hands-on/math_linear_algebra.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Math - Linear Algebra**\n",
    "\n",
    "*Linear Algebra is the branch of mathematics that studies [vector spaces](https://en.wikipedia.org/wiki/Vector_space) and linear transformations between vector spaces, such as rotating a shape, scaling it up or down, translating it (ie. moving it), etc.*\n",
    "\n",
    "*Machine Learning relies heavily on Linear Algebra, so it is essential to understand what vectors and matrices are, what operations you can perform with them, and how they can be useful.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before we start, let's ensure that this notebook works well in both Python 2 and 3:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from __future__ import division, print_function, unicode_literals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Vectors\n",
    "## Definition\n",
    "A vector is a quantity defined by a magnitude and a direction. For example, a rocket's velocity is a 3-dimensional vector: its magnitude is the speed of the rocket, and its direction is (hopefully) up. A vector can be represented by an array of numbers called *scalars*. Each scalar corresponds to the magnitude of the vector with regards to each dimension.\n",
    "\n",
    "For example, say the rocket is going up at a slight angle: it has a vertical speed of 5,000 m/s, and also a slight speed towards the East at 10 m/s, and a slight speed towards the North at 50 m/s. The rocket's velocity may be represented by the following vector:\n",
    "\n",
    "**velocity** $= \\begin{pmatrix}\n",
    "10 \\\\\n",
    "50 \\\\\n",
    "5000 \\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Note: by convention vectors are generally presented in the form of columns. Also, vector names are generally lowercase to distinguish them from matrices (which we will discuss below) and in bold (when possible) to distinguish them from simple scalar values such as ${meters\\_per\\_second} = 5026$.\n",
    "\n",
    "A list of N numbers may also represent the coordinates of a point in an N-dimensional space, so it is quite frequent to represent vectors as simple points instead of arrows. A vector with 1 element may be represented as an arrow or a point on an axis, a vector with 2 elements is an arrow or a point on a plane, a vector with 3 elements is an arrow or point in space, and a vector with N elements is an arrow or a point in an N-dimensional space… which most people find hard to imagine.\n",
    "\n",
    "\n",
    "##  Purpose\n",
    "Vectors have many purposes in Machine Learning, most notably to represent observations and predictions. For example, say we built a Machine Learning system to classify videos into 3 categories (good, spam, clickbait) based on what we know about them. For each video, we would have a vector representing what we know about it, such as:\n",
    "\n",
    "**video** $= \\begin{pmatrix}\n",
    "10.5 \\\\\n",
    "5.2 \\\\\n",
    "3.25 \\\\\n",
    "7.0\n",
    "\\end{pmatrix}$\n",
    "\n",
    "This vector could represent a video that lasts 10.5 minutes, but only 5.2% viewers watch for more than a minute, it gets 3.25 views per day on average, and it was flagged 7 times as spam. As you can see, each axis may have a different meaning.\n",
    "\n",
    "Based on this vector our Machine Learning system may predict that there is an 80% probability that it is a spam video, 18% that it is clickbait, and 2% that it is a good video. This could be represented as the following vector:\n",
    "\n",
    "**class_probabilities** $= \\begin{pmatrix}\n",
    "0.80 \\\\\n",
    "0.18 \\\\\n",
    "0.02\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vectors in python\n",
    "In python, a vector can be represented in many ways, the simplest being a regular python list of numbers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[10.5, 5.2, 3.25, 7.0]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[10.5, 5.2, 3.25, 7.0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since we plan to do quite a lot of scientific calculations, it is much better to use NumPy's `ndarray`, which provides a lot of convenient and optimized implementations of essential mathematical operations on vectors (for more details about NumPy, check out the [NumPy tutorial](tools_numpy.ipynb)). For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 10.5 ,   5.2 ,   3.25,   7.  ])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "video = np.array([10.5, 5.2, 3.25, 7.0])\n",
    "video"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The size of a vector can be obtained using the `size` attribute:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "video.size"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The $i^{th}$ element (also called *entry* or *item*) of a vector $\\textbf{v}$ is noted $\\textbf{v}_i$.\n",
    "\n",
    "Note that indices in mathematics generally start at 1, but in programming they usually start at 0. So to access $\\textbf{video}_3$ programmatically, we would write:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.25"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "video[2]  # 3rd element"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting vectors\n",
    "To plot vectors we will use matplotlib, so let's start by importing it (for details about matplotlib, check the [matplotlib tutorial](tools_matplotlib.ipynb)):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2D vectors\n",
    "Let's create a couple very simple 2D vectors to plot:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "u = np.array([2, 5])\n",
    "v = np.array([3, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These vectors each have 2 elements, so they can easily be represented graphically on a 2D graph, for example as points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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+6pCWpBNn296ZKEnaGcf9ZGLFN8NExEcj4r6IONi7ZU1EnB0Rn4+Ib0TEbRHxjgJNuyPi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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x104eaffd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_coords, y_coords = zip(u, v)\n",
    "plt.scatter(x_coords, y_coords, color=[\"r\",\"b\"])\n",
    "plt.axis([0, 9, 0, 6])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vectors can also be represented as arrows. Let's create a small convenience function to draw nice arrows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_vector2d(vector2d, origin=[0, 0], **options):\n",
    "    return plt.arrow(origin[0], origin[1], vector2d[0], vector2d[1],\n",
    "              head_width=0.2, head_length=0.3, length_includes_head=True,\n",
    "              **options)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's draw the vectors **u** and **v** as arrows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x104f890d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_vector2d(u, color=\"r\")\n",
    "plot_vector2d(v, color=\"b\")\n",
    "plt.axis([0, 9, 0, 6])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3D vectors\n",
    "Plotting 3D vectors is also relatively straightforward. First let's create two 3D vectors:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a = np.array([1, 2, 8])\n",
    "b = np.array([5, 6, 3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's plot them using matplotlib's `Axes3D`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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BFhHTKiESiTgmT7jdr4CFkfJfvW11tDab9ELCDigvzdB9Zt0jxWIR9913H/bu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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105067790>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "\n",
    "subplot3d = plt.subplot(111, projection='3d')\n",
    "x_coords, y_coords, z_coords = zip(a,b)\n",
    "subplot3d.scatter(x_coords, y_coords, z_coords)\n",
    "subplot3d.set_zlim3d([0, 9])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is a bit hard to visualize exactly where in space these two points are, so let's add vertical lines. We'll create a small convenience function to plot a list of 3d vectors with vertical lines attached:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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GIWlGy09tJCp2otmNkyhb0u3s7FQSTVY7c6nJFEbbROolHyJyvjk5LbnNgpb9\nyWTSNq8t+9Ix0svBLNQeUD6JRQ+pLMuKxFGu8gQAxcrGQqv8lydhknHKEfn81FrSDHvekiQpL7HW\n1lZHGpg7hbIl3UAggLq6OsMJLDUQ6ahVetnVZlGWZYVsq6qqsH79eowaNQojR460NC77gvD7/bb0\nXWDnamcvBzNQI2JKlIZCIcfkCbeSXGooJMdIkva2OmqbTZZDwg7IL82w5d70XWezWfz0pz/FwYMH\n0drait27d+Pss8+2/MyeddZZyj0fDAaxefNmS+PxKFvSpW3HrXpsCWR74iu99EJrHiyRs2Nffvnl\n3byTRo3o7LgkI1iRV/JVvBX6rNuwIk+Um3sC0NedS2uzSYqSyxVq555IJBAMBjF+/Hh8/PHH2L17\nNy677DL885//xDPPPIOZM2eaPp7P58O6devQo0cPO6bfDWVLugQrDz3rWxUEwVJky8+DJbBAINDN\nw5vPp1toztStjPZlCwQCStMaM6CHt7W1FT6fufaNVl4gdkGvPFFJ7gk9Vjb6Q70XnEpSurk6ICK+\n6KKL0NbWhlGjRmHp0qVobW3VLCDSC7p+TuG0JF2+SCAcDkMURVt0YZYU8xEY79PVM2e91WlGQGPK\nsoxYLGbLmKUEsx5bAt80xk44RVL8KoDcEYFAwFD/iVIGe+3YRJodCTVBEDBz5kz4/X7ccMMNuP76\n6y2PyaJsSddM9pwnRCoSoGYgViGKItra2gBo958lvPHGG/D5fJg6dWrB81CzapmNlNm5so3aOzo6\nLBEuEVipP6yEQvIEbaxJ170S5AktrdiurZTcjHTZY7W1taF///62jf3mm2+iX79+OHLkCGbOnIlR\no0Zh2rRpto1ftqRL0EM2haJEq8thtgyWxi50802bNq3gkpwlRrsKJqjJDVtNR+RiFtRPGMht75jJ\nZMqKoNiomAi4qqqq7OWJfGSoZmXjk3al1n+Cf07s3qqnX79+AIDevXvjiiuuwObNmz3SBXIjXS39\nhciW3XacNlHkxzJDumwBQjAYhCRJuk3a+W5UtTLjQjd2oXNgXQ5qjgQzEk06nVaq3WKxmPJgplKp\nirE7mZUnyrW4AzC/lZIkdW1K6VbEqyYvWAU9z7FYDIlEAq+88gruuOMOW8YmlC3pEigqYSHL+nra\nEoySrtrGjxTt6gXdxPQw0xwSiYSpvrY0ptq/sTY4O+xfbGkx9SP2+/050ZAgCEoj6UJ2p3IkKDvc\nE25auayfwLB3AAAgAElEQVTCyMsnk8mYKnLQC/662RnpHj58GFdccQUEQYAoirj66qsxa9YsW8Ym\nlC3pakW6bMMYvUtyvaTL915gI1CjxP3+++9j7969+MIXvtBteW+GGPlz1HI55EMhElCrTKOIh58L\n3yvYSGNxlqDUfKduwGyVnRH3BJ0TlYA7uVR3clz6bkVRRDAYVKLdfEUOVhri8N+Nnb10hwwZgh07\ndtgylhbKlnQJRHZGeg1oQYt0nOi9MGrUKIwaNQrJZFLxxQJdTWnMPCDs8fVsKMl/Nh/0VqYZmbce\nglLznRJxu2FJs4Oo8kWI6XS6m+G/nKN/Vs+n83BqKyX2Z/F43CsDdhMUJYmiiHA4bKrXgNbvGykH\nNuuiYKPQEydOWFpyyrKMtrY2w20h841H0oQblWmFlrCs51SWc8uAy0knBk6dq9/vz5FhnCjucNNV\nkA9GVzxaVjb+fLLZrGOboDqB8pmpCuLxuKIf1dXVWbqx2C+T10HtKgdmE3uCICh6MD8HoyDZQ5Ik\nSy8e9vyNVqY5Cf5hpfOlMuBK0omNyBOl5p4wQ+5aL9p8/ScIR48eNd18Px8kSUJTUxMGDhyI5557\nzvbxy5Z0BaGrX2soFFK2Abc6Hj28ZsimEGGyXttIJILjx4/jj3/8I1paWkzPmV32k5xiZcsSNgIX\nBEGXDsyet5tZ63LUic3ADvdEqUS6RqDlKWYtbOvWrcPSpUshiiIuvfRSTJgwAf/+7/+OCy+80NKx\nV69ejdGjRyuee7tRvgXZ6Oq/oOZeMAp6WNvb201v/KhFuqIoKptVhkIhZbPKhoaGboSrN9KlxFtr\nayuAUxtKWgVtyhiJRCyXAhcDRDjsBpTRaDRHaiHvM51rMplUHmI3dGK7QC+dYDCotAatrq5GJBJB\nIBDoZpekBkZOnqvT5E7nTFrx3LlzsX//fowZMwbf/OY3UVtbi08++cTSMQ4ePIgXX3wRCxcutGnW\n3VG2kS5wKgowWyfNFk1QBGqWvOhmoxuvkNdWKxGV72HgtWA2Ejd7DWiesiwjEAiY3g2YP7dSITC9\nOjHvJ6b/d5JI7B5b7VypGCYYDCqrACfkCTe/b/a6dXR0oL6+HrNnz8bs2bMtj33zzTfjvvvuUwIa\nJ1DWpAt0Jzs94KOAaDSq9GGwOhfa0pqcDvX19XkTdXoSU7wWrLbHmVGi44slBEGwpAUTcZFdqNSh\nJU/wWmIikSh7nRhANweLU8Udbl8TO3vpvvDCC2hoaMDEiROxbt06x+7jiiBdtYymFrQ2frSjiTgA\ntLe36ypCEEURP/7xj3HLLbfknAs/B7LCUSReyJFQ6DrwSTKaJ/UaMArSTNvb2+Hz+ZSHFzi1DU0p\nJHn0gCViIiA2QqwUnRiwvzWmm7oxW1RkZ2HEm2++ieeeew4vvvgiOjs70d7ejvnz5+OJJ56wZXxC\nWZMufcl6orxCfQzMLolZEpNlGTU1Nboax/j9fixbtqzb+dAc2Kq3aDRa0Hdc6IZnpRSfz1z7Rn48\nOm+ga/cO2m6IGghRlRobRakRVSlCzXPK/iyfn7iUOnYZIUMr7gk3JSX2nOwsAV65ciVWrlwJAFi/\nfj3uv/9+2wkXKHPSJeT7wvX2MTB606h5bWkpqnfOapAkKacU2I4tcthomQhcbT56k3j8du7xeDzn\nQaWHNxgM5kRRekuBnfQCG4HWdTerE/MvGzZiK1XodU9QL+dEIuG4FMPep3ZWo7mFsibdfJEubemi\nt4+BUdJhm+gQsZjRVVl5hKIIs4UIvMzCvnD0RMuFoEbeFAnpmZseixffzcruun0noUcnZl829HM6\n33I4RwIvT9DzFolEXNm5gz7b1tbmSDXajBkzMGPGDNvHBcqcdAks2ekp2dUao5ADQEsPVpuHHqxZ\nswZf+cpXEIlElCRZKBRCdXW17jH4c6CHnC3b1ds4R2vufDvIQgk3vdeh0HKWr9tnCZh+r9RJKt/L\nJplMAkDOduxqEbGVc3TrGvFyA/vvheQJo30YeHmhb9++jp2XE6gY0iViMLKDLz9GPolCawtyvWPw\nkGUZ1157rUKONTU13RrHGAXprGaugdr5kB+YHA75nBh2gSViVnPmXQWA+lK22BqqHrDnSJFiOenE\nRqBXnjDSPJ0l3ba2NgwfPtzVc7KKsiZdfllupWRXS6LQ6iqmBT2kq9WQxqyDgPRlWe5q/GO1bJfG\nI3+nnvGcTqKwERRF8tFotKCGWsrOCZY8rOjE+c7RzUjXyHGMuCeA3BUAezxP03UZoiji5MmTEISu\nkuBYLGZ6LF6iyNfwO98Y+cBHzHxDdT0SBw+WwH0+H6LRqKWXDo0nCPrLgIuFQhpquTonWOjRialp\nvFpSshw80wS9ThGgq+/C3LlzUV9fj+rqavh8PowfPx41NTWmj59KpTB9+nSk02mk02nMmTNHcTPY\nCaHAl1LS3xhFPKIoIpvNmtZCgS4Cj8fjCIVCSqObSCRiSKIgsuKr2niNNRwOQxAE/P73v8d5552H\nYcOGAYBSpqnn5aEmebS3tyteXqNob29XtFOjHcqOHz+O+vp6pfkQFYhEo1HD89ADup56v2+1oodC\nzgkqlnFyo07yStvRIYuPElkZBugqjnDST0wrLbaBkxOg776qqgpbtmzBAw88gJ49e2Lv3r04fvw4\nPvjgA0vjd3R0IBqNIpvNYurUqbj//vuVfQwNQvMCl3WkKwiCssOp1cIGyipns1nbJIpCEfOXv/xl\nw8fQ29vW6HiZTAbBYNAWi1qpwYxzglYc5VL0oBUlUttTQRDy6sQnT/qwf3/X9Rk2TELPnsaO73ZE\nHQwGcf755+O+++7DI488gpqaGlvmQIECrR569OhheUweZU26BLPGbNZrSzerVYmCokUzrRHznQdl\nu/Ntu2M0kceOV1VVpbp/nF6k02mkUilFW6TrUKpkVWgpSw+d2m7AdhGx09eHPUfWm82+bLLZLA4f\nFvH881Xw+VKQpGPYsaMPLr/cj169jJ1jMbRj0vbtOr4kSWhsbMS+fftw4403YvTo0ZbH5FHWpJvP\np5sPal5baiJuFaIoorW1VdcWOWySAFA/DzNJrXxgdVu2aU4ikTA1HmlsyWRSOVfWXVBOpbIsSVHk\nHwgEcqSJcnMVqBE7rRAJBw74EA5nkEj8HqnUMfh8Z2D79itwwQWnOrcVKnZwq9BD7XzsPK7P58P2\n7dvR1taGWbNmYf369bb7dcuadIHc3gt6wG7rw2qX9Hmz0Ucmk0EymYQsy4jFYrq0wPXr1yMWi6G5\nuVn151Z626qBLW6wug09690Ful5cREbUuyAcDiuRY7mRFQu1KjmrroJSgiQBsnwUqdRRADIk6Riy\n2TiqqxsKFq/QeRajBNjJY9bW1uI//uM/sGXLFo901aCHMNR28OWdAzSOkYeE34Y9m83qTr7827/9\nm+p5aL0YCkHrOthZmcbKEqRTt7a2IplMKmMSEdMDKgiCksjhix/KlYgryTkxfLiM99/vBZ+vBpIU\nh893BsaNO0O3m4C+Q/o3t79Du45x9OhRpQlUZ2cn/vKXv9i+/TpQAaRbKNI1kngyEu3xPR1CoZDS\nVMcsKKogF4KZVoss2EReocq0QufOyxI1NTWKUyESiUAUxZwOYxT9sEtRNptOS1z2BaUVNVrZOdZN\naHlPtcqA6QXLVmfZDT1BREODjC98IYBdu65FMnkU5557BgYOVA8ctPzEdF/QPeFU5M+ejyiKtm4j\n9emnn+LrX/+68p1dc801uOiii2wbn1D2pAucIgz2CzHjtdVDuvy4bJWWGW2Zbkyq/ALMbcHOHp9P\n5FndUJLvuUDyAetrJg2UXhTsJpK0PFUjT9aXTETMPpRqREznmc1my4qICfQd0TUthZ4TffvK6Ns3\nAMB8SW0gENAd+evRidXAlwDb2Xdh3Lhx2LZtm23jaaFiSBc4pfEUyvLnG8ct9wAA7Nq1CwcPHsQF\nF1yAYDCImpoapS+tGdBN3tbWplsHZufOF2bwPReqqqqUhwg41VTI5/Ohuro654ELBALdtt9WI2L2\n4WOPy86LJ2LyZZPDoNx0VCIYQegq6lGTXezq22skwfXBBx+gV69ettmktCJ/vTpxofN0qtmN0yh7\n0mWjTIpAzWb587kH1LbI0fN5NVB0M3jwYAwZMgTRaFTZ18psckAURaWMmJJkduq29KDQOSaTSSVZ\npsdqRuRphIhZ6UjturBG/HLTUXmwy3atnhNOJyNTqVSOBGQEenMhRnVi/jwlSVKuj92Rrlsoe9Kl\npTQRmZXm3LyLgd0ih8/2mwW/VFcjRyPJPFazDgQC3TyZRsDKErxuy/48k8kgFApZbhWph4iJRNmy\nVpIV2IhYluVu5K+mo5YSEev5nrWcE6zPNp8GbgRjx441fA7snMxCSyfWOk9JkvDLX/4Sn332GRKJ\nBNra2ixv2XPw4EHMnz8fhw8fhs/nw/XXX49FixZZGlMLZU+6FJH5fD4lYjQLu9wDag+TmoOA/p3m\nbIQA1KLRdDptOlJhl3pqum0mk1FWEbFYzDFPphYRU/EFAIVwM5lMXo0Y6ErosdeXXdaS40Itaiz1\nwg7+Puf1U7ZZDH2PTr9o7B5X7TwTiQSCwSD69OmDTZs2YevWrejfvz8aGhrw3HPPYcyYMaaOFQgE\nsGrVKkycOBHxeByNjY2YNWsWRo4cacep5B7L9hFdBvUaiMfjln17bKRnxj2g9rv5HAQHDx7E66+/\njnnz5uWMke+BZ+WOQCBguViC1W19vq5tfOjhFYSu0lEt3dYN0PUTRRHhcDjHV836ZEnnVVtu80TM\nLuF5ImYlFJJr3ExomYWWfkrVlnTf5Os58c477+Dss882vCO2Wx5d4NR5fulLX0IikcAll1yCBQsW\n4IMPPsDgwYNNj9u3b1+lL28sFsOoUaNw6NAhj3TVwCYlzH75/BLdSt9Ydh6FEm8DBw7MIdxC0GoJ\nqXbsQqAHklwYkUhE2cWY7F7UsF2vbmsn2Oi2qqpKkToIWq4APUSsphPzREzWP7X2gnY4C9wgKppb\nMBhUrlO+RFZ7e7uymjFzfm7cH2xA0t7ejmHDhsHv99tKjh9++CF27NiByZMn2zYmi4ogXfqv0RuZ\ndyRQBZUdCYlUKlUw8aYGtfPQ20RdD/hImchMFEX4fD4lQQZA0YfdjvBo6W80ujZCxKy+yxMxKxGx\nWzHZ7Sygcd1EvkTWpEmTTJ2fm1IMe6yTJ0/ankiLx+OYO3cuVq9ebakPSz6UPekSjEZ5ag1paBlr\nFrQcTaVSuhJvFHGwuhV7HkY7ihW6BnykTNooq/eJoqhsKEnzo2iIfof9Y+fDRhY0iq7tSFwaJWIi\nV1bbZa8p2bxYks6XcS/1fhOAdeeEW+Dv7fb2dlsbmIuiiLlz5+Kaa67BnDlzbBuXR9mTLhvp8tod\nD9LptLYhN/tgsJGoEadDKpXCmjVr8K1vfSvn30nHpL6hRpqoq5EuXz3H+m1Z3ZbmrhZZskRFDyBr\n77JCxPSiot0/otGooySlRsRsNSG9cGj7JJY8edcEQY2I85U5FyMyLPR7W7ZsQVNTk/L7RpwTwKke\nxE6fI41rt0/3uuuuw+jRo7F48WLbxlRD2ZMuwefz5c3ck/0rn1XLqEShFom2t7fr/nw4HM4hXHpg\nE4mEsvS34sbgdVs1v61e3ZYlKnJe8PYuo0RML0HqUOakK0ILJDHRlkzsfcHqn+wfAN2iWDUiDgQC\nOTsm80QMuEdUekD2yHxQcxSIoqjIaVolwHY4J/gXSGtrq22FHG+++SaefPJJjBs3Dueeey4EQcDK\nlStx6aWX2jI+i4oh3XxRnl49VC/p8lowG4maTeixNrVwOGxq1wU20uLbQQK5vQ+IaKz4bbXsXXqI\nmOZAL0E7dk8wAj2Ezy671SqqeCJmNWLWU8yOR0RMFXUULDjVb8JI0cL06dNNH4eukZ6eE2rOCTOw\nM9KdOnWqabulUZQ96Wol0ozqoWpj8GCTUFpJMqOkS+RH3c8oojQLWZbR1tYGAAqRsFEY+W2diiz1\nEDEVs9DvsqTlRqRHcosZwlcjYgA5GjH94RNRdG+wjYHoGhBJs2OVc7tILS1dyzmhxxnCv0Ao/1Bu\nKHvSBXI7jaktqfUSSz7CZDdszKfZGomWOzs78eijj+KrX/0qzjjjDEVfNQMiEkmSlPmxui39PJ9u\n6xToAaSETDAYVJbdZqUJM2ClBDsq6lgQaeghYgIrZfBRllp1nVaZsxYRG3n5d3Z2Yu/evRg3bpyh\n86bj6I2mtZwTepwhdN7suZX6y0cNFUG6BEmScPLkSVt7L1CChSq1rPQ0ALo7J2666aaceRqNlNmX\nDJEIRY+sbkvyitt+W+DUNVQjfDPShFEiLpZ2zBIxrZKSyaTig2WX3YU6sAHqRCxJubsBqy3b9Vwr\nGscMrFjGjDonAGDHjh3Yvn07gsEgMpmM6bJ3woIFC/D888+joaEBu3btsjSWHpT1bsCEeDyORCIB\nSZJQW1trWh+UZRknTpxAjx49IMtyToctvdVpWjsCA7nRstayNt/n+bmyum0kElGkBdbulM1mFQ+y\n22SrVU1mBGr6qREiZm1o9NJxG/TS8fl8CIfD3YIBrXNUI2L+eWWX4bSioaiRSJuuj1UNVQtu7QSc\nTqchiiJ27NiBX/7yl3jttdfQ2dmJkSNH4uabb8Y111xjatw33ngDsVgM8+fPt5N0NW/0ioh0KQqN\nx+OWls1085qVJ2iMfMUNfLRMRQlGjsGSN6/bxmIxRbelcelmddJjy4JeCFrVZEZgJllHGmkmk3FE\nStCLfM4IFnrOUYuIeR8xLcupax1F92oaqpl+tlrn6da19fv9uOCCCzBx4kR87Wtfw7PPPotdu3ah\npqbG9JjTpk3DgQMHbJxlflQE6Uaj0YIe3UKgZT/QRZJWtmFnG41QMi8Siagm8/74xz/i/PPPx5ln\nnql8Xmv1wTsx8um2sVgsp/RTjaR4ErYjQWO2mswICpEU2/iHXkiZTMa1JBQrZ1CfZKPHNELEarou\n/ZzuJ77xD6uhZrNZtLa24uTJkzj77LNtKXN2AmrVaNFoFFOmTCnyzIyhIkiXXV4RmegFXzBBS3+z\nZEFz6Ozs1NVI/corr1T9PAu1pjms3xZADhnz2l+hB5h8llaI2IlqMiMggqDtb4jw7TxHPWCvg91W\nOCNETGCj63wd2MiuCEA1mVXIVeC2v7pcG5gDFUK6BNZ2owdqvW3b2tpM+WwBKHYg2pjRju3S+XJl\nQN1va7SSyy4idruaTA3sHHgpwemXjZ45OAn2e6Q5kKxDHmA9Hdh69OihFBqwL21aSeVzFZh9XoyC\nJfeTJ0/aWgLsJiqCdLW8ulrgy2LZJJlR9wCBomW6KfVqTKxHlT1+Pt0WgFK6a2c2Xo2I2Qwyu808\nacKiKCIQCBSl7SOQK2fouQ5ORP1soqwYVXX8HGpqalTLd410YCOCpX+jlwibrHN7J2BWXrA70uW1\ncSdREaRLKESY+Xrb6h2DB0vgFOUlk0ndn3/ttdfQu3dvTJw4EcApuYOKJfLptm5UctEDxB6HiI6I\nKZvNIpFIqCbqnIr2SMKxQ84wS8SCICj6cbHseEaSdWrFCkTEBw8eRCaTQd++fXOiWPos/S47HhEx\nlTLT/enUlkLsc2nnVj3z5s3DunXrcOzYMZx55plYsWIFWlpabBlbDacF6bLL9EIaq17S1SJw0hT1\nYubMmTnjUdRWW1urqduatV9ZBXk51RwBbBTFJrLUokUrcEvOyEfEFOGx50jfu5sVY9RPxOxqhyVi\nemHU1taqFnWwHmA2IqZiDSJotvAFUN/N2UqZM/1ua2srevbsaeh8tfDb3/7WlnH0oiJIV0saYJfp\ntOQvFBkWIt1CBG40UubHi0ajSCaTSKfTyg1ZCpopWcC0tuvhK7LYkk8iYlEUcx50o0RsVEqwG2yy\nDjhVZu12so71P9NqyCrIPQMgb0TMShMEsuipRcTAqWReISIuVObMygvt7e0YMmSI5fMuBiqCdAns\nF1tolwU9Y7DgXQ5aBG6EdDOZjLLNEOlw5Nul3gBAF6GFQqGi1JnnqybLByIonojZh5d2ny1U6EBS\nAmuVcxulkqyzakUzA16aoAQ0fbek7xLZ0vdJ378aEWv1m+A3ImWJmCVdOzuMuY2KIF020s1ms2hv\nb1ceUKNZZDXSNLtRpRZYv+0HH3yA9vZ2zJgxQ3kwQ6GQkrAigqGHF7B/ya4GO6rJeBTSFVnfKJsZ\nF0WxaFE+cCrC9vv9RUvWsRq2E1r+vn37UFNTgz59+mj+DvviUbsn+NUN/QEKd2AD1ImYLXMGuiS2\nNWvW4NixY7bdC3/+85+xZMkSSJKEBQsW4NZbb7VlXC1UBOkCp3azpSWXnq5iamB9spIkGS4FJtJW\nq9LhO59VV1ejsbExJwro7OzMS3S8dsov2QOBgKVqMzuryfRAi4iJ6AAUpaoOsHcZb5aIafXjtLzE\nRqpaPy/k0NBa3RQiYnYsnoj56rpMJoOPP/4YGzZswB/+8Af06dMHM2fOxM9//nNT5y1JEr75zW/i\n1VdfRf/+/dHc3Iw5c+Y4siEloSJIV5IktLa2Kje10d1MWRDpUnGDmU5lPHjdVs1vy+q2+YhOTTtl\niTiZTOpasqvBjWqyQuCJjhI87Hmq7VxBLxs7tFO3XjyFiJiCCOCUB92pyroRI0ao/jtLdmZePHqI\nmNWI1Yow6HsHuqTCe++9F1/5ylfw1ltv4ejRozh06JDZ08bmzZtxzjnnKDsJf/WrX8Wzzz7rkW4h\n+P1+1NXVKXYqs2CLGyiTa4Z4iCTIUsTqwGwkDXQ18UgkEkp0biUDzZ4HHxHnI6hiV5PRnPXu/qu2\nc4Vd2il7DxXjxUPnSURELx72heNGsg44Fd3qlVX0Qo2IAe1WmPQ8bdmyBX369MGuXbvw7rvvIhqN\nYsSIEZovDD04dOgQBg0apPx94MCB2Lx5s6XzK4SKIF2gi3jJ0G0GfHGDlZ1AKcNNiSLS4Hi/bTKZ\nxMGDB/Huu+/iS1/6kunjqR1fjYjVCIpuaFohFCO6tZKss0s71et3dRpay3jeK233C2fHjh2orq7G\n4MGDEQwGLUW3ZsGv4kRRRCKRUM79mWeewcsvv4wjR46gubkZ3/nOd/D973+/7BJqFUO6gLlqMjap\nFYlE4PP5FNeDGRDxU/Sq1ieB1W1HjBjh6FKGwBIULVPp4aYXAvvS4fVhJwjIqWSdUSIGulYcxdqn\njeZI94UeorPzhZNOp7FhwwZ0dHSgZ8+emD17NqLRqGvuCB7sC5AClhdeeAHvvPMOHnvsMTQ2NmL7\n9u3YunWrqW2tWAwYMAAfffSR8veDBw9iwIABVk8hLyqGdFnDth7wSS1KvNGSxihY3RZATsMVgl7d\n1knQ8lmWu5rC8FlwN4oceN+vG8k6tfJmPuJnl9OsDu7k3EjSov4aTrbBpNJx9sXq9/tx+PBhxZ1z\n/PhxdHR0oFevXnadoiGw30FNTQ3a2trw7W9/Gz6fD6+88ooS1V588cW4+OKLLR+vubkZe/fuxYED\nB9CvXz88/fTTeOqppyyPmw8VQ7pAfucAgd6iWh3AzBQ38AUYiURCIRQyz2v1SaDPW+1+Xwj5qslY\nFCpyIMmEJAyKhvU6CcxKCXaCrjnpx+RKcauqjuC0DQzQ108jEomgvr4eJ0+eRI8ePdCzZ08lGetW\nYMAn7AKBANatW4c777wT3/nOd3D55Zc7Mhe/34+f/vSnmDVrlmIZGzVqlO3HYVERO0cAp9rRHT9+\nHD169Oj2BfHkqNW+UZa7do/QU2Ko5t9ls85sSXAgEEAwGOy2vGtvb8fvfvc7LFy40Iar0B18VBkK\nhWwpxVVLeuRzTDghJZgBS/qFNOxCdiezRMwnDfXuSmI32O/E7/fj2LFjqK+vV1ZofETsRLIOyNWx\nI5EIOjs7cfvtt+PYsWN4+OGH0bt3b1uP5xI0L1LFkC5le0+cONEtelVr4agFIl014ibwzcmp1pxN\nTNHNTCTHZp8pceV0gYMRgrEKtVJRckyQbEPJumJppnYkyrTsTnq90qw7oliJS+BU34ZgMKi5lZOW\nm8AuIuaLLQKBADZt2oTbbrsNixcvxrx584ryMrIJlb1dDwtWHqAbnAR5PdVprDasFi2z0oRZv22+\nZazR5boaihFVqjkmiPQlSVKSde3t7a5ETwRWM7UjUablOy3klaYih2JuHwScujdYV40WeKmJPs/6\niNU0Yj3fKSWwyaWRTqdxxx13YM+ePXjmmWccT2YVExUT6VLU0draikgkoiRJQqEQIpGIoRv8xIkT\nOR5dXpqg8bT624bDYUMPNruPld7luhpKZdmaL6pUixLpoeVfOFbnzmqmbm9KyRIxX+RgtGjFrvmw\nfRvs3KjUSEQMoFsp8c6dO7Fs2TK0tLRg4cKFRVkJOYDTI9KlF0gikUAwGDS8qSSBjZbVpIlsNtvN\nb0s/N/NgP/roo1i4cGFOMk2rAksroiiFajL+wVaLKtnEDu0ea3cCi122FqtnA0XDfL9dvuLMbPWg\nETidsDMSEQNd1+a1117DiBEj8Mwzz2DTpk148sknMXToUFvnVaqomEg3lUqhtbUVkiQhFApZ8u+1\ntrYiHA4rZFdIt3VrCc8ndUgfpvlQdFuMSIHVKkmfM4t8uimvhfPXnE/KFCtqYnvd5lv50D1F50gv\ndPJPW5Fg2JdgsVc+6XRaeRlns1nMnz8f27dvRzwex5QpUzB58mSsXLmynDVcHpUf6dJDRn1ozYIe\n+EQigXA4bKlPgt3gCxxoPyx6KCVJQjweB2CfPlwIrJRgl1apRzflo0TSTFmtstQ1U8BY9aAR3ZSN\nbou18qF5ULERVXk+9NBDSCaTWL9+Pc444wxs3boV+/fvryTCzYuKiXSpyxjtK2a06Q2r28qyjHA4\njKqqKlXd1u/3IxwO23YjUzWUnpcFG72oRVFm7FxmUGgeboDIKZ1OK/0yZFm2JUo0MxenNFMan/9O\n1cVMe0MAABfySURBVIiYijxKIbql60Ev43/84x9YtGgRPve5z+HWW28tSo8PF1H5kS7dWGZKgXnd\nlm+ZSCWaVnTbfPjTn/6ECy+8EH379s37e2w1mdY8CjXA0aMPF4KeebgBiirZ6jonmuAUAl/l50RU\naaTaDIBSmMPqqG6Bomy6HoIg4Fe/+hWefvppPPTQQzj33HNdm8uePXtw5ZVXKrywf/9+3HXXXVi0\naJFrc+BRMZEuqxtls1lUV1cX/AzfL5d0W4qc