9. Linear Algebra.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Linear Algebra\n",
    "Work in progress. Basic linear algebra to be included here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Simple Array Operations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import seaborn as sns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 2.],\n",
       "       [3., 4.]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.array([[1.0, 2.0], [3.0, 4.0]])\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 3.],\n",
       "       [2., 4.]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.transpose()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 3.],\n",
       "       [2., 4.]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.array([[1,2],[3,6]])\n",
    "b = np.array([[3,1],[2,2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 7,  5],\n",
       "       [21, 15]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dot = np.dot(a,b)\n",
    "dot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 7,  5],\n",
       "       [21, 15]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a@b # Row with col"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 3,  2],\n",
       "       [ 6, 12]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a*b  # Element wise product"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "23"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.vdot(a,b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2,  4,  6,  8, 10, 12])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.array([1,2,3,4,5,6]) # 21\n",
    "np.inner(a,2) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 4],\n",
       "       [ 2],\n",
       "       [ 4],\n",
       "       [ 6],\n",
       "       [ 8],\n",
       "       [10]])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = np.array([2,1,2,3,4,5]) #63\n",
    "np.outer(b,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "## 2+2+6+12+20+30"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "72"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.matmul(a,b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "72"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.inner(a,b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.array([[10,12],[2,4]])\n",
    "y = np.array([[2,4],[6,8]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 68, 156],\n",
       "       [ 20,  44]])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.inner(x,y)  # 10*2+12*4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[20, 40, 60, 80],\n",
       "       [24, 48, 72, 96],\n",
       "       [ 4,  8, 12, 16],\n",
       "       [ 8, 16, 24, 32]])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.outer(x,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](https://miro.medium.com/max/3426/1*-ycNEMicMOIR8WvzcBzERA@2x.jpeg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.0"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.array([[1,2],[2,3]])\n",
    "np.linalg.det(a) # Determinat"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix}^ {-1} =\n",
    "\\frac{1}{ad - bc} \\begin{bmatrix} d & -b \\\\ -c & a \\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-3.,  2.],\n",
       "       [ 2., -1.]])"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.inv(a) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 0.],\n",
       "       [0., 1.]])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = np.eye(2) # unit 2x2 matrix; \"eye\" represents \"I\"\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 0.],\n",
       "       [0., 1.]])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i = np.identity(2)\n",
    "i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0., -1.],\n",
       "       [ 1.,  0.]])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "j = np.array([[0.0, -1.0], [1.0, 0.0]])\n",
    "j @ i        # matrix product"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0., -1.],\n",
       "       [ 1.,  0.]])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(i,j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "matrix_1 = np.matrix([[0,4],\n",
    "                      [2,0]])\n",
    "matrix_2 = np.matrix([[-1,2],\n",
    "                      [1,-2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[-1,  6],\n",
       "        [ 3, -2]])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sums = matrix_1 + matrix_2\n",
    "sums"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x2276ce98748>"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.heatmap(sums,annot=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[-1,  6],\n",
       "        [ 3, -2]])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.add(matrix_1,matrix_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1, 2],\n",
       "        [1, 2]])"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.subtract(matrix_1,matrix_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1, 2],\n",
       "        [1, 2]])"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix_1 - matrix_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "40"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.array([[10,20],\n",
    "             [20,30]])\n",
    "np.trace(a)  # trace"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[[10, 20],\n",
       "        [20, 30]],\n",
       "\n",
       "       [[20, 40],\n",
