Formulas.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Vectors\n",
    "\n",
    "A _vector_ is a series of numbers, usually written like this:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "v = \\begin{bmatrix}5 \\\\ 0\\end{bmatrix} \\\\\n",
    "w = \\begin{bmatrix}3 \\\\ 4\\end{bmatrix}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "although we can write it horizontally as well:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "v = \\begin{bmatrix}5 & 0\\end{bmatrix} \\\\\n",
    "w = \\begin{bmatrix}3 & 4\\end{bmatrix}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "It shows motion in a coordinate space, which is not super-important for what you're learning here, but just to note: a two-element vector is motion on a 2-dimensional plane, with the first number being the movement on the X axis and the second number being the movement on the Y axis. We can visualize this on a grid."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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68HfAV4GTMvOqWYqQpFlEcDCTwfRyJkMJ4G9ZZSCpTAvPoDLzroh4NfBpJt8k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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x118691da0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def draw_vector(x, y, start_x=0, start_y=0, **kwargs):\n",
    "    args = dict(head_width=0.15, head_length=0.3, \n",
    "                overhang=0.5, length_includes_head=True)\n",
    "    args.update(kwargs)\n",
    "    plt.arrow(start_x, start_y, start_x + x, start_y + y, **args)\n",
    "\n",
    "plt.figure(figsize=(7, 7))\n",
    "\n",
    "draw_vector(5, 0, color=\"blue\")\n",
    "draw_vector(3, 4, color=\"red\")\n",
    "\n",
    "plt.grid(True)\n",
    "plt.axis([-1, 7, -1, 7])\n",
    "\n",
    "plt.text(2.5, 0.3, r\"$v = \\left[\\stackrel{5}{0}\\right]$\", fontsize=20)\n",
    "plt.text(1.5, 2.3, r\"$w = \\left[\\stackrel{3}{4}\\right]$\", fontsize=20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For what we will be doing with data, vectors come in very handy, but you need a new style of math to work with them.\n",
    "\n",
    "## Addition and subtraction\n",
    "\n",
    "Adding two vectors of the same size is easy:\n",
    "\n",
    "$$\\\n",
    "\\begin{bmatrix}v_1 \\\\ v_2 \\end{bmatrix} +\n",
    "\\begin{bmatrix}w_1 \\\\ w_2 \\end{bmatrix} =\n",
    "\\begin{bmatrix}v_1 + w_1 \\\\ v_2 + w_2 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "as is subtraction:\n",
    "\n",
    "$$\\\n",
    "\\begin{bmatrix}v_1 \\\\ v_2 \\end{bmatrix} -\n",
    "\\begin{bmatrix}w_1 \\\\ w_2 \\end{bmatrix} =\n",
    "\\begin{bmatrix}v_1 - w_1 \\\\ v_2 - w_2 \\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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cBBw0tKEsnVoP/B2wI3CvGRfTbAZW0YNBLeXa3I75COBF7v6G9vargWe6+wkj\nX5N9x/wNO/jw53PhOe/kHVzN/p88iyO/nnM9Mihb09RkT2lvPwQ8QDO0pwb1ue5cn2d5UqqF7Jhn\n9RPfzFbSXEsB4C7g8qm/JkxdXm+St0/k2X9zB0fwLt7OQ3z7aFh1NCxvl7Oq/bdu63YXty9cBLYN\nLHfgULjwUPjPJ8EJ/xO6+fOg22Xebn99DI01zGBzO+YDgRXufmh7+1Rg/egbgCXsmAEWmy1/WP0Z\nUEZ/VooJd8xGU+39A7Bfe/c64JfAeTTHyn/LvYxriuh1kZSQxUJ2zJcC+5nZMuAm4EjgVWNd3ZgU\ncYV6GZovAi8FrqM5AaeoQSz9NZvD5V5MOlzu4+7+7o0eL2LHLNI1M/YHbtUglvkYxCnZIiJ9os/8\nGxhlkSiLRFkkpWdRzWAWEamFqgwRkQwGUWWIiNSimsFcemfUJWWRKItEWSSlZ1HNYBYRqYU6ZhGR\nDNQxi4j0SDWDufTOqEvKIlEWibJISs+imsEsIlILdcwiIhmoYxYR6ZFqBnPpnVGXlEWiLBJlkZSe\nRTWDWUSkFuqYRUQyUMcsItIj1Qzm0jujLimLRFkkyiIpPYtqBrOISC3UMYuIZKCOWUSkR6oZzKV3\nRl1SFomySJRFUnoW1QxmEZFaqGMWEclAHbOISI9UM5hL74y6pCwSZZEoi6T0LKoZzMABuRdQEGWR\nKItEWSRFZ1HTYN4x9wIKoiwSZZEoi6ToLGoazCIiVahpMC/LvYCCLMu9gIIsy72AgizLvYCCLMu9\ngJmM5XC5Ma1FRGRQNnW43IIHs4iIjFdNVYaISBU0mEVECtP7wWxmh5rZNWb2X2Z2cu715GRme5rZ\nhWZ2pZn9p5m9OfeacjKzxWa22szOzb2WnMxsRzM728yuNrOrzOzA3GvKxcxObf98XGFmnzGzLXOv\naTq9Hsxmthj4EHAo8HjgVWb2uLyrymod8FZ3fwJwIHD8wPM4EbgKGPobKR8AvuLujwP+BLg683qy\nMLNlwBuAp7r7k4DFwCtzrmlTej2YgWcAP3P3Ne6+Dvgs8LLMa8rG3W9x98vbX/+G5g/g7nlXlYeZ\n7QEcBpwODPYiW2a2A3CQu38CwN0fcve7My8rl3toNi9bm9kSYGvgxrxLml7fB/MfANeP3L6hvW/w\n2t3BU4Dv511JNqcBbwPW515IZvsAt5vZGWZ2mZl9zMy2zr2oHNz9V8D7gF8CNwF3ufs38q5qen0f\nzEP/K+oVr93JAAABWklEQVS0zGxb4GzgxHbnPChm9hLgNndfzYB3y60lwFOBj7j7U4H7gFPyLikP\nM9sXeAvNySW7A9ua2VFZF7UJfR/MNwJ7jtzek2bXPFhmthT4PPApd/9i7vVk8izgcDP7OfCvwPPN\n7MzMa8rlBuAGd7+kvX02zaAeoj8FvuPud7r7Q8AXaF4rxen7YL4U2M/MlpnZFsCRwL9nXlM2ZmbA\nx4Gr3P39udeTi7u/3d33dPd9aN7cucDdX5N7XTm4+y3A9Wb22Paug4ErMy4pp2uAA83sUe2flYNp\n3hwuzpLcC1gId3/IzN4EfI3mHdaPu/sg33FuPRt4NfBjM1vd3nequ5+XcU0lGHrldQLw6Xbzci1w\nbOb1ZOHuP2r/5nQpzXsPlwEfzbuq6emUbBGRwvS9yhARqY4Gs4hIYTSYRUQKo8EsIlIYDWYRkcJo\nMIuIFEaDWUSkMBrMIiKF+f9NulZaR1VGogAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x117894e80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(6, 6))\n",
    "\n",
    "draw_vector(5, 0, color=\"blue\")\n",
    "draw_vector(3, 4, color=\"red\")\n",
    "draw_vector(8, 4, color=\"purple\", linestyle=\"dotted\")\n",
    "\n",
    "plt.grid(True)\n",
    "plt.axis([-1, 9, -1, 9])\n",
    "\n",
    "plt.text(2.5, 0.2, r\"$v$\", color=\"blue\", fontsize=20)\n",
    "plt.text(1.5, 2.6, r\"$w$\", color=\"red\", fontsize=20)\n",
    "plt.text(4, 2, r\"$v + w$\", color=\"purple\", fontsize=20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Multiplication by a scalar\n",
    "\n",
    "A _scalar_ is a single number. You can multiply a vector by a scalar.\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}v_1 \\\\ v_2 \\\\ v_3\\end{bmatrix} \\times\n",
    "c =\n",
    "\\begin{bmatrix}v_1 \\times c \\\\ v_2 \\times c \\\\ v_3 \\times c\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Mean of vectors\n",
    "\n",
    "With the ability to add vectors and multiply by a scalar, we can find the mean of several vectors.\n",
    "\n",
    "$$\n",
    "mean = \n",
    "\\left(\n",
    "\\begin{bmatrix}v_{1_1} \\\\ v_{1_2} \\\\ v_{1_3}\\end{bmatrix} +\n",
    "\\begin{bmatrix}v_{2_1} \\\\ v_{2_2} \\\\ v_{2_3}\\end{bmatrix} +\n",
    "\\ldots +\n",
    "\\begin{bmatrix}v_{n_1} \\\\ v_{n_2} \\\\ v_{n_3}\\end{bmatrix} +\n",
    "\\right)\n",
    "\\times\n",
    "\\frac{1}{n}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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vcmgOtRWnIfTczaEM5lAPC1RHelsgMcPcqbbiJGl7cgbVkRjHycCngB1Lu0h3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\npCL9fxreW+2HU8DjAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11050cba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(7, 7))\n",
    "\n",
    "draw_vector(5, 0, color=\"blue\")\n",
