcUFCSmGKIAx+ZKRACAFStWeHZhctuzZ49no34iqj42VyLCjz/+iPT0dM/lzp07e/69dOlSnDhxQkRZRAGLzZVIcSUlJZgxYwZKSkoAAHfffTfsdjtuuukmAEBRURHmzp0rskSigMPmSqS4NWvW4MCBAwAu78IUGhqK8ePHIzhY+1/EN998g88//1xkmUQBhc2VSGFHjhzBypUrPZdHjhyJpk2bAtBWCj/++OOeny1YsAD5+fmm10gUiNhciRTl/qg5917Cle3ClJCQgKioKABAXl4eFi5caHqdRIGIzZVIUR988AF2794NALj22muRmJhY4TZ169b1+mD1Tz75xLOLExH5xuZKpCCn04mlS5d6Lg8dOhQtWrSo9LZdu3ZFr169PJfnzJlT5QerExGbK5GSZs+ejaKiIgBA+/bt0b9//ypv/9xzz3k+WP348eM+P1idiDRsrkSK2bp1q2doNzQ0FBMmTEBISEiV92nUqBFGjRrlubx+/Xrs27fPr3USBTI2VyKFnD592mtR0sCBAxEdHV2t+/bo0QN/+MMfAGgfrD5z5kzPe2OJyBubK5FC5s2bh3PnzgEAbrjhBjz99NM1uv/zzz+PevXqAQAOHjyI999/3/AaiWTAz3MlIiIyGM9ciYiIDMbmSkREZDA2VyIiIoOxuRIRERmMzZWIiMhgbK5EREQGY3MlIiIyGJsrERGRwdhciYiIDMbmSkREZDA2VyIiIoOxuRIRERmMzZWIiMhgbK5EREQG+/9X2SWO/P5XzAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g = VectorGround()\n",
    "g.add_v(0,0,2,3,font_size=20)\n",
    "g.add_v(0,0,4,6,font_size=20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### we can see from geometric graphic that the vector are linear depends\n",
    "#### but lets to check if exists $\\gamma_i$ $\\gamma_2$ ,<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\gamma_i.\\vec{r}_1 + \\gamma_2.\\vec{r}_2 =$ <br> $\\gamma_1.(2.\\vec{e}_1+3.\\vec{e}_2) + \\gamma_2.(4.\\vec{e}_1+6.\\vec{e}_2) = $ <br>$ (\\gamma_1.2+\\gamma_2.4)\\vec{e} +(\\gamma_1.3+\\gamma_2.6)\\vec{e}=0 $ <br>\n",
    "$\t\\Rightarrow$ <br>\n",
    "$(\\gamma_1.2+\\gamma_2.4) =0 $ <br>\n",
    "$(\\gamma_1.3+\\gamma_2.6) =0 $ in matrix form : <br>\n",
    "$$\\begin{pmatrix} \n",
    "2 & 4 \\\\\n",
    "3 & 6 \n",
    "\\end{pmatrix}*\n",
    "\\begin{pmatrix} \n",
    "\\gamma_1 \\\\\n",
    "\\gamma_2\n",
    "\\end{pmatrix}=\n",
    "\\begin{pmatrix} \n",
    "0 \\\\\n",
    "0\n",
    "\\end{pmatrix}\n",
    "\\quad$$\n",
    "\n",
    "the above eqation has a decision only when $\\det{A} = 0$\n",
    "$$\\det{A} = r_{11}.r_{22} -r_{12}.r_{21} =  2*6 - 3*4 =0$$ \n",
    "#### from above theore $\t\\Rightarrow$ $\\gamma_i$ $\\gamma_2$  exists and $\\gamma_1=-2.\\gamma_2$ and moreover we can and from $|\\vec{r}_1|=\\gamma |\\vec{r_2}|$ we can calculate that $\\gamma=r_{11}/r_{21}$=1/2\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Threorem 2:\n",
    "##### if vectors $\\vec{r}_1,\\vec{r}_2,\\vec{r}_3$ satisfy $\\vec{r}_1 \\perp \\vec{r}_3 $  and  $\\vec{r}_2 \\perp \\vec{r}_3 $  \t$\\Rightarrow$  $\\vec{r}_1$ and $\\vec{r}_2$  are linear dependent\n",
    "#### Proof :\n",
    "if $\\vec{r}_1\\perp\\vec{r}_3$ then the angle between the both vector $\\alpha$=90  and moreover the dot product  $<\\vec{r}_1|\\vec{r}_2>=|\\vec{r}_1|.|\\vec{r}_2|.\\cos{90}=r_{ki}.r_{li}=0$ <br>\n",
    "in matrix $r_{ki}.r_{li}=0$ in matrix form form we ca write :\n",
    "\n",
    "$$\\begin{pmatrix} \n",
    "r_{11} & r_{12} \n",
    "\\end{pmatrix}*\n",
    "\\begin{pmatrix} \n",
    "r_{31}\\\\\n",
    "r_{32} \n",
    "\\end{pmatrix}=0\n",
    "\\quad$$\n",
    "\n",
    "\n",
    "$$\\begin{pmatrix} \n",
    "r_{21} & r_{22}  \n",
    "\\end{pmatrix}*\n",
    "\\begin{pmatrix} \n",
    "r_{31}\\\\r_{32}\n",
    "\\end{pmatrix}=0\n",
    "\\quad$$  \n",
    "\n",
    "###### their union\n",
    "$$\\begin{pmatrix} \n",
    "r_{11} & r_{12}  \\\\\n",
    "r_{21} & r_{22} \n",
    "\\end{pmatrix}*\n",
    "\\begin{pmatrix} \n",
    "r_{31}\\\\\n",
    "r_{32} \n",
    "\\end{pmatrix}=0\n",
    "\\quad$$  \n",
    "#### from theorem 1$\\Rightarrow$ $\\vec{r}_1$ and $\\vec{r}_1$ are linear dependent\n",
    "#### ! when $\\vec{r}_1$ , $\\vec{r}_2$ ,$\\vec{r}_3$ $\\in$ $L^2$ then $\\vec{r}_1$ , $\\vec{r}_2$ lie one same line and when$\\vec{r}_1$ , $\\vec{r}_2$ ,$\\vec{r}_3$ $\\in$ $L^3$ then $\\vec{r}_1$ , $\\vec{r}_2 $ are linear independent then $\\vec{r}_1$ , $\\vec{r}_2$  form plane with vectors  $\\vec{z}=a.\\vec{r}_1+b.\\vec{r}_2 $ wich is pependicular to $\\vec{r}_3$ a,b is just a scallar if demension L>3 they can form hyperplane "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "language": "python",
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