01-intro/notebooks/08-linear-algebra.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "3aace4b5",
   "metadata": {},
   "source": [
    "# Machine Learning Zoomcamp\n",
    "\n",
    "## 1.8 Linear algebra refresher"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2cd0b8e5",
   "metadata": {},
   "source": [
    "Plan:\n",
    "\n",
    "* Vector operations\n",
    "* Multiplication\n",
    "    * Vector-vector multiplication\n",
    "    * Matrix-vector multiplication\n",
    "    * Matrix-matrix multiplication\n",
    "* Identity matrix\n",
    "* Inverse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "1317a223",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4052050d",
   "metadata": {},
   "source": [
    "## Vector operations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "e87a01b7",
   "metadata": {},
   "outputs": [],
   "source": [
    "u = np.array([2, 4, 5, 6])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "913795a9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 4,  8, 10, 12])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "24625627",
   "metadata": {},
   "outputs": [],
   "source": [
    "v = np.array([1, 0, 0, 2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "edc95be4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3, 4, 5, 8])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u + v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "62f471c3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2,  0,  0, 12])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u * v"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6cb784ea",
   "metadata": {},
   "source": [
    "## Multiplication"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "6a838e14",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.shape[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "bef565ee",
   "metadata": {},
   "outputs": [],
   "source": [
    "def vector_vector_multiplication(u, v):\n",
    "    assert u.shape[0] == v.shape[0]\n",
    "    \n",
    "    n = u.shape[0]\n",
    "    \n",
    "    result = 0.0\n",
    "\n",
    "    for i in range(n):\n",
    "        result = result + u[i] * v[i]\n",
    "    \n",
    "    return result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "5f212712",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "14.0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vector_vector_multiplication(u, v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "b57c4464",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "14"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.dot(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "b7710217",
   "metadata": {},
   "outputs": [],
   "source": [
    "U = np.array([\n",
    "    [2, 4, 5, 6],\n",
    "    [1, 2, 1, 2],\n",
    "    [3, 1, 2, 1],\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "3f1ee5f1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 4)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "8b4f7530",
   "metadata": {},
   "outputs": [],
   "source": [
    "def matrix_vector_multiplication(U, v):\n",
    "    assert U.shape[1] == v.shape[0]\n",
    "    \n",
    "    num_rows = U.shape[0]\n",
    "    \n",
    "    result = np.zeros(num_rows)\n",
    "    \n",
    "    for i in range(num_rows):\n",
    "        result[i] = vector_vector_multiplication(U[i], v)\n",
    "    \n",
    "    return result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "930f42c3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([14.,  5.,  5.])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix_vector_multiplication(U, v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "0937dafd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([14,  5,  5])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.dot(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "85280363",
   "metadata": {},
   "outputs": [],
   "source": [
    "V = np.array([\n",
    "    [1, 1, 2],\n",
    "    [0, 0.5, 1], \n",
    "    [0, 2, 1],\n",
    "    [2, 1, 0],\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "82039bcf",
   "metadata": {},
   "outputs": [],
   "source": [
    "def matrix_matrix_multiplication(U, V):\n",
    "    assert U.shape[1] == V.shape[0]\n",
    "    \n",
    "    num_rows = U.shape[0]\n",
    "    num_cols = V.shape[1]\n",
    "    \n",
    "    result = np.zeros((num_rows, num_cols))\n",
    "    \n",
    "    for i in range(num_cols):\n",
    "        vi = V[:, i]\n",
    "        Uvi = matrix_vector_multiplication(U, vi)\n",
    "        result[:, i] = Uvi\n",
    "    \n",
    "    return result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "ab0e5aba",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[14. , 20. , 13. ],\n",
       "       [ 5. ,  6. ,  5. ],\n",
       "       [ 5. ,  8.5,  9. ]])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix_matrix_multiplication(U, V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "8d0e3b73",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[14. , 20. , 13. ],\n",
       "       [ 5. ,  6. ,  5. ],\n",
       "       [ 5. ,  8.5,  9. ]])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.dot(V)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb2cdbdd",
   "metadata": {},
   "source": [
    "## Identity matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "ca913560",
   "metadata": {},
   "outputs": [],
   "source": [
    "I = np.eye(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "0614d05f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1. , 1. , 2. ],\n",
       "       [0. , 0.5, 1. ],\n",
       "       [0. , 2. , 1. ],\n",
       "       [2. , 1. , 0. ]])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "aabbf2ad",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1. , 1. , 2. ],\n",
       "       [0. , 0.5, 1. ],\n",
       "       [0. , 2. , 1. ],\n",
       "       [2. , 1. , 0. ]])"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V.dot(I)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e8f786ef",
   "metadata": {},
   "source": [
    "## Inverse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "3e6fc747",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1. , 1. , 2. ],\n",
       "       [0. , 0.5, 1. ],\n",
       "       [0. , 2. , 1. ]])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Vs = V[[0, 1, 2]]\n",
    "Vs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "5265b91e",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.        , -2.        ,  0.        ],\n",
       "       [ 0.        , -0.66666667,  0.66666667],\n",
       "       [ 0.        ,  1.33333333, -0.33333333]])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Vs_inv = np.linalg.inv(Vs)\n",
    "Vs_inv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "3cd1d98b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 0., 0.],\n",
       "       [0., 1., 0.],\n",
       "       [0., 0., 1.]])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Vs_inv.dot(Vs)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1c54ed7d",
   "metadata": {},
   "source": [
    "### Next \n",
    "\n",
    "Intro to Pandas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "64d8bdce",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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