2JJgKm7QG5HaCb3VZEagpQ/n8w87ISWYnbuR7c7p\nXFndVO1cjTomWLdIsa8HWwATCAQKFnM4Ud6spiF/8sknWLRoESZOnIg777xTSZ4WA5IkYeDAgdi0\naVNOZzGH4EW6LFi/LfXLpQcTQI59i25iki+o2KBQQscO8A+T3e311MpDtVwE5JAoZlMYwNx250bP\nVY9jgnymxSxnpnmotaLUchI4Vd7Ma8iCIOCpp57CmjVr8MADD+D8888vuma7du1aDBs2zA3CzYuK\nIV0C759lQSTGNtQg2xeBbmK/36/6MOVL6JjVEROJBI4dO4a+ffsqtrFi9ChQ67tAy1bKqFO7Pbe9\npnbu4gDk7zHBtuXkX7A+n09ZdRSznNlIlF2on4ZVIqZghOx5R44cwdKlSzFw4ED89a9/tdwJzC78\n7ne/w1VXXVXsaVSOvABAkQUSiQTq6upyfsbqtmxjaLJcsTqUUSN9viQH/8CqzfnnP/85UqkUevXq\nha9+9auKhajYD7XaEp7Xh7USOnZppk4nqAodW620GTi1GaWbBQ4EJ0qJ+ZcOnXO+xkasU4Pm8dxz\nz2HVqlW455578LnPfa7o0S0hk8mgf//+eO+999zq5VD58gKBlxdY3ZaKG+jmohuC+pia1eWMLF95\nWeLIkSNIp9OQZRnHjh3Dxx9/jEGDBhW1l2m+slk2Uae1gwMldKz2l9BaOrsFOldyBNAL2e8/tbOD\n1aSkETipIRu16dEKsbW1FXV1dWhvb8fy5csRDoexdu3abkFPsfHSSy+hsbGxJJrnVBTpsn0TCum2\nABzt6VpoLzO6gcPhMHr06KE02RkwYADC4bBt8zACs5VLRjXTQhGi0USZU+CjbPYesdtXWwjsC8gt\nTV3NCUP3iCiKCAQC+MMf/oCVK1ciHA5j4sSJmDNnDv75z3+WHOk+9dRTJSEtABUmL1CN+8mTJxWN\nlZajrM5Lpbs+n89Wv61RUFSYTCaVXqZESIVkCTvhBsnp9Q/TkrVY3l+C1b4Nakt1wLhmynfiKpbc\nBJyS6KhbXDwex3e/+10kEglcf/312L9/P7Zs2YJLL70Ul19+eVHmqIaOjg4MHjwY+/fvR01NjVuH\nrfzWjkCXTNDe3o5sNotYLJZXty3WBoxA994ArIFdjZgA5yrM2J4NbpfN8vpwJpMBAEtJSTvm5FTT\noHzfrRoRk0Oi2CXNrFWQXkBvvvkmvve972Hp0qWKD9ZDDk4PTZeST4lEQtGc6GYoFX8pW8WltkzU\nK0tYJSY2kivWC4g0RGr4Tt8NS8R26cN64LSGrOUioGiYdUxQkFDMfhpA9+1zkskkvv/97+PAgQN4\n9tln0a9fP9fm0traioULF+Lvf/87fD4fHn30UUyePNm149uFiop0qZIskUhAFMUcwZ92TSi2n9KM\nO4KHFbdEvijbbejNwjsd/ZeKhgwgxwtOcosTZdyFwEe3wWAQW7duxfLly3HDDTfg2muvdf1FcO21\n12LGjBloaWlRpA7qjVKCOD3kheuuuw6ffvopzjvvPMRiMbzzzju4++67EY1GNauQnL5xnKgm0zpO\nIWKiB6mUlqtmdEo7+0uUSleyfNdEb98Fu2QY/pqIooh7770X27Ztw89//nOcddZZlo9hFG1tbTj3\n3HOxb98+149tEqcH6cqyjLfeegvf+ta3cPDgQUyfPh2HDh3COeecg+bmZkyZMgXDhg0DcGp7H/ZB\npRJfu/yldu9NZvT47DKdbQPJekzd1EsB/S0PjcKof1hNpyxmgoqW8HqviR4ipnPSe15qSbv33nsP\nN998M6688kp84xvfKNpLaefOnbjhhhswevRo7Ny5E01NTVi9erXhxlYu4vQgXQB4+eWXsXv3btx0\n001K787du3djw4YN2LhxI9577z2EQiGcd955aG5uxqRJk1BfX69647LEZARu7k2WD3xSiNVLjcoS\nVlGMXRy0+ktQG0y/31/074dfwluB0UQdCz5pJ0kSfvKTn2Dt2rV45JFHMGLECEtzs4qtW7diypQp\n2LBhA5qamrBkyRLU1dVhxYoVRZ1XHpw+pFsIsiwjHo9jy5Yt2LBhAzZt2oTDhw/jzDPPRFNTEyZP\nnowxY8bA5/MZ1g9LZcdbIHeJmM8W54ZeWgpbCAGnCmVozzbWt+2UO0QLZqJbM9BDxOQcoXt27969\nWLJkCS655BLccsstResix+Lw4cM4//zzsX//fgDAG2+8gXvvvRf/+7//W+SZacIj3XyQJAkHDhxQ\nouGdO3dClmWMHz8eTU1NmDJlChoaGnJuYNY9QBn4UkhO8T0KjC6bC+mlRmQJvcTvNNS6X7F6qRv9\nh9m52Bndmjm+mk3vjTfewNNPP41oNIqdO3dizZo1JecMmDFjBtasWYPhw4djxYoV6OjowL333lvs\naWnBI10jIG1r+/bt2LhxIzZu3IgDBw6gV69eaG5uxuTJkzFx4kRUVVXhk08+Qc+ePbvVqLutEToZ\nURbSD3kZxmqizE6Y6VPgVH8JVs82ulmqneDLiYPBIHbs2IH7778fR48eRWdnJ9577z3cdNNNuP/+\n+4syRzXs3LkTCxcuRCaTwdChQ/HYY4+VXOUbA490rUKWZRw+fFgh4ddffx0ffvghgsEgli9fjgsu\nuABDhgzJ8V06laTjUQwNWWvZSv5SIpZiugHstIGZ6T/MfpZKZ4sR3bJgt88h4n/yySfx+OOP48c/\n/rES3dKeg3369CnaXMscHunaia1bt+KSSy7BsmXLcPHFF2Pr1q3YuHEj9uzZg+rqajQ2NmLSpElo\nampCTU2NrUk6FqWmIVMLSLKnmZUl7JiLGzYwPXo4fUdud0jjwUos9BI6fPgwbr75ZgwdOhQrV64s\nZSdAOcIjXTshSRIOHz7crRpHlmW0trZi8+bNSpLu+PHjGDJkiGJZGzFihFKwQQ+p0YbovB2t2A+z\nVkRpVJawYy7FlDUK2fTcKmzgwbct9fl8eOaZZ/Dggw/ihz/8IWbMmOH6/XPWWWehrq5OqdDbvHmz\nq8d3AR7pFguSJGHfvn1Kku6dd96B3+/HhAkTFH24V69eOVFTPu2QIkrAvl6qZmEmomTlFzvdEk75\nf82A5kL+bL75jZutIPkE4okTJ7Bs2TLU1dXhRz/6UdEquoYOHYqtW7eiR48eRTm+C/BIt1QgyzI6\nOjoUSWLz5s04dOgQ+vbtq/iGx48fn7PPFYCcJii0U3G5OiRY8P0HjLol7N5Rwgr4pt5aVisr+rCR\nubBtOn0+H15++WXcfffdWLFiBS677LKiNqkZMmQItmzZgjPOOKNoc3AYHumWMmRZxsGDB5Uk3bZt\n25BOpzF27Ficd955SCQSSKfTaGlpUaSJYmilbu3ioEeWIJteqUgsVq+LnX5pdvucUCiE9vZ23Hbb\nbchkMnjwwQfRs2dPU+dpJ4YOHYr6+nr4/X7ccMMNuP7664s9JbvhkW65IZ1O4/e//z2+973vQRRF\njB07FgDQ2NiIyZMno7GxEZFIxLXKMjPWKzvBRsP0X+AUKdF5u028TlXamfEPq22f87e//Q233347\nvv3tb2Pu3Lkl04Lx008/Rb9+/XDkyBHMnDkTP/3pTzFt2rRiT8tOeKRbjvj+97+PM888E9dddx0E\nQcCxY8ewadMmbNiwAW+//Tba2tqUvhKTJ0/G2WefDQCWknQ8SqkDF59ArKqqsqWIw+xctAounEI+\n/zBtKZROp9GjRw+k02nceeed+OSTT/Czn/0MDQ0Njs7NClasWIGamhosXbq02FOxEx7pViL09pWg\njS6NNkQpZoNzHnoibSIlVh9Wc0sYaQKjBl4vLWYyk3y3lID97//+bzzxxBOKdbGlpQXTpk0rib3B\nCFSKHYvFkEgkMGvWLNxxxx2YNWtWsadmJyqLdP/85z9jyZIlkCQJCxYswK233lrsKZUEtPpKDBo0\nSCHhsWPHqvaVYEmJrFelkJyyul1NPrcET8R6xnI7us0HfvucdDqNu+++G7t378bll1+ODz/8EJs3\nb8bcuXOxYMGCos2Txz/+8Q9cccUVSnR+9dVX47/+67+KPS27UTmkK0kShg8fjldffRX9+/dHc3Mz\nnn76aYwcObLYUytJ5Osr0djYiClTpqBv3745ESLt6FBVVeVoJV0hONEUppBbQqt6kPe6FjO6Vevf\nsGvXLixduhRXX301brrppqKuSjwAqCTS3bhxI1asWIGXXnoJAHDPPfdAEAQv2tUJrb4SVVVVOHbs\nGMaPH49Vq1YhHA673v6RnaObZbOFZAmKcEOhUElEt/QiogbjP/7xj/H666/jkUcewTnnnOP6nCRJ\nQlNTEwYOHIjnnnvO9eOXKCpnj7RDhw5h0KBByt8HDhxYidUsjkEQBITDYZx//vk4//zzAXQlMn7y\nk5/gqquuQjQaxTXXXIOOjg6MHDlSSdJRXwnaEsmJKiuKQKmwgN3y3EmobTXOJu2oqoy2kjcqS9gB\nteh29+7dWLJkCT7/+c/jlVdeKVr0vXr1aowePRptbW1FOX65oexI14P9uOCCC3DjjTfmZLhFUcS7\n776LDRs24MEHH8zpK9Hc3Izm5maEQiFIkoR0Om151wI+OVXMHq58F66qqirl3ykaJmuWG02N2Mq/\nWCwGWZbx8MMP49lnn8XPfvYzxU5YDBw8eBAvvvgivvvd72LVqlVFm0c5oexId8CAAfjoo4+Uvx88\neBADBgwo4ozKHzNnzuz2b4FAABMmTMCECRNw4403dusr8atf/Sqnr8TkyZMxcuRI+Hw+JdkEFE5Y\n8S0po9FoUZfvrEuC3xFYEASFgIHusoQdLx8WaknEAwcOYNGiRZg2bRpee+21oiY5AeDmm2/Gfffd\nh9bW1qLOo5xQdqTb3NyMvXv34sCBA+jXrx+efvppPPXUU7YfZ8GCBXj++efR0NCAXbt22T5+uUEQ\nBNTX12PWrFmKtYftK/Hkk0+q9pXo3bu3amRIZJRMJiEIgiNbnhuBWnRbiCi1ZAm2Qbjelw8Pdvuc\nWCwGAPif//kf/OY3v8Hq1avR3Nxs8Yyt44UXXkBDQwMmTpyIdevWoUB+yMO/UHaJNKDLMrZ48WLF\nMuaE3eSNN95ALBbD/PnzPdLVCb6vxKZNm/DJJ5+gb9++aGpqwqRJkzBhwgQIgoCPPvoI/fv3B9C9\n1NXtzDtFt9SP2M7jk1uCLWjIJ0uoRbefffYZFi9ejFGjRuGuu+5COBy2bX5W8J3vfAe/+c1vEAgE\n0NnZifb2dnzxi1/EE088UeyplQIqx73gJg4cOIDZs2d7pGsBfF+Jv/71r/j4449xzjnnYOHChWhs\nbMTgwYNzlulObZWjNjcrHmArx1VzS/h8PoWgjx8/jrPOOgt/+tOf8PDDD+NHP/oRpk2bVjJlvDzW\nr1+P+++/33MvnELluBc8lBcEQcCgQYMwaNAg+P1+PPXUU3jggQcwfPhwbN68Gffddx/27duHuro6\nJRpuampSSnzt1kkJ/PLdzeialyWI/FOpFAKBAD799FNceumlyGQyqK2txfz58xWZwkP5w4t088CL\ndO1FPB5HOp3u1uVKlmXNvhK0Q/Pw4cNzuo8B5jpwFSu61QK/fY7P58MLL7yAH/7wh1i6dKnS4Hv/\n/v344x//WLR5ejAMT14wA490iwc9fSV69OjRraqM14ZZQnVrGx89UNs+p62tTSnyWb16dSU3+D4d\n4JGuGXz44YeYPXs23nnnHUfGP3jwIObPn4/Dhw/D5/Ph+uuvx6JFixw5VrlDlmW0t7djy5YtSpLu\ns88+w5lnntmtrwTppdQYnP03KiwodnTLb5+zbt063HnnnbjtttuUvgRuIZVKYfr06Uin00in05gz\nZw5Wrlzp2vErFB7pGsW8efOwbt06HDt2DA0NDVixYgVaWlpsPcZnn32Gzz77DBMnTkQ8HkdjYyOe\nffZZr4+ETmj1lRg3bpwiS5w4cQLJZBJjxoyBLMuu7dCsBrWGOR0dHbj99ttx7NgxPPzww0XrBtbR\n0YFoNIpsNoupU6fi/vvvx9SpU4sylwqBl0gzit/+9reOH6Nv377o27cvACAWi2HUqFE4dOiQR7o6\n4fP5MGTIEAwZMgTz5s3L6Suxfv16zJkzB//85z9xySWXYMyYMWhubsZ5550Hv9+vmqSze6NMFmzF\nXXV1NXw+HzZu3IjbbrsNixcvxrx584oafUejUQBdUa8kSZ604SC8SLdE8OGHH+LCCy/E3//+d8UM\n78E8rr32WkiShAceeADpdFqRJLZs2ZLTV2LSpEkYOnSoLUk6LfDb56RSKfzgBz/Anj178Mgjj5RE\nRaUkSWhsbMS+fftw44034oc//GGxp1Tu8OSFUkY8HseFF16I22+/HXPmzCn2dCoCyWRSs4iA7Sux\nceNG7NmzB9FoFI2NjZg0aRKam5tRW1trKEmnBrWNKnfs2IFly5ahpaUFCxcuLLkWjG1tbZg1axbu\nvfdezJgxo9jTKWd4pFuqEEURn//853HZZZdh8eLFjhzDS5TkB99XYtOmTTl9JSZNmoRRo0Ypzd9F\nUQSAbgUcLIGy27CHw2GIoogf/ehH2LhxIx555BEMGzasWKdbEHfddRei0SiWLVtW7KmUMzzSLVXM\nnz8fvXr1crxDk5coMQZJkrB3716FhHft2gW/34+JEyfm9JVQq6SjCrOqqipEIhG8//77WLJkCb74\nxS9i0aJFRe0xoYajR48iGAyirq4OnZ2duOSSS3DHHXfgoosuKvbUyhke6ZYi3nzzTUyfPh3jxo1T\nKqxWrlyJSy+91LFjdnR04MILL8Tjjz+O0aNHO3acSoNaX4lDhw6hb9++SqvLbDaLw4cP49JLL8XJ\nkyfR1NSEc845B0ePHsXy5csxd+5cpd9EKeGdd97B17/+daU8+ZprrsEtt9xS7GmVOzzSPd3hJUrs\nB/WVWLduHVatWoV9+/Zh+vTpGDBgAAYPHoy1a9di9OjR6N27N95++21s3boV+/fvRyQSKfbUPTgP\nzzJ2usPn82H79u1KomT9+vVeosQiqK/E3r17MW7cOLz22muorq7Gzp078etf/xo333wzZs+erfw+\n7UDh4fSGF+mehvASJfaCtvApNrwKx5KC5tu1tPwqHhzB0aNHlc7+nZ2d+Mtf/oKJEyc6djxJknDe\neefhC1/4gmPHKCWUAuECXbt9rFq1SrHDPfTQQ/i///u/Yk/LAwdPXjgN8Omnn3ZLlDiZmfY2KiwO\nvArH8oBHuqcBxo0bh23btrlyLG+jwtLAhx9+iB07dmDy5MnFnooHDp684MFW0EaFXsKoeIjH45g7\ndy5Wr17tlZSXIDzS9WAb2I0KaW8wD+5CFEXMnTsX11xzjVdSXqLw3AsebIO3UWHx4VaFo4eC8Ioj\nPLgLNzYqPOuss1BXVwefz6dsa3M6oxgVjh404RVHeKg80I4LXu/XLkydOlVpTemhdOFFuh7KFkOG\nDMGWLVtwxhlnFHsqHjzw8IojPFQeBEHAzJkz0dzcjDVr1hR7OrZhwYIFaGhowPjx44s9FQ8OwCNd\nD2WLN998E9u2bcOLL76Ihx56CG+88Uaxp2QLWlpa8PLLLxd7Gh4cgke6HsoW/fr1AwD07t0bV1xx\nRcUk0qZNm+bp1BWMQpquBw8lCUEQogB8sizHBUGoBvAKgBWyLL/i0PHqAPwSwFgAEoDrZFne5MSx\n/nW8wQD+V5ZlT2OoMHjuBQ/ligYAzwiCIKPrPn7SKcL9F1YDeFGW5S8LghAAEHXwWB4qGF6k68FD\nAQiCUAtguyzLrm1s5kW6lQtP0/XgoTCGADgqCMJjgiBsEwThF4IgOL39g4A8tiMP5QuPdD14KIwA\ngPMAPCTL8nkAOgD8l1MHEwThtwDeAjBcEISPBEFocepYHtyHJy948FAAgiA0ANggy/LQf/19GoBb\nZVmenf+THjx0hxfpevBQALIsHwbwsSAIw//1TxcBeK+IU/JQxvj/ATo4DBHS6l7zAAAAAElFTkSu\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105166e90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_vectors3d(ax, vectors3d, z0, **options):\n",
    "    for v in vectors3d:\n",
    "        x, y, z = v\n",
    "        ax.plot([x,x], [y,y], [z0, z], color=\"gray\", linestyle='dotted', marker=\".\")\n",
    "    x_coords, y_coords, z_coords = zip(*vectors3d)\n",
    "    ax.scatter(x_coords, y_coords, z_coords, **options)\n",
    "\n",
    "subplot3d = plt.subplot(111, projection='3d')\n",
    "subplot3d.set_zlim([0, 9])\n",
    "plot_vectors3d(subplot3d, [a,b], 0, color=(\"r\",\"b\"))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Norm\n",
    "The norm of a vector $\\textbf{u}$, noted $\\left \\Vert \\textbf{u} \\right \\|$, is a measure of the length (a.k.a. the magnitude) of $\\textbf{u}$. There are multiple possible norms, but the most common one (and the only one we will discuss here) is the Euclidian norm, which is defined as:\n",
    "\n",
    "$\\left \\Vert \\textbf{u} \\right \\| = \\sqrt{\\sum_{i}{\\textbf{u}_i}^2}$\n",
    "\n",
    "We could implement this easily in pure python, recalling that $\\sqrt x = x^{\\frac{1}{2}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "|| [2 5] || =\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "5.3851648071345037"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def vector_norm(vector):\n",
    "    squares = [element**2 for element in vector]\n",
    "    return sum(squares)**0.5\n",
    "\n",
    "print(\"||\", u, \"|| =\")\n",
    "vector_norm(u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, it is much more efficient to use NumPy's `norm` function, available in the `linalg` (**Lin**ear **Alg**ebra) module:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.3851648071345037"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy.linalg as LA\n",
    "LA.norm(u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot a little diagram to confirm that the length of vector $\\textbf{v}$ is indeed $\\approx5.4$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Wc4HdbFZyFfxwno/Gp59moqkJSktDR3FTjI+PmxzQzmWSd87TKD95kpV33snpV19lYvXq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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105198250>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "radius = LA.norm(u)\n",
    "plt.gca().add_artist(plt.Circle((0,0), radius, color=\"#DDDDDD\"))\n",
