Tutorial 1 & PCA.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numpy import linalg\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn.decomposition import PCA\n",
    "from sklearn import preprocessing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.dot(np.random.random(400).reshape(200,2),np.random.random(4).reshape(2,2))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(200, 2)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x18be84eebc8>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x[:,0],x[:,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x= preprocessing.scale(x)\n",
    "plt.scatter(x[:,0],x[:,1])\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "PCA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PCA(copy=True, iterated_power='auto', n_components=2, random_state=None,\n",
       "    svd_solver='auto', tol=0.0, whiten=False)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pca = PCA(n_components=2)\n",
    "pca.fit(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.70710678 -0.70710678]\n",
      " [-0.70710678  0.70710678]]\n"
     ]
    }
   ],
   "source": [
    "print(pca.components_)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.6459901  0.36406015]\n"
     ]
    }
   ],
   "source": [
    "print(pca.explained_variance_)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Linalg Eig"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "covar = np.cov(x.T)\n",
    "w, v = linalg.eig(covar)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.6459901 , 0.36406015])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.70710678, -0.70710678],\n",
       "       [ 0.70710678,  0.70710678]])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Linalg SVD"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "u, s, vh = np.linalg.svd(covar)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.70710678, -0.70710678],\n",
       "       [-0.70710678,  0.70710678]])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.6459901 , 0.36406015])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.70710678, -0.70710678],\n",
       "       [-0.70710678,  0.70710678]])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vh"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "PLOT"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ek9q//zts3Rrz1ff3z3//3nuhuTnnsUjG7HXtk2sUGRUnMhuGCL5S6htKqWGl1LEM7yul1FeUUn1KqaNKqeuMaFeoPpY9+KccYM880R+9aRevvvWdvOJsmie8pfS1G/UjT1iu/Rf9HB/wMhOO4rJbUUrN/j/pD6f923wnnnzGI+9JbetW+MhHYPPm+e+tWhVz5xSIWYLkSxmj0jK/CXwV+McM778DuDL+7wbgb+P/C0uNq6/Gc+e7Ofjt3ezmIE+zg0d5J7/xjJPHfnADe1ob5v1Jucvj5ktyeudKqyW2YbNvBrvVQmOtnWA4CjpWkyUT+eSjl7Rg2tq18L73wd/8zfz3vvAFqCl8zM28dWShpKa8bmyro3do2rTrOoza4vAJYHyBQ94N/KOO8TTQqJRqN6JtoQr5whdocAQ5yG62qacB+M+ZXdzScQTviHfe4WYPqCVbrkrBqqZamuvs+GbC+IIRnHYrG9vcbF5pzA+/pONx/jx0dcGpU7HnbfFzbtoEv/VbRZ16sWXCpLqozoz6+OJPTnJ6xGdKlxWAMmovcaXUOuDHWustad77MfBFrfVT8ec/Az6jtX42zbF7gb0AbW1t1+/fv9+Q/hWKz+fDXeRqwsVI0ePS3w8jI4Su2sQrPRH81ALgVlNcubUOi22uLRIIRfAGwsyEozhsFhpcttlqhpWm/6Ifu9WCUmCPBvFrO9MzEbSObUgS1ZpIVNPidhjW55KMRygEJ09CMF4Go7U1lm/f2wtXXAGewicsn8+HzVnDqG8Gq0VhUaok41JOhqeCRKIaqyVW7dMXDM8+dztjd3OJ58vrM68wN1pjdu7c+ZzWelu698q10laleS3tTKO13gfsA9i2bZvu6uoqYbey093dTaX7YEaKHpfR0Vi2xz330Ln+ArduucDR6HYAbnIc4rH+bTSkce8kY5ayCA8e6o25WFx2VgVOM+Razxm/j0FvkDWWWlPe2s8j1bL/vd+DT34SolH45jdjj1W6n3FuJK4Xs3xnRvDpf32Jdo9rtrzzT0+9htthZXomyq5NsTujqNYMTgZ44NevzniecmpMuQS/H0hO3O0ABsrUtmBGWlrgK18BoKNzFY8eg1u3/JKj0e1x984hHuu/IaPoL1QWASirqCRXfVwJTPpDWCwW/vRdmxZsN5v4lU0c04n9V78KStEz7OPx3/48Z//tqCF9MEvtHCNIjaU0uOxM+kM0mDTWBOVLy3wE+K14ts6NwKTWerBMbQtmpbZ29mFM9Fex1fJLYGGfPmTO+Pj24bOGpf7lmjOenAkTikRzSu/MlqJYthTGhcQ+3ocxbTOtT7oSJK6LE4OTPP3qGKdHfES1ZkWDk+lgmBX1TlPGmsC4tMzvAIeBq5RS/Uqpu5RSH1NKfSx+yGPAKaAP+L/A7xnRrrC4yEf0M6UmvnB+wpDUv3wFN5He2dFUk1N6Z7YUxbKkMC4g9rn0wayLqEpJ8nXx+hUNXLXCTe+wj55BL+ta3Hz2HVexvtVt2sVbhrh0tNa/meV9DXzciLaExUeq6+Irv2jg93cu7N7JlJqoUIbk7KdWT0z8b9Q2d9lSFEuewphF7LP1oVSVRs3u48+lquatlexgFmSlrVBR0lnSPxwM8JVfNCxo6WdKTbx2tceQ1L9SV0/MlqJY0hTGHMQ+Wx9KcQdi9pW4UP1VNUXwhYqSSTieD1oWdO9kWkH6wR1rDclRL3XOeLZc+kJy7XNyseQo9tn6UArhq4aVuNW+lkAEX6goCwlHNp9+clmEPVvaOHBsiK8/dYYau4VQODLPj5qPz7nUi72ylTzItyRCTtZxiti/+Ot38Olf++88+PgracdioT6kE75zY9OcG79UsE8/07VwwkSxArMvAsyG7HglVJRsZQIypWz+1bOv58nX/FyY8OOwKoa8QVY319Lucc1uhp0skPn6nBNil+xPNmKXrdQ2FjpfPimMWWMOKWL/1Nvfx+N33Ut7jX3BscjUh9QNyM+NTfP8uQmuXd2YdXwz+enTXQvnxqY5P+ZnVWNtxXclg/JcF6VEBF+oKKnCkRDr5P1q04n+3dcd5G3faKK9zcMTvSNMBcKs8DixKHvaAGshQdhK5oznG7xcMMibxrJ//K578dQ6gMIC0qnCN+gNcu3qRtwuG0dOj+MLhLFbFd8+fJb/ddvWOZ8r08Sb7lo4OeRjY5u7ZMHzQqjmtQTi0hEqSq6ui1T3zpHwbn7+kYtEJi8RimjcTit9w5e3REz1J1dTsK2Q4GUm3/JVMxPzfPb/9MH/SX2SFQ2FjUWyS21Ncy21TivPn5sgGIrgdlpBa57qG5vT74X89OmuhY6mGta21BXdVyGGWPhCxcnVYkpY+rdsPsKv9A08Hd4Ndxxk1f+yEnI58QZCs8emBtKMrrhZyvTBQu5G0lnH6vx5PvTl34WzZ2IHxQO0qx5/xfDqo6saa+g+OYzTZrlcF0cpmmpjYn5tvKls6aap18Js2YoyrF41e0qoEYiFL5ia1EDrVKOb9+8L8wYV2y7x6fBu+j8XYXpsCofVkjGQZmSwrdTpg4XcjaRax6umRrn3y7+LI0XsUaokgcc9W9q4eCmE1hqtNYFQhGA4yqaV9XP6nW+WS7mCpNWQEmoEIviCacn0I7xm+zre/OWpWdE/Et6N78+sXFVnzegWMnJ3q1zTBwtdiVpo6t+si+XGZv775z+SVuwTxxm901dnu4c3X7EMpRRTwTAuu5Xr1zbitNnm9DtfAS/X1oPVkBJqBOLSEUxLJtdG79A0d9/xRlZ6jsHemHvnSGQ3to8d4rH+zRkLrhkVbFvILXFta+x5MStRcwlkZyRLbZxSuiw+uGPt7GdO7ffQyVitxEKyXMoRJE18pyNTAfpGpvEFwrid1jmF0BYDYuELpmUh10Znu4f7PnoTjx3vyLngmlHkYoEXYzEWbNXmUAitlC6LXPttxr1jVzXWcHZ0ek7Q2RsI03/Rv6jcOmLhC6Yll0BrIaWVFyIXK3ghCzxhyRZbCydvqzaPQmhQuvTGak1Z3LOljXv2vwYKnDYLwXCUQCiC02rhkw+/xNs3rVgUQVyx8AXTkqu/N9/SypnI1QrOxZIt6xL8HAuhVUtaaiXobPewelkNDS4bU/GdqxTgtFtAs2iCuGLhCyXBCH9xrv7ensFJDvRP82sPTKE/FfPpF2Lp52MFZ7Nki/LD50MehdDMvBG8GdjU7pkdo6dPjc1uXVhfYzPFgi8jEMEX8iIh5K0+Pw8e6k0r5EaWzs0mrMltrd3cwa0PvQafKEz0jSxJnMtkVfSkmGchtLJMQCWk1EHn5DGa9M/gtFoIRjSbV8auncVwRyQuHSFnEuJ6esTHTDjKo0cHuWf/Szx69MKc48qZ4pba1vKN7dz6UGA2ZTMf947RbpiFgpNFB1HzEPtEX8qR3lgqksfLboXuk8P8zref53PfP2qYmyV5jCzKAkpx3ZpGWutjRsBiuCMyaserPUqpk0qpPqXUZ9O8/9tKqRGl1Ivxfx81ol2hvBw4NkQkEqV32IdG01xnBwUP/fzVOT+6cvqL07XVcuUKuv63L2+ffjkrIRY1KeYp9gnMmB2TK4nxCkUivHAudq011dg4PuA11LeeGKP//b6tbGh147BZq7IqZiaKFnyllBX4a+AdwCbgN5VSm9Ic+l2t9TXxf18rtl2h/FyY8POaN4DTZkEphVKKBpeNUCQ6R6jKGbDM1Nb6LavzDuSW0woueFIsUOyrncR49Q1Pz5ZvcNqthCK6JHeP1X5HlAkjfPjbgT6t9SkApdR+4N