    "draw_vector(3, 4, color=\"red\")\n",
    "draw_vector(1, 5, color=\"green\")\n",
    "draw_vector(3, 3, color=\"purple\")\n",
    "\n",
    "plt.grid(True)\n",
    "plt.axis([-1, 7, -1, 7])\n",
    "\n",
    "plt.text(1.5, 1.4, r\"mean\", color=\"purple\", fontsize=20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dot Product\n",
    "\n",
    "The _dot product_ is hard to describe geometrically. You may hear it called the length of the projection of _v_ onto _w_.\n",
    "\n",
    "Dot products are used a lot in prediction and analysis.\n",
    "\n",
    "The formula is easy:\n",
    "\n",
    "$$\n",
    "v \\cdot w = \\left(v_1 w_1 + v_2 w_2 + \\ldots + v_n w_n\\right)\n",
    "$$\n",
    "\n",
    "\n",
    "This formula helps us figure out our next and last vector formula.\n",
    "\n",
    "## Magnitude\n",
    "\n",
    "Magnitude is the length of a vector. Its formula is $\\sqrt{v \\cdot v}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Matrices\n",
    "\n",
    "A _matrix_ is a two-dimensional collection of numbers. You can think of it as a collection of vectors.\n",
    "\n",
    "$$\n",
    "A = \n",
    "\\begin{bmatrix}\n",
    "2 & 9 & 4 \\\\\n",
    "7 & 5 & 3\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Variables for matrices generally are in uppercase.\n",
    "\n",
    "The _shape_ of a matrix is its number of rows and columns, always in that order. The number of rows is usually called _n_ and the number of columns called _k_. The matrix _A_ above is a 2 x 3 matrix.\n",
    "\n",
    "## Addition and Subtraction\n",
    "\n",
    "Matrix addition and subtraction works like vector addition and subtraction.\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "A_{1,1} & A_{1,2} \\\\\n",
    "A_{2,1} & A_{2,2} \\\\\n",
    "A_{3,1} & A_{3,2}\n",
    "\\end{bmatrix}\n",
    "+\n",
    "\\begin{bmatrix}\n",
    "B_{1,1} & B_{1,2} \\\\\n",
    "B_{2,1} & B_{2,2} \\\\\n",
    "B_{3,1} & B_{3,2}\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "A_{1,1} + B_{1,1} & A_{1,2} + B_{1,2} \\\\\n",
    "A_{2,1} + B_{2,1} & A_{2,2} + B_{2,2} \\\\\n",
    "A_{3,1} + B_{3,1} & A_{3,2} + B_{3,2} \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "## Multiplication by a scalar\n",
    "\n",
    "This also works the same way it did with a vector.\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "A_{1,1} & A_{1,2} & A_{1,3} \\\\ \n",
    "A_{2,1} & A_{2,2} & A_{2, 3} \n",
    "\\end{bmatrix} \n",
    "\\times\n",
    "c =\n",
    "\\begin{bmatrix}\n",
    "A_{1,1} \\times c & A_{1,2} \\times c & A_{1,3} \\times c \\\\ \n",
    "A_{2,1} \\times c & A_{2,2} \\times c & A_{2, 3} \\times c \n",
    "\\end{bmatrix} \n",
    "$$"
   ]
  },
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   "metadata": {},
   "source": [
    "## Multiplication by a vector\n",
    "\n",
    "When you multiply a matrix by a vector, you get a new vector.\n",
    "\n",
    "In order to multiply a matrix by a vector, you must obey this _shape rule_:\n",
    "\n",
    "*Shape Rule: The number of rows of the vector must equal the number of columns of the matrix.*\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "A_{1,1} & A_{1,2} \\\\\n",