    "plot_vector2d(u, color=\"red\")\n",
    "plt.axis([0, 8.7, 0, 6])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looks about right!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Addition\n",
    "Vectors of same size can be added together. Addition is performed *elementwise*:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  [2 5]\n",
      "+ [3 1]\n",
      "----------\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([5, 6])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(\" \", u)\n",
    "print(\"+\", v)\n",
    "print(\"-\"*10)\n",
    "u + v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at what vector addition looks like graphically:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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PYGzSWIZfM7zOyzuWf4xDJw+Vfh08eZBiXUx+UX6Znx86eYhTBWfmQV9+3uV0\nv6A78zbMK7O8OVlzUEpx3xX3+Rdk1y7o0gXatavz7yREfRTWhTojI8O6UaZV9cmmrJrC6OWjGXHt\nCMb0HOPKc/VL68d5k84r/frZpJ+xM2cn8zfMr/DzSSsnlXlsyhUp7Dm+h6XbzwxDmrdhHp3P60zX\nNl2dh9Aa4uLOtJ8csrWfaGMuyeSMjZmcCutTcZWOMA3yKFN/Tc+czrAlwxhy1RAm9q5+18Hq+tCF\nxWXnM0++aTJH8soOOhz43kASWifwxDVPlP4sKyuLO6+4s8z9+nfpz/Alw5n79Vx6X9ybz378jO1H\ntjPpxrIFvUZvvgnffCOHiwtRF1prV77Moixjtue8TlGtOVlzNM+iB70/yNH9X/3yVR0xNkKv27uu\nzM/zCvJ0g7ENdK+3elX7+Pi/xOv7F93v6Ll+m/Zb3eyFZjr3dK7+w/t/0A3HNdR7j+919FittdZF\nRVrfdpvW69c7f4wQ9YivbtZYX8O39eHhKFOnFmxaQMqiFJI7JzOr7yxHj+l4bke01mVaEgCTV02m\nWLs7nznlihROFpzkb1//jYXfLOSmS27y/+QEQ4ea/rQQotbCt1CXjDLt6kc/NQhK+mTpW9NJXphM\nnw59SLsrzfHje1/cm06xnRizfAwjl45keuZ07v3nvczOml3r03BV1bu7rcNttGrSihFLR3As/xgp\nV6Q4X2jJsP/rr6/Vvuu29hNtzCWZnLExk1PhWaj37TOXlh4qvuT7JfRN60vPdj35cMCHfj02QkWQ\n3j+dpPgkpn01jSeXPUlhcSErUlfQtFHTGvelVr7/nIhsEEn/Lv05fvo4LRu3pF8nP46O3LgRXn3V\n+f2FEFUKz1kf7dqZKXm25DnLiuwVJM1Jolvrbqx5cI3XcQJnxAjo0AH+8AevkwhhLaezPsKvUOfm\nQrNmZpSpZVPyVu9aTeKsRDq06sC2Idu8jhNYubkQHe35IftC2Kz+DmU6a5SpTT2pdXvXkTgrkXP2\nnWNdkXZ9PY0cCTt31qlI2/Tanc3GXJLJGRszORVehbqwEDIyzNtui2z8aSPdZnQjOjKa95Lf8zpO\nYO3YAZs2Qdu2XicRImyEV+tj0CAzJa+42Jq33N8e+paO0zoCUDym2JXBSVbbtAm+/NK8FkKIatW/\nHrXWEBFhRpkuWOBdjrNkH82m/dT2QD0p0idPmj+SzZp5nUSIkFD/etSVjDL1sie169iuSou0jX0y\n1zK99ppc4VOZAAATKklEQVRrh4rbuJ7AzlySyRkbMzkVPrM+LBpluv/EftpOMT3aojFF4b8lDZCf\nD3/7G3zyiddJhAg74dH6mDfPTMk7etTzKXmHTh4idpI5QrBwdKF/c5tDWVERfP89dOzodRIhQkb9\n6lErZUaZHj7szfP75OTlEPNiDACnnz5NZINIT/MEzalTkJYG99/vdRIhQkr96VGX9J0qGWUazJ7U\nidMnSot03lN5VRZpG/tkdc70ySewxt2jLG1cT2BnLsnkjI2ZnKqxUCul4pRSnyilNimlNiilHglG\nMMd69TKXF13kWYRTBadoPsGcZip3VC6NGzb2LIsnVqyAfu6cJV0IUVGNrQ+lVGugtdY6SynVDFgD\n9NNabyl3v+C3PjZvhs6dzShTj6bk5RfmE/V8FADHRh6jeWPn5wUMC1qb3fKaNvU6iRAhx7XWh9Z6\nn9Y6y/f9CeAb4MK6R3RB587m0qMiXVBUUFqkj4w4Uj+LdL9+cPCg10mECGt+9aiVUvFAAvBlIML4\npWSU6UcfVXmXQPakioqLaDTe7Ap44IkDxETFOHqcjX2yWmf68kuIiQnI4eI2riewM5dkcsbGTE45\n3o/a1/ZYCDzq27KuIDU1lfj4eABiYmJISEgoPUV7yUpy7foVV5jrt95a5f2zsrIC8vzFupiGg8yq\n2zttL7HRsY4fX8L19eHF9R9/JOn++yEiwvXlZ/k+HLbq9z2LLXlsvW7j6xeoeuDP9ZLvs7Oz8Yej\n3fOUUg2BfwH/p7WeWsV9gtejLhllOn06PPBAcJ7TR2tNxDjzRmTnYzuJaxEX1Oe3xr59EBkJ557r\ndRIhQpbbu+e9CWyuqkgHXckoUw+L9PZHttffIg3m7C1pzk8hJoSoPSe7510LDASuV0qtU0qtVUrd\nEvhoVfBjlGn5t6t1cXaR3vLQFtqf075Wy3Ezk1v8znT0KPzjHwH9Q2njegI7c0kmZ2zM5FSNPWqt\n9ReAPcdBlxSHCROC+rQtJ5pD09f/cT2dYjsF9bmt06SJOcAlsp4ceSmEx0LrEHKPRpnGTY5j9/Hd\nZA7OpPsF3YP2vFbaswfmz4fhw71OIkTIC89DyCsZZRpol027jN3Hd7Ny0Eop0gDp6ab9JIQImtAq\n1H6OMq1rT6rHX3uw5dAWlqcsJ7FtYp2W5VamQPAr086d8NvfBixLCRvXE9iZSzI5Y2Mmp0JnHvW8\neeZy2bKgPN0Nc24gc08miwcuJik+KSjPab2CAvPHMirK6yRC1Cuh06MO4ijTvvP7kr4tnUXJi+h3\nqQwbAky7IyEBPv0UWrXyOo0QYSG8etTVjDJ124B3B5C+LZ2036VJkT7bggVwyy1SpIXwQGgU6lqO\nMvW3JzU4fTDzN85ndr/ZJHdJ9uuxgcoUDI4yXXMNDB4c8CwlbFxPYGcuyeSMjZmcsr9Qb95sLteu\nDejTDF08lJlrZ/Jan9dITUgN6HOFnA0bzG6Rner5/uNCeMT+HnXJiWED2P8etWwUEz6fwEs3vsTw\na2T/4AoefBD69oXbbvM6iRBhJTx61A5GmdbV+E/HM+HzCYxNGitFujK7dpl3M336eJ1EiHrL7kJ9\n9dXm0jfK1F819aSmrJrC6OWjGXHtCMb0HFOr53A7kxeqzdSqFaxefeadTZDYuJ7AzlySyRkbMzll\nb6HOzYUdO8wo0wCYnjmdYUuGMeSqIUzsPTEgzxHyMjNhyhRoYM+oFyHqI3t71L16md3yAtCbnvv1\nXFIWpTCo6yBm9Z3l+vLDxtNPw+WXw4ABXicRIiyFdo/aj1Gm/lqwaQEpi1JI7pwsRbomzZpJb1oI\nC9hZqF0aZVq+J5W+NZ3khcn06dCHtLu8GXpvY5+s0kyHD5sJeTHOzgXpNhvXE9iZSzI5Y2Mmp+wr\n1FrD7NlmlKmLH2At+X4JfdP60rNdTz4c8KFryw1LubnQowcUFXmdRAiBjT3q5583vdH8fMdT8mqy\nInsFSXOS6Na6G2seXOPKMsPa+PHmpAABaD0JIc5w2qO2r1ArZUaZZmbWfVnA6l2rSZyVSIdWHdg2\nZJsrywx7Bw6Y1yE21uskQoS10PwwsWSU6dKlrizur+/+lcRZibRu1tqaIm1jn6xMpo8+gpwcz4u0\njesJ7MwlmZyxMZNTdhXqe+4xo0xd+ABr00+beCD9AaIjo9k7fK8L4eqBggLz+UBentdJhBBnsaf1\nkZFh9p3+8Ue/p+SV9+2hb+k4rSMAxWOKUUE+qi5kbd5sTncWxFOdCVGfhV6P2qXhS9lHs2k/tT0g\nRdpvubnQtKnXKYSoN0KrR+3SKNNdx3aVKdIrVqyoazLX2dgny8jIgHfegZde8jpKKRvXE9iZSzI5\nY2Mmp2os1EqpWUqp/Uqp9QFL0bmzuezatdaL2H9iP22ntAWgaEyRbEn768MPz5ygQQhhlRpbH0qp\n64ATwFyt9S+quV/tWh/79kGbNmZvg1pOyTt08hCxk8xeCoWjC2kQIUOE/FJcDHPmwL33QsPQOd+x\nEKHO1R61UqodkB6QQt2unZmSV8vedE5eDjEvmr1ETj99msgGkbVaTr32zTfm7C0RdnTChKgvnBZq\nbzef6jjK9MTpE6VFOu+pvApFOiMjg6SkpLqmdFUwMo0bBz/84Oy+zfMOcNuq33Lz95sDmslfNr52\nYGcuyeSMjZmccrVQp6amEh8fD0BMTAwJCQmlK6akkV/m+tChJAE88EDlt1dz/d9L/80tb98C7SF3\nVC6rPl9V4f5ZWVmOlxes6yUCtfxf/zqJqVPh8OGS50sqecZKr7dqlUTn598g47PPApKnttezfGec\ntyVPsF6/cLlu4+tnQz0o+T47Oxt/eNf6KCw8M09ion+D+/ML84l6PgqAYyOP0bxxc78eH+5GjjTz\n/k+frvo+TZrA738Pf/kLtGgRvGxCiDPcbn0o35d7ajnKtKCooLRIHxlxRIp0OUePwvbtVRfpJk3g\n3HNh/ny47rrgZhNC1I6T3fP+DqwEOiqldiil7q/zs9ZylGlRcRGNxpuJegeeOEBMVPWHmpd/u2oD\ntzNpDcuWmT0clTJH4L/zTuX3bdIEhgyBb78tW6Trw3pyi425JJMzNmZyqsZCrbUeoLW+QGvdWGt9\nkdZ6dp2f9YUXzKUfhyoX62IaPmfeAOwdvpfY6Po72e3oUXj8cVOYIyKgd29zzNCdd8J335niPX++\nOUELQHS0OaPWl1/Ciy9CVJS3+YUQ/vHmEHI/R5lqrYkYZ/6m7HxsJ3Et4mobMyRpDZ98Ao88cuYg\nTjAF95VXYNCgiuefzcszA/C0hrFj4bHH5By1QtjG3t3z/BxlenaR/v6R7+tNkT561Mzvf/nlsj+/\n80743/+FSy6p/vFRUfD559CyJbRvH7icQojAC/4RDn6MMj27SG95aAsXn3OxX09lY0+qqkyV9Zpf\nftkU3BkzzE4yWsO779ZcpEskJDgr0qG0nrxmYy7J5IyNmZwKbqEuWVG+fSxr0nJiSwDW/3E9nWI7\nBSiUd5z0mk+dgsGDpW0hRH0W3B61H6NM4ybHsfv4bjIHZ9L9gu4uJPRebXrNQojwZd+YUz9GmV42\n7TJ2H9/NykErQ75Iy1azEKKugleoHY4y7fHXHmw5tIXlKctJbJtYp6f0oidVU6956dIMv3vNgWZj\n787GTGBnLsnkjI2ZnApOod63z1x+9FG1d7thzg1k7slk8cDFJMUnBT6XS2SrWQgRSMHpUTsYZdp3\nfl/St6WzKHkR/S7t50qmQJFesxDCDfb0qB2MMu3/bn/St6WT9rs0a4t0VVvNd9whW81CiMAKfKG+\n/XZzWTKEqZzB6YNJ25jGm33fJLlLsqtPXZeelNP9mt97z79es419MsnknI25JJMzNmZyKrCFurDQ\n7Ds9YkSlNw9dPJSZa2fyWp/XuL9r3Wc91ZVbW82LF5vHT5tW+e2JiXD++VBUFJjfQwgRZrTWrnyZ\nRZVz//1ag9bFxRVuenLpk5pn0S998VLFxwVJcbHWS5dq3bmziVnyFRWl9YwZWhcW1m65RUVat2mj\ndY8eFW/79lutldL6scfqll0IEfp8dbPG+hq4LepqRpmO/3Q8Ez6fwNiksQy/ZnjAIlSmsq3mTZvc\n7TVHRJgj5desgS1byt42Z4557vvuq/vvIoSoJ5xUcydflN+iHj/ebJ7m55f58eSVkzXPokd8PMLl\nv00VLV++PGBbzTXZuNFsOT/5ZNmft2lzUv/iF4F5ztpavny51xEqsDGT1nbmkkzO2JgJz7eon37a\njDJt1Kj0R9MzpzNsyTCGXDWEib39O/2WP0q2mnv1CtxWc006d4Zu3c4MCwRYsQL27YsiJSUwzymE\nCFNOqrmTL87eon77bbPZeuRI6Y/mZM3RPIse9P4g1/8qebXVXJOpU7WOiNB62TJzfdAgrSMjtd63\nz5s8Qgi74HCLOjAHvJTsz3b4MAALNi0geWEyyZ2TSbsrzZXnq2pe8x13wKRJdhyeffAgXHgh9O8P\nb7wBrVubU2D9619eJxNC2MC7A17KjTL9YOsHJC9Mpk+HPnUq0iX7NXfp4ny/Zq/3m4yNhVtvNZnm\nzYNjx+DKKzd5mqkyXq+nytiYCezMJZmcsTGTU+4X6l69zOVFF7Hk+yX0S+vHry/6NR8O+NDvRQVj\nD41AS0mBEydg+HBzroRrrz3odSQhRIhxt/WxaZP5FG3tWlacc4ykOUl0a92NNQ+ucbSMkhkajz5q\nCnKJUJ6hUVAAF1xgukCDB5sWiBBCgPPWh7uF2vf96p2rSJyVSIdWHdg2ZFu1jwuFXrMQQgSCqz1q\npdQtSqktSqltSqnKjwf3WffOqyTOSqR1s9aVFuna9Jpry8aelGRyxsZMYGcuyeSMjZmcqrFQK6Ui\ngGnAzUBnoL9S6tLK7rvpPOi2aQjRkdHsHb639Ode9ZqzHJ6bMZgkkzM2ZgI7c0kmZ2zM5FRDB/e5\nCvhWa/0jgFIqDegHbCl/xy4PmcvjI0+wbJn3veajR48G/kn8JJmcsTET2JlLMjljYyannBTqC4Gd\nZ13fhSnelXu2mAbPnmm5SK9ZCCHqxkmhdu7ZYqKilDV7aGRnZ3sboBKSyRkbM4GduSSTMzZmcqrG\nvT6UUr8EntVa3+K7PhJz2OOL5e7nzu4jQghRj7iye55SqgGwFbgB2At8BfTXWn/jRkghhBDVq7H1\nobUuUko9DCzB7CUyS4q0EEIEj2sHvAghhAiMOs/68OdgmGBRSs1SSu1XSq33OksJpVScUuoTpdQm\npdQGpdQjFmRqrJT6Uim1zpfrBa8zlVBKRSil1iqlPvA6C4BSKlsp9bVvXX3ldR4ApVRLpdQ7Sqlv\nfK/f1RZk6uhbR2t9lzmW/Ft/0reO1iul5imlGtX8qIBnetRXC2quB05moVb1hSn03wHtgEggC7i0\nLst04wu4DkgA1nud5axMrYEE3/fNMH1/G9ZVtO+yAbAauNbrTL48jwFvAx94ncWXZztwjtc5ymV6\nC7jf931DoIXXmcrliwD2AG09ztHO9/o18l3/B3Cfx5k6A+uBxr7/95YAF1d1/7puUZceDKO1LgBK\nDobxlNb6c+CI1znOprXep7XO8n1/AvgGs4+6p7TWJ33fNsb8j+X5elNKxQF9gJleZzmLIhDTJmtJ\nKdUC+JXWejaA1rpQa33M41jl9Qa+11rvrPGegXUMOA00VUo1BKIxf0C8dBnwpdY6X2tdBHwK3FnV\nnev6D6+yg2E8Lz62U0rFY7b4v/Q2SWmLYR2wD8jQWm/2OhMwBXgCsOkDFA18rJT6j1JqsNdhgPbA\nQaXUbF+bYYZSqonXocpJBuZ7HUJrfQR4GdgB7AaOaq2XepuKjcCvlFLnKKWiMRsmbau6szVbCPWF\nUqoZsBB41Ldl7SmtdbHWuisQB/xaKdXTyzxKqduA/b53H8r3ZYNrtdbdMP9DPaSUus7jPA2BbsBr\nvlwngZHeRjpDKRUJ9AXesSDLxZhWWjvgAqCZUmqAl5m01luAF4GPgY+AdUBRVfeva6HeDVx01vU4\n389EJXxvuxYCf9Nav+91nrP53jZ/CFzpcZRrgb5Kqe2YrbFeSqm5HmdCa73Xd3kA+CfVjVEIjl3A\nTq11pu/6QkzhtsWtwBrf+vLalcAXWuvDvjbDe8A1HmdCaz1ba32l1joJOApUORO6roX6P8DPlVLt\nfJ+i/h6w4lN67NoaK/EmsFlrPdXrIABKqVilVEvf902AGzEfCHtGaz1Ka32R1vpizL+nT7TW93mZ\nSSkV7XsnhFKqKXAT5q2rZ7TW+4GdSqmOvh/dANjQtirRHwvaHj5bgV8qpaKUUgqzrjw/FkQpdZ7v\n8iLgDuDvVd23TrM+tKUHwyil/g4kAecqpXYAz5R86OJhpmuBgcAGX09YA6O01os9jNUGmOP7xxuB\n2dJf5mEeW50P/NM3JqEhME9rvcTjTACPAPN8bYbtwP0e5wHMHzbMB4kPeJ0FQGv9te9d2RpMe2Ed\nMMPbVAC8q5RqBRQAf67uw2A54EUIISwnHyYKIYTlpFALIYTlpFALIYTlpFALIYTlpFALIYTlpFAL\nIYTlpFALIYTlpFALIYTl/j8PXIftdqEtdAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1052840d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_vector2d(u, color=\"r\")\n",
    "plot_vector2d(v, color=\"b\")\n",
    "plot_vector2d(v, origin=u, color=\"b\", linestyle=\"dotted\")\n",
    "plot_vector2d(u, origin=v, color=\"r\", linestyle=\"dotted\")\n",
    "plot_vector2d(u+v, color=\"g\")\n",
    "plt.axis([0, 9, 0, 7])\n",
    "plt.text(0.7, 3, \"u\", color=\"r\", fontsize=18)\n",
    "plt.text(4, 3, \"u\", color=\"r\", fontsize=18)\n",
    "plt.text(1.8, 0.2, \"v\", color=\"b\", fontsize=18)\n",
    "plt.text(3.1, 5.6, \"v\", color=\"b\", fontsize=18)\n",
    "plt.text(2.4, 2.5, \"u+v\", color=\"g\", fontsize=18)\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vector addition is **commutative**, meaning that $\\textbf{u} + \\textbf{v} = \\textbf{v} + \\textbf{u}$. You can see it on the previous image: following $\\textbf{u}$ *then* $\\textbf{v}$ leads to the same point as following $\\textbf{v}$ *then* $\\textbf{u}$.\n",
    "\n",
    "Vector addition is also **associative**, meaning that $\\textbf{u} + (\\textbf{v} + \\textbf{w}) = (\\textbf{u} + \\textbf{v}) + \\textbf{w}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you have a shape defined by a number of points (vectors), and you add a vector $\\textbf{v}$ to all of these points, then the whole shape gets shifted by $\\textbf{v}$. This is called a [geometric translation](https://en.wikipedia.org/wiki/Translation_%28geometry%29):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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EmzaVpK2UA7RGHYecOBi3oKSEpMQ4e6OVny/dFwsWwIwZsurviy+k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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1054941d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t1 = np.array([2, 0.25])\n",