3ACQPOLZiIVY01vHhuImbZxwmGoyyrc8wRqnL6ixdqq6Pdk3fKZrnSCgsKoi5RsYfL4+UNhKh3xmQrGI7idtmKvnvMFBuo1hTThVCx7WaLOIFS7wX2aK0/Gn9+J3CD1vrupGN+G/hzYAToBe7RWp/PcL69wF6Atra26/fv319U/4rF5/Phdrsr2gezEAhFOH8x9sNqtkeYCFuJanDZLDjtVpbXO+cc6w2EmQlHcdgsNLhsl4tqlaBfC7UVCoR45XgIP7UAuNUUV26tw2IzPoSV6/USCEUY9c1gtSgsShHVmkhU0+J2pB+nUAhOnoRgMPa8tRXWrDG495f7ZvR3V+zvKDFewXAErWNzXFRDrcOKRSmsFjXn+sv3vDl/DyXAaI3ZuXPnc1rrbeneM8LCT2depM4i/w58R2sdVEp9DPgW8NZ0J9Na7wP2AWzbtk13dXUZ0MXC6e7uptJ9MBOPHr3AQz9/ldvavTx+sZl2jwuLtrB3h7lvdzvXXuDWLRc4Gt0OwE2OQzzWv62gxVkLkc/1knPWSTrL/pOfLIlln5z1VF8Xv2MaDxXtzjDid9QzOMk/HT7Lk31jNNXa6Wyvx2WxzdvsJh8ePNTLpHXundakP8QIdu7p2lhUf3OlnBpjhOD3A6uTnncAA8kHaK3Hkp7+X+AvDGhXKDM9g5P0Dk2zurkGm3WKhho761rcs0Jl5vKyRq/INYKcXAZlduOUa0VuIXS2e7j/tq1zrjNPjb2oHaeMTMWtBowQ/GeAK5VS64ELwAeAO5IPUEq1a60H40/fBfQY0K5QRpItv9evaKDGP0a9yz5H7I3KvTeyz6kT0KPHVplK9BekAj77ahBAI33rS21BWtFOTK11GLgbOEhMyB/WWh9XSn1BKfWu+GG/r5Q6rpR6Cfh94LeLbVcoL6lphFaLmpMdYbbyspny3Kca3fHsnct5+rtXHubZX6UNKVWOCgVoy1oSwgRU+6bk+WJI1Epr/ZjWeqPW+nVa6/vjr31ea/1I/PG9WuvNWuurtdY7tdYvG9GuUD6ypRGarVbLQhPQVKObN31pas4mKndfd8I8ol/BbByjN4pJ7AcwPBU0ZR2axZp+mQkprSCkJdUd4rQqpgLhjLe+Zrs1TrgmRn0Bjp6fZNAbQOtYzvboVIDWK9t5/76Jy/X0w7v5xHUHOTjgqax7p8Kpl4XUq09Hqosv4tcVd/FlYjGmX2ZCBF+YRzp//MBkAItS0FxLvctGJKqZDFzOra9ErZaFgsSrGms4M+rjaP8kFy+FcNgUUQ3j0yF+9vIwO1/fimdtK2//q370HxzhWHyP3Hes+ik/uXBjZUTfJHn2RghgavA32QW4VMTVjEgtHWEe6dwha5fV0dbgnL31tVrUHGut3LfG2WrR7NnSxskhH1OBMA6rQgEaRWu9A63hxMAUo74Ax6I21t87xJa4e+e/Qm/nHaueLvkmKvMwidgbRSYX3/GBwrZ9FIxBLHxhHpkyNQYnw9yzK5ab3N19Me+SwUaSLX2ws91DR1MNI1MBoiicVgut9Q5qHVZC4SgXL4U4dsEbmwzam+n4zGuov4i5d/4r9PbyZu9USOxLmUabzsV3dnSa/ot+au1WXvMGePHcBAePvcYn3vY6bt26ypB2hYURwRfmYTZ/fDpySR/cvNLDyFRsZWpi1WQgFMFT62DzygZeOD8BOlbvfPMb1/OmfVOzPv10KZs9g5N8+/BZXjg/gUJx7WoPH9yxtjiRLEDs0wk1kJd4lzqNNtXFF4lqeod8rGhw0Dvsw2mz0FxnxxsI89DPX2VDq1tcPWVAXDrCPKohVS2X9ME9W9pocTuZCoTxz4QJzITxBcI01zm4c8da3r5pBTdsWMaODctorXdRu7Y1Y2nlnsFJvnTgJEdOjeOwKGwWOHxqnAcOFuGWKFDsU11ZDxzs5UsHTuZVarnUabSpLj6rRbF6WQ2XQtHZ4meZiu/lS3I2kLiJFkYEX5hHNaSq5TIpdbZ7+PTujezY0Ew4CjNRzQ0bmvnDPVfR2e5Je45Ai4eHuj3zqmz++1OnGJ+ewe2y4XLYqHHEinaN+oKFiVWBbpx0Qj3qCzI+PZOXeJcjjTa5HPPyeieb2j2MT4dwJtUwSld8Lx/KsTn7YkJcOkJazJ6qlmv6YGI5fr7nePTYhTkrcsN3HKTpjzX1LZd9+k6bBW8glL9YnT/PzJvfguPsGQBe/PU7cN53P505+OzTubKC4QgqpaRVNvGuhNtuz5Y2Dh57DW8gTIPLRjAcJRiOsm5ZbcHtmrkUhBkRwReqFiMmpUznSK29cyS8mxv+5CC2/2+K2uaY6AfDUZw2a35ilUbsf3jXvUw+eSanu1k7byIAACAASURBVKh0Qu20za/qmE28K5FG29nu4RNvex0P/fxVxqZnWFbnYN2yWiwWS8HuwmooBWEmxKUjCBmIif7lTVSORHYz/P/CpTEv/pkwU4EwLW5n7mIVd+Mki/0v7v48nlpHzv7zjW11PH1qjB8fHeDwq6OcGfXR4nbSXOfIK+ZSKbfdrVtX8eD7r+adW1eyZlkd61rcRbW71EpBFItY+IKpqXQFzlRL/5eR3Wz//EGa//QSOzrbc8/SSfHZJ8Q+4bPPxSrtGZzk8Z4RNi5385o3wPh0CK8/zCfe9jo2tLrzXh1bKbedke0uhs3Zy4kIvmBazFKBM53o3/SFQ3ymf11uefppxP6Hd92LJ8lnn4tVmuyvXt8a2zBj0h+id2iaW7euWpI+a6NKQSwVRPAF02KmgFzB9fTTZOM477ufySfPgFJ5WaXir06P2RMMzIT48AXTYrYKnKk+/awbo2dIvexc2ViQ/1z81UKxiIUvFEQgFOHBQ70l9a1XcsVvpthBzpZ+ljz7QqxS8VcLxSIWvpA3PYOTjPpmSr7YpVIrfrMt5slq6ZeoNk41LIgTzI0hgq+U2qOUOqmU6lNKfTbN+06l1Hfj7x9RSq0zol2hMhw4NjRb7raUu1tVSuByKTtwWfTn7pzl914qaSG05NWr9+zaKGIv5EXRLh2llBX4a2AXsQ3Nn1FKPaK1PpF02