{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = Matrix([[1, 10],\n",
    "            [1, 8],\n",
    "            [1, 13],\n",
    "            [1, 9],\n",
    "            [1, 11],\n",
    "            [1, 14],\n",
    "            [1, 6],\n",
    "            [1, 4],\n",
    "            [1, 12],\n",
    "            [1, 7],\n",
    "            [1, 5]])\n",
    "\n",
    "y = Matrix([[8.04],\n",
    "            [6.95],\n",
    "            [7.58],\n",
    "            [8.81],\n",
    "            [8.33],\n",
    "            [9.96],\n",
    "            [7.24],\n",
    "            [4.26],\n",
    "            [10.84],\n",
    "            [4.82],\n",
    "            [5.68]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 10\\\\1 & 8\\\\1 & 13\\\\1 & 9\\\\1 & 11\\\\1 & 14\\\\1 & 6\\\\1 & 4\\\\1 & 12\\\\1 & 7\\\\1 & 5\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  10⎤\n",
       "⎢     ⎥\n",
       "⎢1  8 ⎥\n",
       "⎢     ⎥\n",
       "⎢1  13⎥\n",
       "⎢     ⎥\n",
       "⎢1  9 ⎥\n",
       "⎢     ⎥\n",
       "⎢1  11⎥\n",
       "⎢     ⎥\n",
       "⎢1  14⎥\n",
       "⎢     ⎥\n",
       "⎢1  6 ⎥\n",
       "⎢     ⎥\n",
       "⎢1  4 ⎥\n",
       "⎢     ⎥\n",
       "⎢1  12⎥\n",
       "⎢     ⎥\n",
       "⎢1  7 ⎥\n",
       "⎢     ⎥\n",
       "⎣1  5 ⎦"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}8.04\\\\6.95\\\\7.58\\\\8.81\\\\8.33\\\\9.96\\\\7.24\\\\4.26\\\\10.84\\\\4.82\\\\5.68\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡8.04 ⎤\n",
       "⎢     ⎥\n",
       "⎢6.95 ⎥\n",
       "⎢     ⎥\n",
       "⎢7.58 ⎥\n",
       "⎢     ⎥\n",
       "⎢8.81 ⎥\n",
       "⎢     ⎥\n",
       "⎢8.33 ⎥\n",
       "⎢     ⎥\n",
       "⎢9.96 ⎥\n",
       "⎢     ⎥\n",
       "⎢7.24 ⎥\n",
       "⎢     ⎥\n",
       "⎢4.26 ⎥\n",
       "⎢     ⎥\n",
       "⎢10.84⎥\n",
       "⎢     ⎥\n",
       "⎢4.82 ⎥\n",
       "⎢     ⎥\n",
       "⎣5.68 ⎦"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "v = Inverse(X.T * X) * X.T * y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}3.00009090909091\\\\0.500090909090909\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡3.00009090909091 ⎤\n",
       "⎢                 ⎥\n",
       "⎣0.500090909090909⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "#v = (b, m), or (intercept, slope)\n",
    "#So, the best fit line is given by (approximately) y = 0.5x + 3"
   ]
  }
 ],
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