    "t2 = np.array([2.5, 3.5])\n",
    "t3 = np.array([1, 2])\n",
    "\n",
    "x_coords, y_coords = zip(t1, t2, t3, t1)\n",
    "plt.plot(x_coords, y_coords, \"c--\", x_coords, y_coords, \"co\")\n",
    "\n",
    "plot_vector2d(v, t1, color=\"r\", linestyle=\":\")\n",
    "plot_vector2d(v, t2, color=\"r\", linestyle=\":\")\n",
    "plot_vector2d(v, t3, color=\"r\", linestyle=\":\")\n",
    "\n",
    "t1b = t1 + v\n",
    "t2b = t2 + v\n",
    "t3b = t3 + v\n",
    "\n",
    "x_coords_b, y_coords_b = zip(t1b, t2b, t3b, t1b)\n",
    "plt.plot(x_coords_b, y_coords_b, \"b-\", x_coords_b, y_coords_b, \"bo\")\n",
    "\n",
    "plt.text(4, 4.2, \"v\", color=\"r\", fontsize=18)\n",
    "plt.text(3, 2.3, \"v\", color=\"r\", fontsize=18)\n",
    "plt.text(3.5, 0.4, \"v\", color=\"r\", fontsize=18)\n",
    "\n",
    "plt.axis([0, 6, 0, 5])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, substracting a vector is like adding the opposite vector."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Multiplication by a scalar\n",
    "Vectors can be multiplied by scalars. All elements in the vector are multiplied by that number, for example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.5 * [2 5] =\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 3. ,  7.5])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(\"1.5 *\", u, \"=\")\n",
    "\n",
    "1.5 * u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Graphically, scalar multiplication results in changing the scale of a figure, hence the name *scalar*. The distance from the origin (the point at coordinates equal to zero) is also multiplied by the scalar. For example, let's scale up by a factor of `k = 2.5`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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s8qRqqi8dqaYH169nk9/Psq5d6d+8OXPbXsE5iyew85k34cYbq3xvmlJM7NAB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\ntSAIgsf5f+8HlIWQVm5RAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10553e990>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "k = 2.5\n",
    "t1c = k * t1\n",
    "t2c = k * t2\n",
    "t3c = k * t3\n",
    "\n",
    "plt.plot(x_coords, y_coords, \"c--\", x_coords, y_coords, \"co\")\n",
    "\n",
    "plot_vector2d(t1, color=\"r\")\n",
    "plot_vector2d(t2, color=\"r\")\n",
    "plot_vector2d(t3, color=\"r\")\n",
    "\n",
    "x_coords_c, y_coords_c = zip(t1c, t2c, t3c, t1c)\n",
    "plt.plot(x_coords_c, y_coords_c, \"b-\", x_coords_c, y_coords_c, \"bo\")\n",
    "\n",
    "plot_vector2d(k * t1, color=\"b\", linestyle=\":\")\n",
    "plot_vector2d(k * t2, color=\"b\", linestyle=\":\")\n",
    "plot_vector2d(k * t3, color=\"b\", linestyle=\":\")\n",
    "\n",
    "plt.axis([0, 9, 0, 9])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you might guess, dividing a vector by a scalar is equivalent to multiplying by its inverse:\n",
    "\n",
    "$\\dfrac{\\textbf{u}}{\\lambda} = \\dfrac{1}{\\lambda} \\times \\textbf{u}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Scalar multiplication is **commutative**: $\\lambda \\times \\textbf{u} = \\textbf{u} \\times \\lambda$.\n",
    "\n",
    "It is also **associative**: $\\lambda_1 \\times (\\lambda_2 \\times \\textbf{u}) = (\\lambda_1 \\times \\lambda_2) \\times \\textbf{u}$.\n",
    "\n",
    "Finally, it is **distributive** over addition of vectors: $\\lambda \\times (\\textbf{u} + \\textbf{v}) = \\lambda \\times \\textbf{u} + \\lambda \\times \\textbf{v}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Zero, unit and normalized vectors\n",
    "* A **zero-vector ** is a vector full of 0s.\n",
    "* A **unit vector** is a vector with a norm equal to 1.\n",
    "* The **normalized vector** of a non-null vector $\\textbf{u}$, noted $\\hat{\\textbf{u}}$, is the unit vector that points in the same direction as $\\textbf{u}$. It is equal to: $\\hat{\\textbf{u}} = \\dfrac{\\textbf{u}}{\\left \\Vert \\textbf{u} \\right \\|}$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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5/Qk5Oyi/b77yx2Z4i4gMJLEZ3rW1tb4jpET5/Qk5Oyi/b77yZ/RQwYy8kIhI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\nIiIB+v/muDwvCNxNZwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105494050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.gca().add_artist(plt.Circle((0,0),1,color='c'))\n",
    "plt.plot(0, 0, \"ko\")\n",
    "plot_vector2d(v / LA.norm(v), color=\"k\")\n",
    "plot_vector2d(v, color=\"b\", linestyle=\":\")\n",
    "plt.text(0.3, 0.3, \"$\\hat{u}$\", color=\"k\", fontsize=18)\n",
    "plt.text(1.5, 0.7, \"$u$\", color=\"b\", fontsize=18)\n",
    "plt.axis([-1.5, 5.5, -1.5, 3.5])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dot product\n",
    "### Definition\n",
    "The dot product (also called *scalar product* or *inner product* in the context of the Euclidian space) of two vectors $\\textbf{u}$ and $\\textbf{v}$ is a useful operation that comes up fairly often in linear algebra. It is noted $\\textbf{u} \\cdot \\textbf{v}$, or sometimes $⟨\\textbf{u}|\\textbf{v}⟩$ or $(\\textbf{u}|\\textbf{v})$, and it is defined as:\n",
    "\n",
    "$\\textbf{u} \\cdot \\textbf{v} = \\left \\Vert \\textbf{u} \\right \\| \\times \\left \\Vert \\textbf{v} \\right \\| \\times cos(\\theta)$\n",
    "\n",
    "where $\\theta$ is the angle between $\\textbf{u}$ and $\\textbf{v}$.\n",
    "\n",
    "Another way to calculate the dot product is:\n",
    "\n",
    "$\\textbf{u} \\cdot \\textbf{v} = \\sum_i{\\textbf{u}_i \\times \\textbf{v}_i}$\n",
    "\n",
    "### In python\n",
    "The dot product is pretty simple to implement:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def dot_product(v1, v2):\n",
    "    return sum(v1i * v2i for v1i, v2i in zip(v1, v2))\n",
    "\n",
    "dot_product(u, v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But a *much* more efficient implementation is provided by NumPy with the `dot` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(u,v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Equivalently, you can use the `dot` method of `ndarray`s:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.dot(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Caution**: the `*` operator will perform an *elementwise* multiplication, *NOT* a dot product:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   [2 5]\n",
      "*  [3 1] (NOT a dot product)\n",
      "----------\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([6, 5])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(\"  \",u)\n",
    "print(\"* \",v, \"(NOT a dot product)\")\n",
    "print(\"-\"*10)\n",
    "\n",
    "u * v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Main properties\n",
    "* The dot product is **commutative**: $\\textbf{u} \\cdot \\textbf{v} = \\textbf{v} \\cdot \\textbf{u}$.\n",
    "* The dot product is only defined between two vectors, not between a scalar and a vector. This means that we cannot chain dot products: for example, the expression $\\textbf{u} \\cdot \\textbf{v} \\cdot \\textbf{w}$ is not defined since $\\textbf{u} \\cdot \\textbf{v}$ is a scalar and $\\textbf{w}$ is a vector.\n",
    "* This also means that the dot product is **NOT associative**: $(\\textbf{u} \\cdot \\textbf{v}) \\cdot \\textbf{w} ≠ \\textbf{u} \\cdot (\\textbf{v} \\cdot \\textbf{w})$ since neither are defined.\n",
    "* However, the dot product is **associative with regards to scalar multiplication**: $\\lambda \\times (\\textbf{u} \\cdot \\textbf{v}) = (\\lambda \\times \\textbf{u}) \\cdot \\textbf{v} = \\textbf{u} \\cdot (\\lambda \\times \\textbf{v})$\n",
    "* Finally, the dot product is **distributive** over addition of vectors: $\\textbf{u} \\cdot (\\textbf{v} + \\textbf{w}) = \\textbf{u} \\cdot \\textbf{v} + \\textbf{u} \\cdot \\textbf{w}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating the angle between vectors\n",
    "One of the many uses of the dot product is to calculate the angle between two non-zero vectors. Looking at the dot product definition, we can deduce the following formula:\n",
    "\n",
    "$\\theta = \\arccos{\\left ( \\dfrac{\\textbf{u} \\cdot \\textbf{v}}{\\left \\Vert \\textbf{u} \\right \\| \\times \\left \\Vert \\textbf{v} \\right \\|} \\right ) }$\n",
    "\n",
    "Note that if $\\textbf{u} \\cdot \\textbf{v} = 0$, it follows that $\\theta = \\dfrac{π}{4}$. In other words, if the dot product of two non-null vectors are orthogonal, it means that they are orthogonal.\n",
    "\n",
    "Let's use this formula to calculate the angle between $\\textbf{u}$ and $\\textbf{v}$ (in radians):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Angle = 0.868539395286 radians\n",
      "      = 49.7636416907 degrees\n"
     ]
    }
   ],
   "source": [
    "def vector_angle(u, v):\n",
    "    cos_theta = u.dot(v) / LA.norm(u) / LA.norm(v)\n",
    "    return np.arccos(np.clip(cos_theta, -1, 1))\n",
    "\n",
    "theta = vector_angle(u, v)\n",
    "print(\"Angle =\", theta, \"radians\")\n",
    "print(\"      =\", theta * 180 / np.pi, \"degrees\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note: due to small floating point errors, `cos_theta` may be very slightly outside of the $[-1, 1]$ interval, which would make `arccos` fail. This is why we clipped the value within the range, using NumPy's `clip` function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Projecting a point onto an axis\n",
    "The dot product is also very useful to project points onto an axis. The projection of vector $\\textbf{v}$ onto $\\textbf{u}$'s axis is given by this formula:\n",
    "\n",
    "$\\textbf{proj}_{\\textbf{u}}{\\textbf{v}} = \\dfrac{\\textbf{u} \\cdot \\textbf{v}}{\\left \\Vert \\textbf{u} \\right \\| ^2} \\times \\textbf{u}$\n",
    "\n",
    "Which is equivalent to:\n",
    "\n",
    "$\\textbf{proj}_{\\textbf{u}}{\\textbf{v}} = (\\textbf{v} \\cdot \\hat{\\textbf{u}}) \\times \\hat{\\textbf{u}}$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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F3op0Q96fQVBOf0UlpxfBHMouWUJn4MQFCwL5+Ibik0/gsstcYS3Xrx889hi0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\nFhEJORVqEZGQU6EWEQk5FWoRkZD7f9FA/lHa1t0VAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x104e9d9d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "u_normalized = u / LA.norm(u)\n",
    "proj = v.dot(u_normalized) * u_normalized\n",
    "\n",
    "plot_vector2d(u, color=\"r\")\n",
    "plot_vector2d(v, color=\"b\")\n",