F3ARa31FUqpDxDbxPz9xbYtVIYLE35WOvNb2ZlPemWlUzFzDY4mds7S/zNWcO3p8G76X/kRwVP9OKHqSxwLiw8jLPztQJ/W+pTWegbYD7w75Zh3A9+KP/434G1Kya+gWlnVWENU6zmvLeRbz6feiRlqo+QaHD1wbCi+c1aYq9QLAISxEcIuYi+YEqVTfrh5n0Cp9wJ7tNYfjT+/E7hBa3130jHH4sf0x5+/Gj9mNM359gJ7Adra2q7fv39/Uf0rFp/Ph9vtrmgfzEYgFOHS9DRB5cCiFFGtiUQ1LW7HbBniZIangkSiGqvlsvglni+vdxZ8bKkIhCKM+mawWtSCn6//oh+71cJMIMzweTvgoKPDi8M7QdOVa8rS12pBfkeZMXpsdu7c+ZzWelu694zI0klnwqTOIrkcE3tR633APoBt27bprq6uojpXLN3d3VS6D2bkwKGfcZbVObldPv2vL9HuccW2SIwT1ZrByQAP/PrVBR9bSnJxKz14qJfjLw7xD/euIRpZC8BffeURwhtv4FNdV5Wtr9WA/I4yU86xMULw+4HVSc87gIEMx/QrpWyABxg3oG2hQrjsVu7p2pjTsfmkV5pl85Vc0ibbg5pPf3YN0WhM7FvX/JTmVXa2bV2R8W8qHZ8QljZG+PCfAa5USq1XSjmADwCPpBzzCPCh+OP3Aj/XxfqShFnMvgFEPumV2Y41y2c9cmSAO9/jmhX7ZasP8pl9a2mtdywYjK50fEJY2hQt+FrrMHA3cBDoAR7WWh9XSn1BKfWu+GFfB5YppfqATwLzUjeFwqgGEcknvXKhY83yWY8cGeBNbwoRDsfEfsuW/2D4zNv51O6r0sYwEuSzy5RZJjZhcWHISlut9WPAYymvfT7pcQC43Yi2hLmYqd7MQuSzsjTTsWb4rOnE/qWX3oLFUtjmJenSPc1SNE5YfEhphSqn2gtq5ePTrvRnLUbsIff4hBkmNmFxIoJf5ZglyFkI+Vqyic8aikToG57GGwjhsFrYsjKHEsVFspDYJ09a211BegYn0/Y/11o4hUxsEgwWckFq6VQ5lao3YwT5+LQh9lnPj1/i8Kvj+GfCOCwKXyDMwGQgJx93oX7xbGKfHFeIRHXGuEKusYx8q2KaJbYhmB+x8KucUmwAUS5rMV9LtrPdQ1uDk1FfkFBE43bZ2LzKg8NmzeruKNQvns2Nk+p+SdQYytSfXGIZ+VbFLLcLaKHrQ+40zI0I/iLAyA0gyhkwzNcd1TM4yQvnJ0BDfY2dK1rraK13EdU6qx+/EFHMxWdfirhCZ7uHmztb+dbhcwx5A7Q1uPjQjjWmiG0sdH0AC76X7lwyOZQXcekIc8jXzZKgEHdJPu6ohNA4rBbsVkUwFOH5cxOMTAVyilnku5lKrgHaUmxKkti7dlN7A++6eiWb2ht4vGck45iWc2OUha6P1PdCkQinRnx88uGXGJ4Kzum/uKEqgwi+MIdCdpkq9MebT35+Qky2rGpgJhJbs+e0Ko4PeHOKWTisiid6Rzh0YojDp8YWnCjyycZJnbQiUV10DCVZOMd8QU4MevlV/yR//MiJtGNazjjOQtdH8nujvgDPnZ1Aaw2aebGNQg0LoThE8IU5FGItFvPjzbW+e0JMWtwurl/biMtuJRiJEororO6mnsFJhrxBpgJhbBYIzoQ5cmqcs2PT80Qx39TL1EnLalFFu78Sn3VkKsDz5yYIhiI01doY8wXTTqTl3Ddgoesj+b2+4WmcNgtKKepr7HNiG8mfMZlqSieuVsSHL8yhkG30yuFDTvb3t7hdtLhds8+zCduBY0Osbq5lhcc5m87pdtlY6XHN+dtC8+yTYyjd3RfzEtp0fuzEZ+0biYmmy24lEIrQ4nZmDAiXayPvbNdH4r1J/wxOq4VgRLN5ZQPgnXNNVHM6cTUjFr4wh0KsxXL4kItxWyTfHdy4YRlv37SCt2xsJRi5XM6p2EVVhZDOFfalAyd5ZcjLz18e5pXXpgiFIwRCEYLhKFcsr6u4FZx6fcyEI9TaLXz9qTMcODbEzZ2t8Ts9CyjFdWsaaa13EYpEeaJ3hBMDXh481MvGtrqqTSeuZsTCF+aRr7VYjs21i0k/zWZNVkLsYX7m0Ew4wrmxS7hdNm66opmf9Qxz7qKfdcvquH5t4+xdTaWt4MT1kZyx0+y2MekP8XjPCHvfsn72mnDYrAx5/dSFI0wFwtywoWn2uJs7W+kdmjYsnVjIjgi+UDSlWAuQqZ1CzrnQhFQpsYf5rrC+kWncTiszkSjL62vYtamNw6+OY7Uomuucs1awkRNpMSyU6nrPro2z18SR02NcuRJ2vK6ZFvflz9s7NM09u3IrsS0Ygwi+YAjl8iEXQqYJyXtuumJiD/PvPHzxoHKDK/a8xe3ihg1NvHTey+BkwHRWcLbYTeKauDDhx+3001LjSnucUD5E8IUlQeqEVEnLPkHqnYfdqpgKhHlDx+V+Om02dm1qM6UlnGvgdVVjDdFLue+BLJQOCdoKSw4ziD3MD4BuXtnA+pY67FZrVQQycw2k79nSNrs+oRo+12JGLHyhaKppibxZxD5B6p1H6lgW4sIp1/eRa+yms93DWbeDEewSoK0wRQm+UqoZ+C6wDjgDvE9rfTHNcRHgV/Gn57TW70o9RqhOqmmzjnKJfTGCW2wspNzfR679zWcPZKF0FGvhfxb4mdb6i0qpz8affybNcX6t9TVFtiWYkGrZrKNcYh8IReYJ7gMHe2lrcDIT0SW/A6qW70OoDMX68N8NfCv++FvAbxR5PqHKqIYl8uV043gD4XkFxE6PTnN8wFuWImHV8H0IlUNprbMflemPlZrQWjcmPb+otW5Kc1wYeBEIA1/UWv9wgXPuBfYCtLW1Xb9///6C+2cEPp8Pt9td0T7kSyAUwRsIMxOO4rBZaHDZFtxcuxAS4zI8FSQS1ViTxDPxfHm909A2C2F6OsTJkxqtHQDU1PjYtOny95ltrPIdy/EJLyGLExUfDl8wTDQa+401xK3tUo6PWb+PavwdlQujx2bnzp3Paa23pXsvq0tHKfU4sCLNW