       "        [ 2,  7]]])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.array([\n",
    "    [[10,20],[20,30]],\n",
    "    [[20,40],[2,7]]\n",
    "            ])\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([12, 27])"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.trace(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Singular Value Decomposition  or (SVD)\n",
    "\n",
    "we have to find the SVD of a matrix i.e., break up a matrix into the porduct of three matrices\n",
    "![](https://s3-ap-south-1.amazonaws.com/av-blog-media/wp-content/uploads/2019/07/svd.jpg1.jpg)\n",
    "1. $U$, \n",
    "2. $\\sigma$,\n",
    "3. $V^*$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1, 4],\n",
       "        [2, 0]])"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mx = np.matrix([[1,4],[2,0]])\n",
    "mx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(matrix([[-0.98888296, -0.14869598],\n",
       "         [-0.14869598,  0.98888296]]),\n",
       " array([4.15941525, 1.92334728]),\n",
       " matrix([[-0.30924417, -0.95098267],\n",
       "         [ 0.95098267, -0.30924417]]))"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "svd = np.linalg.svd(mx)\n",
    "svd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.matrix_rank(mx)  # Rank of a Matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1, 4],\n",
       "        [2, 0]])"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[9, 4],\n",
       "        [2, 8]])"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.matrix_power(mx,2) # power of Matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0, 0],\n",
       "       [0, 2, 0],\n",
       "       [0, 0, 3]])"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.diag((1,2,3))\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "eig is the funcation which is used to comput the eigenvalues and eigenvectors of square arrya"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "c,d = np.linalg.eig(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1., 2., 3.])"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 0., 0.],\n",
       "       [0., 1., 0.],\n",
       "       [0., 0., 1.]])"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 1],\n",
       "       [2, 3]])"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.arange(4).reshape(2,2)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.trace(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solve** Which is used for linear Matrix equation or system of linear equation.\n",
    "\n",
    "Computes the `exact` soluation `x` of the well-determined i.e., full rank , linear martix equation is **ax=b** "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.array([[3,1],[1,2]])\n",
    "b = np.array([9,8])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ 3*X_0+X_1=9$\n",
    "\n",
    "$ X_0+2*X_1 = 8$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2., 3.])"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.solve(a, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "jupyter": {
     "source_hidden": true
    }
   },
   "source": [
    "## Solve the banded system a x = b, where:\n",
    "\n",
    "\n",
    "    [5  2 -1  0  0]       [0]\n",
    "    [1  4  2 -1  0]       [1]\n",
    "a = [0  1  3  2 -1]   b = [2]\n",
    "    [0  0  1  2  2]       [2]\n",
    "    [0  0  0  1  1]       [3]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "jupyter": {
     "source_hidden": true
    }
   },
   "source": [
    "There is one nonzero diagonal below the main diagonal (l = 1), and two above (u = 2). The diagonal banded form of the matrix is:"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "jupyter": {
     "source_hidden": true
    }
   },
   "source": [
    "     [*  * -1 -1 -1]\n",
    "ab = [*  2  2  2  2]\n",
    "     [5  4  3  2  1]\n",
    "     [1  1  1  1  *]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### lSTSQ() [Least-squares solution to a linear Matrix Equation ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.arange(0,9)\n",
    "A = np.array([x,np.ones(9)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0., 1., 2., 3., 4., 5., 6., 7., 8.],\n",
       "       [1., 1., 1., 1., 1., 1., 1., 1., 1.]])"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = [19,20,20.5,21.5,22,23,23,25.5,24]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.71666667, 19.18888889])"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We have to obtaining the parameters of regression line \n",
    "w = np.linalg.lstsq(A.T,y,rcond=None)[0]\n",
    "w"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$y = m_x+b$\n",
    "\n",
    "$Y =  \\beta_0 + \\beta_1 * X$\n",
    "\n",
    "\n",
    "**$\\beta_1 , m $**: Slope of line equation\n",
    "\n",
    "**$\\beta_0 , b$** : Interception point"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([19.18888889, 19.90555556, 20.62222222, 21.33888889, 22.05555556,\n",
       "       22.77222222, 23.48888889, 24.20555556, 24.92222222])"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "line = w[0]*x+w[1]  # Regression Line y = mx+b\n",
    "line"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x2276e42b358>]"
      ]
     },
     "execution_count": 76,
     "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,y)\n",
    "plt.plot(x,line,'y-')\n",
    "#plt.plot(x,y,'o')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}