    "\n",
    "plot_vector2d(proj, color=\"k\", linestyle=\":\")\n",
    "plt.plot(proj[0], proj[1], \"ko\")\n",
    "\n",
    "plt.plot([proj[0], v[0]], [proj[1], v[1]], \"b:\")\n",
    "\n",
    "plt.text(1, 2, \"$proj_u v$\", color=\"k\", fontsize=18)\n",
    "plt.text(1.8, 0.2, \"$v$\", color=\"b\", fontsize=18)\n",
    "plt.text(0.8, 3, \"$u$\", color=\"r\", fontsize=18)\n",
    "\n",
    "plt.axis([0, 8, 0, 5.5])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Matrices\n",
    "A matrix is a rectangular array of scalars (ie. any number: integer, real or complex) arranged in rows and columns, for example:\n",
    "\n",
    "\\begin{bmatrix} 10 & 20 & 30 \\\\ 40 & 50 & 60 \\end{bmatrix}\n",
    "\n",
    "You can also think of a matrix as a list of vectors: the previous matrix contains either 2 horizontal 3D vectors or 3 vertical 2D vectors.\n",
    "\n",
    "Matrices are convenient and very efficient to run operations on many vectors at a time. We will also see that they are great at representing and performing linear transformations such rotations, translations and scaling."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrices in python\n",
    "In python, a matrix can be represented in various ways. The simplest is just a list of python lists:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[10, 20, 30], [40, 50, 60]]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[\n",
    "    [10, 20, 30],\n",
    "    [40, 50, 60]\n",
    "]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A much more efficient way is to use the NumPy library which provides optimized implementations of many matrix operations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 20, 30],\n",
       "       [40, 50, 60]])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([\n",
    "    [10,20,30],\n",
    "    [40,50,60]\n",
    "])\n",
    "A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By convention matrices generally have uppercase names, such as $A$.\n",
    "\n",
    "In the rest of this tutorial, we will assume that we are using NumPy arrays (type `ndarray`) to represent matrices."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Size\n",
    "The size of a matrix is defined by its number of rows and number of columns. It is noted $rows \\times columns$. For example, the matrix $A$ above is an example of a $2 \\times 3$ matrix: 2 rows, 3 columns. Caution: a $3 \\times 2$ matrix would have 3 rows and 2 columns.\n",
    "\n",
    "To get a matrix's size in NumPy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 3)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Caution**: the `size` attribute represents the number of elements in the `ndarray`, not the matrix's size:\n",
    "\n",
    "$M_{i,j}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.size"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Element indexing\n",
    "The number located in the $i^{th}$ row, and $j^{th}$ column of a matrix $X$ is sometimes noted $X_{i,j}$ or $X_{ij}$, but there is no standard notation, so people often prefer to explicitely name the elements, like this: \"*let $X = (x_{i,j})_{1 ≤ i ≤ m, 1 ≤ j ≤ n}$*\". This means that $X$ is equal to:\n",
    "\n",
    "$X = \\begin{bmatrix}\n",
    "  x_{1,1} & x_{1,2} & x_{1,3} & \\cdots & x_{1,n}\\\\\n",
    "  x_{2,1} & x_{2,2} & x_{2,3} & \\cdots & x_{2,n}\\\\\n",
    "  x_{3,1} & x_{3,2} & x_{3,3} & \\cdots & x_{3,n}\\\\\n",
    "  \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "  x_{m,1} & x_{m,2} & x_{m,3} & \\cdots & x_{m,n}\\\\\n",
    "\\end{bmatrix}$\n",
    "\n",
    "However in this notebook we will use the $X_{i,j}$ notation, as it matches fairly well NumPy's notation. Note that in math indices generally start at 1, but in programming they usually start at 0. So to access $A_{2,3}$ programmatically, we need to write this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "60"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[1,2]  # 2nd row, 3rd column"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The $i^{th}$ row vector is sometimes noted $M_i$ or $M_{i,*}$, but again there is no standard notation so people often prefer to explicitely define their own names, for example: \"*let **x**$_{i}$ be the $i^{th}$ row vector of matrix $X$*\". We will use the $M_{i,*}$, for the same reason as above. For example, to access $A_{2,*}$ (ie. $A$'s 2nd row vector):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([40, 50, 60])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[1, :]  # 2nd row vector (as a 1D array)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, the $j^{th}$ column vector is sometimes noted $M^j$ or $M_{*,j}$, but there is no standard notation. We will use $M_{*,j}$. For example, to access $A_{*,3}$ (ie. $A$'s 3rd column vector):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([30, 60])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[:, 2]  # 3rd column vector (as a 1D array)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the result is actually a one-dimensional NumPy array: there is no such thing as a *vertical* or *horizontal* one-dimensional array. If you need to actually represent a row vector as a one-row matrix (ie. a 2D NumPy array), or a column vector as a one-column matrix, then you need to use a slice instead of an integer when accessing the row or column, for example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[40, 50, 60]])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[1:2, :]  # rows 2 to 3 (excluded): this returns row 2 as a one-row matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[30],\n",
       "       [60]])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[:, 2:3]  # columns 3 to 4 (excluded): this returns column 3 as a one-column matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Square, triangular, diagonal and identity matrices\n",
    "A **square matrix** is a matrix that has the same number of rows and columns, for example a $3 \\times 3$ matrix:\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  4 & 9 & 2 \\\\\n",
    "  3 & 5 & 7 \\\\\n",
    "  8 & 1 & 6\n",
    "\\end{bmatrix}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An **upper triangular matrix** is a special kind of square matrix where all the elements *below* the main diagonal (top-left to bottom-right) are zero, for example:\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  4 & 9 & 2 \\\\\n",
    "  0 & 5 & 7 \\\\\n",
    "  0 & 0 & 6\n",
    "\\end{bmatrix}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, a **lower triangular matrix** is a square matrix where all elements *above* the main diagonal are zero, for example:\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  4 & 0 & 0 \\\\\n",
    "  3 & 5 & 0 \\\\\n",
    "  8 & 1 & 6\n",
    "\\end{bmatrix}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A **triangular matrix** is one that is either lower triangular or upper triangular."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A matrix that is both upper and lower triangular is called a **diagonal matrix**, for example:\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  4 & 0 & 0 \\\\\n",
    "  0 & 5 & 0 \\\\\n",
    "  0 & 0 & 6\n",
    "\\end{bmatrix}\n",
    "\n",
    "You can construct a diagonal matrix using NumPy's `diag` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4, 0, 0],\n",
       "       [0, 5, 0],\n",
       "       [0, 0, 6]])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.diag([4, 5, 6])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you pass a matrix to the `diag` function, it will happily extract the diagonal values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 5, 9])"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = np.array([\n",
    "        [1, 2, 3],\n",
    "        [4, 5, 6],\n",
    "        [7, 8, 9],\n",
    "    ])\n",
    "np.diag(D)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, the **identity matrix** of size $n$, noted $I_n$, is a diagonal matrix of size $n \\times n$ with $1$'s in the main diagonal, for example $I_3$:\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  1 & 0 & 0 \\\\\n",
    "  0 & 1 & 0 \\\\\n",
    "  0 & 0 & 1\n",
    "\\end{bmatrix}\n",
    "\n",
    "Numpy's `eye` function returns the identity matrix of the desired size:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.,  0.,  0.],\n",
       "       [ 0.,  1.,  0.],\n",
       "       [ 0.,  0.,  1.]])"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.eye(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The identity matrix is often noted simply $I$ (instead of $I_n$) when its size is clear given the context. It is called the *identity* matrix because multiplying a matrix with it leaves the matrix unchanged as we will see below."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Adding matrices\n",
    "If two matrices $Q$ and $R$ have the same size $m \\times n$, they can be added together. Addition is performed *elementwise*: the result is also a $m \\times n$ matrix $S$ where each element is the sum of the elements at the corresponding position: $S_{i,j} = Q_{i,j} + R_{i,j}$\n",
    "\n",
    "$S =\n",
    "\\begin{bmatrix}\n",
    "  Q_{11} + R_{11} & Q_{12} + R_{12} & Q_{13} + R_{13} & \\cdots & Q_{1n} + R_{1n} \\\\\n",
    "  Q_{21} + R_{21} & Q_{22} + R_{22} & Q_{23} + R_{23} & \\cdots & Q_{2n} + R_{2n}  \\\\\n",
    "  Q_{31} + R_{31} & Q_{32} + R_{32} & Q_{33} + R_{33} & \\cdots & Q_{3n} + R_{3n}  \\\\\n",
    "  \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "  Q_{m1} + R_{m1} & Q_{m2} + R_{m2} & Q_{m3} + R_{m3} & \\cdots & Q_{mn} + R_{mn}  \\\\\n",
    "\\end{bmatrix}$\n",
    "\n",
    "For example, let's create a $2 \\times 3$ matric $B$ and compute $A + B$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2, 3],\n",
       "       [4, 5, 6]])"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = np.array([[1,2,3], [4, 5, 6]])\n",
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 20, 30],\n",
       "       [40, 50, 60]])"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[11, 22, 33],\n",
       "       [44, 55, 66]])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A + B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Addition is *commutative***, meaning that $A + B = B + A$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[11, 22, 33],\n",
       "       [44, 55, 66]])"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B + A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**It is also *associative***, meaning that $A + (B + C) = (A + B) + C$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[111, 222, 333],\n",