/fl0Yc1WusBpdQG4OdKqV9prV9Nd6DWeh+wD2Dbtm26q6srj2aMp7u7m0r3IR+Sfbj1dfFFRuMhw324iXGZ015iUVPA+PYK4ciRAXbtSrXsb53dqerbh8/yVN8YTbV1bFpZj1Pb5oxVIWP58L8fpEd34Inn0v/01Gs4LAqnw8aOpmUARLVmcDLAA79+teGfuVzfR75ximr7HZWTco5NVpeO1vpmrfWWNP9+BAwppdoB4v8PZzjHQPz/U0A3cK1hn0AALtej/+TDL3FqxEcoEilL2dlyVmrMh1y2JTw+4KWpJmbzvHBuklAkMmesCqkC2uCyzUlVdFgt+IIRltXaOXxqjEMnhniidwSHtTRZQeX4PqSWffVSbND2EeBDwBfj//8o9QClVBNwSWsdVEq1ADcBXyqyXSGJZKsODVprnjs7MVt/pdQ+XLOtss11W8JQRON2WlFx/0vf8DTb1zfPjlUhVUBdduucVMUtKxs4OTRF75APt9OKzRJbdDTkzbzZebGU+vuQwHD1UqzgfxF4WCl1F3AOuB1AKbUN+JjW+qNAJ/D3SqkosTuKL2qtTxTZrpBE8g+wvsZOMBTBaVP0DU/T4nYtqVWN+WxL6HbZCIYiuOxWnDYL3kBozlgVWsI3VXDv+/5Rjg14mYlEaXDZeUOHB7s1+z68ZqWcWyoKxlKU4Gutx4C3pXn9WeCj8cf/BbyhmHaEhUn+AV7RWsfz5yZwWhWT/hnTFdwqJflsSzjpD82OFcTuihxWy5yxMqoKaDCiecvGVizqcj9y2Yc3H8q5+E1q2VcvUlphEZBcj7613sV1axpBKSzKUlKfeiH72JaKQrYldNisXLM6Ni4T/jBbVjbMGSuj/OHF7heQbZzL7VMv55aKgrFIaYVFQKol6rBZ2dDqLmnwNN0Co0qtsC10W8IDx4bwBcN0XbU8o0VshD+8mDuFXFbOltunXq5y2ILxiOAvAirxA0xeYASVC9wZsS0hXLaiS+ESKeb7yUXMK+FTN1ugXsgNEfxFQrl/gDPhKPV1lV3RadQK2nLUnyn0+0mI+agvMLsfb73Thqf2sv9cfOpCrojgCwXhsFmYilv5CcopMvmIfbaAppnTDFc11nBm1MfJIR9Om4V6pw1vIIzXH55N6yzHFpPC4kAEvwzkkkFRTSWGIb7AaDwElF9k8hX7bNa7mdMM92xp457vvgaA02YhGI4CsLHNPTshiU9dyBUR/BKTi+BUQ4nh1AmpEyoiMvmK/R8/coJx3wzNbgdXtNbRWh8T9mTr3cwukc52Dx1NNXj9IXzBCFYL2CyK3uEp+kamZw0D8akLuSCCX2JycReY2aUA6Sek0cgMa6GsW+8VYtmP+YI018YWoz1/boLr1jSyzO2cY72b3SWyeaWHSX+IUCTCc2cncNosWJUFlDKdYSCYG8nDLzG5lKs1e0nbdDVlrBZVsvo86cg3QJvoc4s7Voc+sZq2b2Sas6PTnBu/NJvXDpiyHlCCRN77sQve2Ro8wYhm88qGouokmWkdhVAexMIvMbm4C8zsUgA4MTjJ5KUQU8EwDS47VyyvY6VVlW1CKiQbJ+GXv2J5Hc+dja2mdVgVAxf9DE74uW5N4zz3mRk3CofLaZ2ffPgl0FBfY2PzygZa610Fr9gtxo1YbfEm4TJi4ZeYXFYlmnnlYs/gJOfH/HgDYeqdNgKhmFthJhwpy4RUaOplYnVri9vF9WsbcdmtjF8KEUVz3ZpG1rW4y1JN1Cg62z28fdMKbtiwjB0bls3GIgo1DAqpBApzV/XardB9cpjf+fbzfO77R+UOoQoQwS8xuSzPN2uJYYgJw8a22OYMwXAUpy12yQTC0ZJPSMXk2SdPos11TjrbG9ja0cjmlQ2sWVY3e9zIVIATA5P88MULpndrGGkYFOpGvFxpNMIL52Jj1VRj4/iAV0okVwHi0ikDuWRQlDPLIp9b8gsTfta21OF22egbmcYXCNPgsuGwWkra32IXVWVKVTxwbGjWfTYyFZgtntZW7zRldlSCxHfmC4a4MOGnwWVj80pPwZlRhboRE66yX5724rRZcNmtaK3xBS/vJWC2sRMuI4K/xMjXd5sQhtZ616wbYdIfokZNlayPxYh9LpNZIiOnb9g3+9qVbW7TZUclSP7OXr+iYTaLqBjfeSIz6eJ0kMHJAGPTM9itFj7x1tct+HeJ6yGx4hdid35ul81UiQZCesSls8TI13ebyY3Q4CqNrVCs2GerGpnsPuu/6GfSP0MwHKFveJpRX8CUolWov30hOts93NzZysnXfIxPh1hW52DjcjeP94ws6JaZrTRqtRAIRQiEIgTDUa5orTNVooGQHhH8PKn2VLZ8fbeZ4gtGb4oOxbtxchXGRDmCBpcdT42DVrdzNhh9bmzadKJVqrTd3qFpbnzdMravbwKl6BuZ5tSIj28fPpvxbxLXw5aVDUz4YyWfr1ntwWGzmibRQMhMUWaaUup24E+I7Wq1Pb7xSbrj9gD/B7ACX9Naf7GYdivFQu4QoCpS1Qrx3aaLLwydNLZfRhRCS1ciIRAKc+T02LzvJRGM7h32zQajg+EoJ4d8fKxrYbdGKUnnkjIybTf5/McHJlnTVMPZcT9OmwW300owFOGpvrEFt1/sbPdw/21b55zLU2OXcg5VQLEW/jHgNuCJTAcopazAXwPvADYBv6mU2lRkuxUhkwX5T4fPVs2mzmZMATWq6mXqRiMjUwGeOX0Rh9Uy73tJBKOvW9OI027FF4zQ4LLR0VRTMdHK5JLa2FZnyHeWen6H1cKR0xeJRmML05RSoBRNtbm5izrbPdyzayMP3H419+zaKGJfBRS7xWEPMLsJdAa2A31a61PxY/cD7waqbl/bTEW2ftYzzvb1zaYtjZCM2QptGSX2ML9EwvEBLxrYsqphdoKG2PeSKRidbEWXm0wlNnqHptN+Z0BeNfxTz79lVQOvjkwzcWmGxlo7wXCUYDjKtWs8nCjh/gBC5VBa6+JPolQ38Ol0Lh2l1HuBPfENzVFK3QncoLW+O8O59gJ7Adra2q7fv39/0f0rBp/Ph9sdy0