    "A_{2,1} & A_{2,2} \\\\\n",
    "A_{3,1} & A_{3,2}\n",
    "\\end{bmatrix}\n",
    "\\times\n",
    "\\begin{bmatrix}\n",
    "v_1 \\\\\n",
    "v_2\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "A_{1,1} \\times v_1 + A_{1,2} \\times v_2 \\\\\n",
    "A_{2,1} \\times v_1 + A_{2,2} \\times v_2 \\\\\n",
    "A_{3,1} \\times v_1 + A_{3,2} \\times v_2 \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "or, maybe more clearly:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "a & b \\\\\n",
    "c & d \\\\\n",
    "e & f\n",
    "\\end{bmatrix}\n",
    "\\times\n",
    "\\begin{bmatrix}\n",
    "m \\\\\n",
    "n\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "am + bn \\\\\n",
    "cm + dn \\\\\n",
    "em + fn \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Example:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "3 & 2 \\\\\n",
    "4 & 1 \\\\\n",
    "0 & 4\n",
    "\\end{bmatrix}\n",
    "\\times\n",
    "\\begin{bmatrix}\n",
    "2 \\\\\n",
    "3\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "3 \\times 2 + 2 \\times 3 \\\\\n",
    "4 \\times 2 + 1 \\times 3 \\\\\n",
    "0 \\times 2 + 4 \\times 3\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "12 \\\\\n",
    "11 \\\\\n",
    "12\n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Multiplication by another matrix\n",
    "\n",
    "When you multiply a matrix by a matrix, you get a new matrix.\n",
    "\n",
    "*Shape Rule: The number of columns of the first matrix must equal the number of rows of the second matrix.*\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "A_{1,1} & A_{1,2} \\\\\n",
    "A_{2,1} & A_{2,2} \\\\\n",
    "A_{3,1} & A_{3,2}\n",
    "\\end{bmatrix}\n",
    "\\times\n",
    "\\begin{bmatrix}\n",
    "B_{1,1} & B_{1,2} \\\\\n",
    "B_{2,1} & B_{2,2}\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "A_{1,1}B_{1,1} + A_{1,2}B_{2,1} & A_{1,1}B_{1,2} + A_{1,2}B_{2,2} \\\\\n",
    "A_{2,1}B_{1,1} + A_{2,2}B_{2,1} & A_{2,1}B_{1,2} + A_{2,2}B_{2,2} \\\\\n",
    "A_{3,1}B_{1,1} + A_{3,2}B_{2,1} & A_{3,1}B_{1,2} + A_{3,2}B_{2,2} \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "or, maybe more clearly:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "a & b \\\\\n",
    "c & d \\\\\n",
    "e & f\n",
    "\\end{bmatrix}\n",
    "\\times\n",
    "\\begin{bmatrix}\n",
    "w & x \\\\\n",
    "y & z\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "aw + by & ax + bz \\\\\n",
    "cw + dy & cx + dz \\\\\n",
    "ew + fy & ex + fz \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Example:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "3 & 2 \\\\\n",
    "4 & 1 \\\\\n",
    "0 & 4\n",
    "\\end{bmatrix}\n",
    "\\times\n",
    "\\begin{bmatrix}\n",
    "2 & 1 \\\\\n",
    "3 & 0\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "3 \\times 2 + 2 \\times 3 & 3 \\times 1 + 2 \\times 0 \\\\\n",
    "4 \\times 2 + 1 \\times 3 & 4 \\times 1 + 1 \\times 0 \\\\\n",
    "0 \\times 2 + 4 \\times 3 & 0 \\times 1 + 4 \\times 0\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "12 & 3 \\\\\n",
    "11 & 4 \\\\\n",
    "12 & 0\n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  }
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