       "       [444, 555, 666]])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = np.array([[100,200,300], [400, 500, 600]])\n",
    "\n",
    "A + (B + C)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[111, 222, 333],\n",
       "       [444, 555, 666]])"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A + B) + C"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Scalar multiplication\n",
    "A matrix $M$ can be multiplied by a scalar $\\lambda$. The result is noted $\\lambda \\times M$ or $\\lambda \\cdot M$ or simply $\\lambda M$, and it is a matrix of the same size as $M$ with all elements multiplied by $\\lambda$:\n",
    "\n",
    "$\\lambda M =\n",
    "\\begin{bmatrix}\n",
    "  \\lambda \\times M_{11} & \\lambda \\times M_{12} & \\lambda \\times M_{13} & \\cdots & \\lambda \\times M_{1n} \\\\\n",
    "  \\lambda \\times M_{21} & \\lambda \\times M_{22} & \\lambda \\times M_{23} & \\cdots & \\lambda \\times M_{2n} \\\\\n",
    "  \\lambda \\times M_{31} & \\lambda \\times M_{32} & \\lambda \\times M_{33} & \\cdots & \\lambda \\times M_{3n} \\\\\n",
    "  \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "  \\lambda \\times M_{m1} & \\lambda \\times M_{m2} & \\lambda \\times M_{m3} & \\cdots & \\lambda \\times M_{mn} \\\\\n",
    "\\end{bmatrix}$\n",
    "\n",
    "In NumPy, simply use the `*` operator to multiply a matrix by a scalar. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 20,  40,  60],\n",
       "       [ 80, 100, 120]])"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Scalar multiplication is **commutative**, meaning that $\\lambda \\times M = M \\times \\lambda$. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 20,  40,  60],\n",
       "       [ 80, 100, 120]])"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is also **associative**, meaning that $\\alpha \\times (\\beta \\times M) = (\\alpha \\times \\beta) \\times M$, where $\\alpha$ and $\\beta$ are scalars. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 60, 120, 180],\n",
       "       [240, 300, 360]])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * (3 * A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 60, 120, 180],\n",
       "       [240, 300, 360]])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(2 * 3) * A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, it is **distributive over addition** of matrices, meaning that $\\lambda \\times (Q + R) = \\lambda \\times Q + \\lambda \\times R$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 22,  44,  66],\n",
       "       [ 88, 110, 132]])"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * (A + B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 22,  44,  66],\n",
       "       [ 88, 110, 132]])"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * A + 2 * B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrix dot product\n",
    "So far, matrix operations have been rather intuitive. But multiplying matrices is a bit more involved.\n",
    "\n",
    "A matrix $Q$ of size $m \\times n$ can be multiplied by a matrix $R$ of size $n \\times q$. This is also called a **dot product** and it is noted $Q \\cdot R$ or simply $QR$. The result $P$ is an $m \\times q$ matrix where each element is computed as a sum of products:\n",
    "\n",
    "$P_{i,j} = \\sum_{k=1}^n{Q_{i,k} \\times R_{k,j}}$\n",
    "\n",
    "The element at position $i,j$ in the resulting matrix is the sum of the products of elements in row $i$ of matrix $Q$ by the elements in column $j$ of matrix $R$.\n",
    "\n",
    "$P =\n",
    "\\begin{bmatrix}\n",
    "Q_{11} R_{11} + Q_{12} R_{21} + \\cdots + Q_{1n} R_{n1} &\n",
    "  Q_{11} R_{12} + Q_{12} R_{22} + \\cdots + Q_{1n} R_{n2} &\n",
    "    \\cdots &\n",
    "      Q_{11} R_{1q} + Q_{12} R_{2q} + \\cdots + Q_{1n} R_{nq} \\\\\n",
    "Q_{21} R_{11} + Q_{22} R_{21} + \\cdots + Q_{2n} R_{n1} &\n",
    "  Q_{21} R_{12} + Q_{22} R_{22} + \\cdots + Q_{2n} R_{n2} &\n",
    "    \\cdots &\n",
    "      Q_{21} R_{1q} + Q_{22} R_{2q} + \\cdots + Q_{2n} R_{nq} \\\\\n",
    "  \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "Q_{m1} R_{11} + Q_{m2} R_{21} + \\cdots + Q_{mn} R_{n1} &\n",
    "  Q_{m1} R_{12} + Q_{m2} R_{22} + \\cdots + Q_{mn} R_{n2} &\n",
    "    \\cdots &\n",
    "      Q_{m1} R_{1q} + Q_{m2} R_{2q} + \\cdots + Q_{mn} R_{nq}\n",
    "\\end{bmatrix}$\n",
    "\n",
    "You may notice that each element $P_{i,j}$ is the dot product of the row vector $Q_{i,*}$ and the column vector $R_{*,j}$:\n",
    "\n",
    "$P_{i,j} = Q_{i,*} \\cdot R_{*,j}$\n",
    "\n",
    "So we can rewrite $P$ more concisely as:\n",
    "\n",
    "$P =\n",
    "\\begin{bmatrix}\n",
    "Q_{1,*} \\cdot R_{*,1} & Q_{1,*} \\cdot R_{*,2} & \\cdots & Q_{1,*} \\cdot R_{*,q} \\\\\n",
    "Q_{2,*} \\cdot R_{*,1} & Q_{2,*} \\cdot R_{*,2} & \\cdots & Q_{2,*} \\cdot R_{*,q} \\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "Q_{m,*} \\cdot R_{*,1} & Q_{m,*} \\cdot R_{*,2} & \\cdots & Q_{m,*} \\cdot R_{*,q}\n",
    "\\end{bmatrix}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's multiply two matrices in NumPy, using `ndarray`'s `dot` method:\n",
    "\n",
    "$E = A \\cdot D = \\begin{bmatrix}\n",
    "  10 & 20 & 30 \\\\\n",
    "  40 & 50 & 60\n",
    "\\end{bmatrix} \\cdot \n",
    "\\begin{bmatrix}\n",
    "  2 & 3 & 5 & 7 \\\\\n",
    "  11 & 13 & 17 & 19 \\\\\n",
    "  23 & 29 & 31 & 37\n",
    "\\end{bmatrix} = \n",
    "\\begin{bmatrix}\n",
    "  930 & 1160 & 1320 & 1560 \\\\\n",
    "  2010 & 2510 & 2910 & 3450\n",
    "\\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 930, 1160, 1320, 1560],\n",
       "       [2010, 2510, 2910, 3450]])"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = np.array([\n",
    "        [ 2,  3,  5,  7],\n",
    "        [11, 13, 17, 19],\n",
    "        [23, 29, 31, 37]\n",
    "    ])\n",
    "E = A.dot(D)\n",
    "E"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's check this result by looking at one element, just to be sure: looking at $E_{2,3}$ for example, we need to multiply elements in $A$'s $2^{nd}$ row by elements in $D$'s $3^{rd}$ column, and sum up these products:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2910"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "40*5 + 50*17 + 60*31"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2910"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E[1,2]  # row 2, column 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looks good! You can check the other elements until you get used to the algorithm.\n",
    "\n",
    "We multiplied a $2 \\times 3$ matrix by a $3 \\times 4$ matrix, so the result is a $2 \\times 4$ matrix. The first matrix's number of columns has to be equal to the second matrix's number of rows. If we try to multiple $D$ by $A$, we get an error because D has 4 columns while A has 2 rows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ValueError: shapes (3,4) and (2,3) not aligned: 4 (dim 1) != 2 (dim 0)\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    D.dot(A)\n",
    "except ValueError as e:\n",
    "    print(\"ValueError:\", e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This illustrates the fact that **matrix multiplication is *NOT* commutative**: in general $Q \\cdot R ≠ R \\cdot Q$\n",
    "\n",
    "In fact, $Q \\cdot R$ and $R \\cdot Q$ are only *both* defined if $Q$ has size $m \\times n$ and $R$ has size $n \\times m$. Let's look at an example where both *are* defined and show that they are (in general) *NOT* equal:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[400, 130],\n",
       "       [940, 310]])"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F = np.array([\n",
    "        [5,2],\n",
    "        [4,1],\n",
    "        [9,3]\n",
    "    ])\n",
    "A.dot(F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[130, 200, 270],\n",
       "       [ 80, 130, 180],\n",
       "       [210, 330, 450]])"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F.dot(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On the other hand, **matrix multiplication *is* associative**, meaning that $Q \\cdot (R \\cdot S) = (Q \\cdot R) \\cdot S$. Let's create a $4 \\times 5$ matrix $G$ to illustrate this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[21640, 28390, 27320, 31140, 13570],\n",
       "       [47290, 62080, 60020, 68580, 29500]])"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G = np.array([\n",
    "        [8,  7,  4,  2,  5],\n",
    "        [2,  5,  1,  0,  5],\n",
    "        [9, 11, 17, 21,  0],\n",
    "        [0,  1,  0,  1,  2]])\n",
    "A.dot(D).dot(G)     # (A⋅B)⋅G"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[21640, 28390, 27320, 31140, 13570],\n",
       "       [47290, 62080, 60020, 68580, 29500]])"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.dot(D.dot(G))     # A⋅(B⋅G)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is also ***distributive* over addition** of matrices, meaning that $(Q + R) \\cdot S = Q \\cdot S + R \\cdot S$. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1023, 1276, 1452, 1716],\n",
       "       [2211, 2761, 3201, 3795]])"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A + B).dot(D)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1023, 1276, 1452, 1716],\n",
       "       [2211, 2761, 3201, 3795]])"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.dot(D) + B.dot(D)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The dot product of a matrix $M$ by the identity matrix (of matching size) results in the same matrix $M$. More formally, if $M$ is an $m \\times n$ matrix, then:\n",
    "\n",
    "$M \\cdot I_n = I_m \\cdot M = M$\n",
    "\n",
    "This is generally written more concisely (since the size of the identity matrices is unambiguous given the context):\n",
    "\n",
    "$MI = IM = M$\n",
    "\n",
    "For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 10.,  20.,  30.],\n",
       "       [ 40.,  50.,  60.]])"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.dot(np.eye(3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 10.,  20.,  30.],\n",
       "       [ 40.,  50.,  60.]])"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.eye(2).dot(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Caution**: NumPy's `*` operator performs elementwise multiplication, *NOT* a dot product:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 10,  40,  90],\n",