MfngoSiWqsSWIUiWp88RWoyfOe1hCKROloMpc/2CiSx6VQpqdDnDyp0doBQE2Nj02bijtnIBTBGwgzE44SCEWocVixWy/fyCa+lxa3g1HfDFaLwqIUUa2JRDUtbkdR8YlixqX/oh+lYlkvievMabOgNfOuo0Aoknf/+y/6sVstc65Trz9EKBLFZrVgtShcNguRqCYQjuJ22gwbGyOul8WK0WOzc+fO57TW29K9l9XCV0o9DqxI89Z9Wusf5dB+OlMt4yyjtd4H7APYtm2b7urqyqGJ0tHd3U2iD8k+/NkiW4EQNXYLDqx4XJetw0l/CI/bzge78l+uXw1L15PHpRCOHBlg165Uy/7Wgiz7TDx4qJd+fyj997JrY0nGuZhxue/7Rzn86jj1LttsTGEqEGbHhmY+2LV1zrEPHupl0jr3jmTSH2IEO/dkuOYePNQb+/xJ43F6ykfvsI8bNyybvaafPjPGxuVu1jddFqFs585GsdfLYqacY5NV8LXWNxfZRj+wOul5BzBQ5DkrQiZ3CGBYtcVqKJVcLEa6cRYiWxVMs5UU1sy3jhTpraNCavinjsfZ0Wl6h3w01tk5MeidXczV0VTD2pa6OX9rxnRVIX/KsfDqGeBKpdR64ALwAeCOMrRbEjKJhFF+cbOXSi6Wcok9mC9ekY2ZiOaN65s4NXoJXyCM22Wjs72emch8yS802yoxHscHJum/6OeqFW7WLKubs5gL5sczJMd+cVBsWuZ7gIeAVuBRpdSLWuvdSqmVxNIvb9Fah5VSdwMHiaVlfkNrfbzonpsMo6xFM+++VCzlFPsEpbbi528MEyn4XAkR37Fh2exrmQLJhdbwT4zHg4d66WiqTWtYmH1/AKFwikrL1Fr/QGvdobV2aq3btNa7468PaK1vSTruMa31Rq3167TW9xfb6cVMamohLA7rqhJiX2rSpVGO+mYKTsfNJ2W22IJ7Cy3mMnMxP6E4pJaOyViM1tViEftUa35kKjDP/Wa9pAp2v+Xrgirm7iWbS8hs8Q3BGETw86TUGTTV5nfOxmIS+9Rg+lN9Y/y31zXPEU2LKm5jmHIJ7WI0LITsiODnQbkyaHL50VdD6uZiEXtIH0xvqrXTMzhFW8Nld1tU66pwvy02w0LIDRH8PChnBs1Cgl4NqZuLSewhfTB908p6/rNvnEl/aNZKbo3qqikgJm6bpYcIfh6UK4Mmm6DnMvFU8g5gsYk9gNOqeKJ3hJlIdHZfX6fNxpuvWIanxj47zi04REQF0yKCnwfl2mw8m6Bnm3jSTRhfOnCSlR4XwYgueAJInkS2u4JpKypWQuxLPbn1DE4yMBmI5cY7rfhnwhx+dZz1LXV8evfcomHd3VW5plBYIkg9/DwoV6XJbPXPs6VuJk8YY74gz5wZ59kz4xzqGcJujeV2P3Cwl899/2jOdf1TUxAjUT2vImilxL7UlUoPHBti7bI6btjQjNNhIxyNfR9tDU6x5oWqQgQ/D8qVn5xN0LNNPIkJI7Fn6+hUkBqHlVBE88K5SYa9fk6PTnN8wJuzSKaWhrZa1JzNRSrlxinFblCpJMaztd7Fjg3L2LWpjbdsbE27AlYQzIy4dPKkHIGuXGrALJRhkXA99Y1M47RZiGiNBah1WHHaLLzYP0lLnYNQRM8rG7zQZubtHhejvgB9w9Pc3BTixPgknlp7RX325YirlMuVJwilRgTfpNTYLRw5PYZCce1qz7w7iYUmnsSEMe6boanWhlUpguEobR4HTpuF6WCY1joH7iS3UTaRXNVYw5lRHyeHfDhtFqxK4Q2EOdtzkZvuWk6kQgHahBjPhCP0jUzjC4SxWxWbVzYY1obkrAuLBXHpmIyET9phs3JzZxvb1zdzKRTN6xyJO4Bmt4PxSyFa65001cZcHoFQBKfNgi8Y4YrWyxURs1mse7a0cXLIBxCr0Y7m4pkpnvnq5oqJfaJfZ8emOXJqnOBMGJsl9lmGvEHD/PhSakBYLIiFbzKMyvXvbPfwp+/aNJutEwiF6RmcYsIfZtvaJvyhKA6blajWOVmsne0epn81Sfsbl+ELRojMRHn2q1ejo5VNvexs97DS42J8emY2ZfINHR7sVquh6yMkZ11YDIjgmwwjfdLJvn5fMEzXVctnUxZTUxmzrbLs7/fyn/9wA79zTR/tys7Uay6ikZjYr1h3iJdeurliefbBiOYtG1uxJG3lFNV6UVQYFQQjEcE3GUYHCDNZpvlarHfc8Txad/Gjh55h6NQWvvQXAQCWrT7I40/dUNFFVRJUFYTcEB++yShXrn8+HD06zJNPvhGAwVdunWPZ/8fhG9i8qrFifQNzjpkgmBERfJNhxgDhHXe8DMzd8s5mu8T5V95WcbEHc46ZIJiRYne8uh34E6AT2K61fjbDcWeAKSAChDPtqC7EMFOA8NChMxw/vmPe6+FwLS7XALfd9ir791e+To6ZxkwQzEqxPvxjwG3A3+dw7E6t9WiR7Qll5q67BoB1ad6J8K53vcrXvnZjwWJfDSWeBWExUZTga617AJSq3iqIQma+9a0TnD//31JenebNb36GrVsVn/rUrxV87p7BSR442MuoL0gwHOGVoSmOXZicV4xMEATjUFoXXw9EKdUNfHoBl85p4CKggb/XWu9b4Fx7gb0AbW1t1+/fv7/o/hWDz+fD7XZXtA8LEQhF8AbCzISjOGwWGlw2XHarIed+8UU/kcjlTJeaGh/r17uoqbEVPS4DE358wQhWCygUGk0kCm6nlZVVnF1j9uulUsi4ZMbosdm5c+dzmdzmWS18pdTjwIo0b92ntf5Rjn24SWs9oJRaDhxSSr2stX4i3YHxyWAfwLZt23RXV1eOTZSG7u5uKt2HTCSXQa6viy/5Hw8ZErC8//5n+aM/6gKgo+MwX/vaCnbvXj/7fq7jksltc8v/eRKbBWocly9B/0yYcBQe+403F9X3SmLm66WSyLhkppxjk1XwtdY3F9uI1nog/v+wUuoHwHYgreALufu2E6tyQ5EIvzztxRsI4bBa+KfDZ7n/tq0Ftx8OR/mzP6unsfEF/vIvHXz4w/ODtrl+jkwbuWg0irmuQAVopAKlIJSKkqdlKqXqlFL1icfA24kFe01Nz+AkDx7qpf+iP6d68Ua2m2t99wsTfoLhMM+dnSAQilDvtKG15sm+saL6+/3vv8J9900xNnYNH/7w5oLPs1Dp4mtXN+ILRgiEImitCYQi+IIRrl1d+TRPQVisFJuW+R7gIaAVeFQp9aLWerdSaiXwNa31LUAb8IN4YNcG/IvW+kCR/S4pyZbpSqulqD1j881EyaeWzqrGGrpPDuO0WWb99kopmmrt/NPhs7TUuwrKgHnf+67K6zNmYqEyEXe9aR2DkwHGp2eYCoRx2CysWVbLnTvWGtK2IAjzKTZL5wfAD9K8PgDcEn98Cri6mHbKTbLoqgB4XIUVMCtks/F8auns2dLGD164QFNNzLIPhqMEw1HWLavhyb4x3vr65Xltcm50muRCJQ862z384Z6rJC1TEMqI1NJJg1EFzAqpfJlPXZjOdg9vumIZxwe8+IIR3C4bm1c2cHzAS1Ntfu0WMjllI5eNXETgBaF8LLrSCgnfe657taYj2xaDuZJtb9p05FsX5s4da9nQ6mb7+mZuWN+Mw2bl4qUQne31ebVbiq0CpeSBIJiLRWXhG2WlJlumK2FWdPPd4aiQKo7Zti/M5fg3XbEMh21uLn62dpPvakamAvSNTDPlDxEIRxiZCjAT0QW5XcSKFwTzsKgE38jNQxIiGvJF8bjtWevFp6PQrfHyFcnU4xMTXz7tJm8V+Py5CZw2C+FIlIvTIY6cGueN65vmTKCCIFQfi8qlU4gLJR3JwUuHzVJwMLFSLo1C2k24ko4PeHFaY/nx45dCtDU4cbtsnBq9ZIibRxCEyrGoLHwjNsJIdQtF/Lqo4GWlXBqF3CXsfct6PvXwUaI6iqfGgafGNjuWvnhMIzGBXttakm4LglBCFpWFb8RGGKnBS6tFLRmrtrPdw65Nbdy4oYUbNyyjtd41m+rpjt85yU5SglC9LCrBN8KFYpRbqFpJnjQ3tNYyFQjjC4TZ0FIrO0kJQpWzqFw6kN6Vkc+CIjPvj1qO+vFzs37C7NjQjAZmIhpPzeXg9dBJsJcw0wAABCZJREFUQ5sVBKEMLDrBTyXfVM3UzJpIVDMZyD8l02hKsTAqE5JKKQiLk0Xl0klHvguKUt1CVosyxWKhUiyMEgRhabHoLfxCyiQkW7jd3RcrLvZgXLkHQRCWLovewjeqTEKlWSyfQxCEyrHoBd+IVE0zsFg+hyAIlWPRC/5iKeC1WD6HIAiVY9H78GHxZJ0sls8hCEJlKMrCV0p9WSn1slLqqFLqB0qptPvTKaX2KKVOKqX6lFKfLaZNQRAEoTCKdekcArZorbcCvcC9qQcopazAXwPvADYBv6mU2lRku4IgCEKeFCX4Wuufaq0TqSNPA+lWJ20H+rTWp7TWM8B+4N3FtCsIgiDkj5E+/I8A303z+irgfNLzfuCGTCdRSu0F9gK0tbXR3d1tYBfzx+fzVbwPZkTGJT0yLumRcclMOccmq+ArpR4HVqR56z6t9Y/ix9wHhIF/TneKNK/pTO1prfcB+wC2bdumu7q6snWxpHR3d1PpPpgRGZf0yLikR8YlM+Ucm6yCr7W+eaH3lVIfAt4JvE1rnU7I+4HVSc87gIF8OikIgiAUj0qv0Tn+sVJ7gL8Efk1rPZLhGBuxgO7bgAvAM8AdWuvjOZx/BDhbcAeNoQUYrXAfzIiMS3pkXNIj45IZo8dmrdY67RZFxQp+H+AExuIvPa21/phSaiXwNa31LfHjbgH+CrAC39Ba319wo2VGKfWs1npbpfthNmRc0iPjkh4Zl8yUc2yKCtpqra/I8PoAcEvS88eAx4ppSxAEQSiORV9aQRAEQYghgp+dfZXugEmRcUmPjEt6ZFwyU7axKcqHLwiCIFQPYuELgiAsEUTwBUEQlggi+DmQa1XQpYZS6nal1HGlVFQpteRT7qQq7HyUUt9QSg0rpY5Vui9mQim1Win1C6VUT/w39D/K0a4Ifm5krQq6RDkG3AY8UemOVBqpCpuRbwJ7Kt0JExIGPqW17gRuBD5ejutFBD8HcqwKuuTQWvdorU9Wuh8mQarCpkFr/QQwXul+mA2t9aDW+vn44ymgh1ihyZIigp8/HwF+UulOCKYjXVXYkv+AhepHKbUOuBY4Uuq2lsQWh7lgQFXQRUku4yIAeVaFFQQApZQb+B7wB1prb6nbE8GPY0BV0EVJtnERZpGqsEJeKKXsxMT+n7XW3y9Hm+LSyYF4VdDPAO/SWl+qdH8EU/IMcKVSar1SygF8AHikwn0STIpSSgFfB3q01n9ZrnZF8HPjq0A9cEgp9aJS6u8q3SEzoJR6j1KqH9gBPKqUOljpPlWKeFD/buAgsQDcw7mUAF/sKKW+AxwGrlJK9Sul7qp0n0zCTcCdwFvjmvJivKpwSZHSCoIgCEsEsfAFQRCWCCL4giAISwQRfEEQhCWCCL4gCMISQQRfEARhiSCCLwiCsEQQwRcEQVgi/P/BULIuEdw+AgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x[:,0],x[:,1],alpha=0.5)\n",
    "\n",
    "plt.quiver(0,0,u[:,0],u[:,1],scale=3)\n",
    "\n",
    "plt.quiver(0,0,v.T[:,0],v.T[:,1],scale=4,color='red')\n",
    "\n",
    "plt.quiver(0,0,pca.components_[:,0],pca.components_[:,1],scale=3,color='blue')\n",
    "\n",
    "\n",
    "plt.grid()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = np.array([[1,2],[3,4]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2],\n",
       "       [3, 4]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 3])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data[:,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 4])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data[:,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3, 4])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "l = [[1,2],[2,3],[3,4]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[1, 2], [2, 3], [3, 4]]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 2]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'1.18.4'"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np \n",
    "np.__version__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'1.0.3'"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "pd.__version__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "b = np.random.randint(5,size=4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 1, 1, 4])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "d = np.random.randint(6,size=(2,3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[5, 5, 4],\n",
       "       [4, 1, 4]])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d.ndim"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "l = [[1,2,3,4],[5,6,7,8]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = np.random.randint(12,size=(2,6))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[11,  7,  2,  8,  2,  2],\n",
       "       [ 1,  3,  4,  2,  7, 11]])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[11,  7,  2,  8],\n",
       "       [ 2,  2,  1,  3],\n",
       "       [ 4,  2,  7, 11]])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.reshape(3,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[[1, 2, 3],\n",
       "        [4, 5, 6],\n",
       "        [1, 2, 4]],\n",
       "\n",
       "       [[1, 2, 3],\n",
       "        [7, 9, 4],\n",
       "        [1, 5, 6]]])"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = np.array([[[1,2,3],[4,5,6],[1,2,4]],[[1,2,3],[7,9,4],[1,5,6]]])\n",
    "C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "D = np.array([[1,2,3],[5,6,7]])\n",
    "Q = np.array([[2,3,6],[8,2,4]])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2, 3],\n",
       "       [5, 6, 7]])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 3, 6],\n",
       "       [8, 2, 4]])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "F = np.concatenate([D,Q],axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2, 3, 2, 3, 6],\n",
       "       [5, 6, 7, 8, 2, 4]])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "a=np.random.seed(410)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 6,  1, 10,  4,  3, 11],\n",
       "       [ 5, 11,  9,  0, 11,  5]])"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "q = np.random.randint(12,size=(2,6))\n",
    "q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2], dtype=int32)"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.bitwise_not(np.array([1]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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