       "       [160, 250, 360]])"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * B   # NOT a dot product"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**The @ infix operator**\n",
    "\n",
    "Python 3.5 [introduced](https://docs.python.org/3/whatsnew/3.5.html#pep-465-a-dedicated-infix-operator-for-matrix-multiplication) the `@` infix operator for matrix multiplication, and NumPy 1.10 added support for it. If you are using Python 3.5+ and NumPy 1.10+, you can simply write `A @ D` instead of `A.dot(D)`, making your code much more readable (but less portable). This operator also works for vector dot products."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python version: 2.7.11\n",
      "Numpy version: 1.10.4\n"
     ]
    }
   ],
   "source": [
    "import sys\n",
    "print(\"Python version: {}.{}.{}\".format(*sys.version_info))\n",
    "print(\"Numpy version:\", np.version.version)\n",
    "\n",
    "# Uncomment the following line if your Python version is ≥3.5\n",
    "# and your NumPy version is ≥1.10:\n",
    "\n",
    "#A @ D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note: `Q @ R` is actually equivalent to `Q.__matmul__(R)` which is implemented by NumPy as `np.matmul(Q, R)`, not as `Q.dot(R)`. The main difference is that `matmul` does not support scalar multiplication, while `dot` does, so you can write `Q.dot(3)`, which is equivalent to `Q * 3`, but you cannot write `Q @ 3` ([more details](http://stackoverflow.com/a/34142617/38626))."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrix transpose\n",
    "The transpose of a matrix $M$ is a matrix noted $M^T$ such that the $i^{th}$ row in $M^T$ is equal to the $i^{th}$ column in $M$:\n",
    "\n",
    "$ A^T =\n",
    "\\begin{bmatrix}\n",
    "  10 & 20 & 30 \\\\\n",
    "  40 & 50 & 60\n",
    "\\end{bmatrix}^T =\n",
    "\\begin{bmatrix}\n",
    "  10 & 40 \\\\\n",
    "  20 & 50 \\\\\n",
    "  30 & 60\n",
    "\\end{bmatrix}$\n",
    "\n",
    "In other words, ($A^T)_{i,j}$ = $A_{j,i}$\n",
    "\n",
    "Obviously, if $M$ is a $m \\times n$ matrix, then $M^T$ is a $n \\times m$ matrix.\n",
    "\n",
    "Note: there are a few other notations, such as $M^t$, $M′$, or ${^t}M$.\n",
    "\n",
    "In NumPy, a matrix's transpose can be obtained simply using the `T` attribute:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 20, 30],\n",
       "       [40, 50, 60]])"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 40],\n",
       "       [20, 50],\n",
       "       [30, 60]])"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you might expect, transposing a matrix twice returns the original matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 20, 30],\n",
       "       [40, 50, 60]])"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.T.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Transposition is distributive over addition of matrices, meaning that $(Q + R)^T = Q^T + R^T$. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[11, 44],\n",
       "       [22, 55],\n",
       "       [33, 66]])"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A + B).T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[11, 44],\n",
       "       [22, 55],\n",
       "       [33, 66]])"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.T + B.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Moreover, $(Q \\cdot R)^T = R^T \\cdot Q^T$. Note that the order is reversed. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 930, 2010],\n",
       "       [1160, 2510],\n",
       "       [1320, 2910],\n",
       "       [1560, 3450]])"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A.dot(D)).T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 930, 2010],\n",
       "       [1160, 2510],\n",
       "       [1320, 2910],\n",
       "       [1560, 3450]])"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D.T.dot(A.T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A **symmetric matrix** $M$ is defined as a matrix that is equal to its transpose: $M^T = M$. This definition implies that it must be a square matrix whose elements are symmetric relative to the main diagonal, for example:\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  17 & 22 & 27 & 49 \\\\\n",
    "  22 & 29 & 36 & 0 \\\\\n",
    "  27 & 36 & 45 & 2 \\\\\n",
    "  49 & 0 & 2 & 99\n",
    "\\end{bmatrix}\n",
    "\n",
    "The product of a matrix by its transpose is always a symmetric matrix, for example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  87,  279,  547],\n",
       "       [ 279,  940, 1860],\n",
       "       [ 547, 1860, 3700]])"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D.dot(D.T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Converting 1D arrays to 2D arrays in NumPy\n",
    "As we mentionned earlier, in NumPy (as opposed to Matlab, for example), 1D really means 1D: there is no such thing as a vertical 1D-array or a horizontal 1D-array. So you should not be surprised to see that transposing a 1D array does not do anything:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 5])"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 5])"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We want to convert $\\textbf{u}$ into a row vector before transposing it. There are a few ways to do this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 5]])"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u_row = np.array([u])\n",
    "u_row"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice the extra square brackets: this is a 2D array with just one row (ie. a 1x2 matrix). In other words it really is a **row vector**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 5]])"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u[np.newaxis, :]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This quite explicit: we are asking for a new vertical axis, keeping the existing data as the horizontal axis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 5]])"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u[np.newaxis]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is equivalent, but a little less explicit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 5]])"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u[None]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the shortest version, but you probably want to avoid it because it is unclear. The reason it works is that `np.newaxis` is actually equal to `None`, so this is equivalent to the previous version.\n",
    "\n",
    "Ok, now let's transpose our row vector:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2],\n",
       "       [5]])"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u_row.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Great! We now have a nice **column vector**.\n",
    "\n",
    "Rather than creating a row vector then transposing it, it is also possible to convert a 1D array directly into a column vector:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2],\n",
       "       [5]])"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u[:, np.newaxis]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting a matrix\n",
    "We have already seen that vectors can been represented as points or arrows in N-dimensional space. Is there a good graphical representation of matrices? Well you can simply see a matrix as a list of vectors, so plotting a matrix results in many points or arrows. For example, let's create a $2 \\times 4$ matrix `P` and plot it as points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x105492790>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P = np.array([\n",
    "        [3.0, 4.0, 1.0, 4.6],\n",
    "        [0.2, 3.5, 2.0, 0.5]\n",
    "    ])\n",
    "x_coords_P, y_coords_P = P\n",
    "plt.scatter(x_coords_P, y_coords_P)\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course we could also have stored the same 4 vectors as row vectors instead of column vectors, resulting in a $4 \\times 2$ matrix (the transpose of $P$, in fact). It is really an arbitrary choice.\n",
    "\n",
    "Since the vectors are ordered, you can see the matrix as a path and represent it with connected dots:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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FgAHw05/C/PlhPZmBA2u3BaMeukgFmTYtzLyU3A44IDxyWbEirCuz777ljamc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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10502f390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x_coords_P, y_coords_P, \"bo\")\n",
    "plt.plot(x_coords_P, y_coords_P, \"b--\")\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or you can represent it as a polygon: matplotlib's `Polygon` class expects an $n \\times 2$ NumPy array, not a $2 \\times n$ array, so we just need to give it $P^T$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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83EIOfRrwmqouDGx7gcGqesxdhyyHbkz0XHTR5bz//oP4bxsQqj3AQrKzy4Bd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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1057458d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from matplotlib.patches import Polygon\n",
    "plt.gca().add_artist(Polygon(P.T))\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Geometric applications of matrix operations\n",
    "We saw earlier that vector addition results in a geometric translation, vector multiplication by a scalar results in rescaling (zooming in or out, centered on the origin), and vector dot product results in projecting a vector onto another vector, rescaling and measuring the resulting coordinate.\n",
    "\n",
    "Similarly, matrix operations have very useful geometric applications."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Addition = multiple geometric translations\n",
    "First, adding two matrices together is equivalent to adding all their vectors together. For example, let's create a $2 \\times 4$ matrix $H$ and add it to $P$, and look at the result:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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TU5+ytymOT5b+DZ65a8jpdZPsdgXkGsEmGmu5CJcDd6pp4NhqbWPOnKmn+qta\nLv4zLVbop0+fpqmpibVr144aKy8vp6Ojw6tDP3PmDMeOHWP79u0+XWf37t288MIL/Nd//Repk+jG\n093drTJEJ4HT6aSlpZXKyla6uuJJSMghOzsDIQS9+QtwLSzBePQDDE11OLPyQGV3BhaXC0+KVsdF\nS4hrJzt7dCSZIvjE9Aq9oqKCLVu2sHnzZoQQPPvsszz88MOA5uS3bNnCli1bEELwzDPP8OijjwLQ\n2trK5s2baWtrQwjBl7/8ZQ4dOjTh9TZt2sRvf/tbHnroId5+++1x546lD8ZclyIfmKxWarVaOXfu\nEnv3nuaTTxwIsYDc3MWkpw/fjHNlz6T9s3diLV2Dvr0VvblJqxUTgUSjhi7EYFJRIFP9lYbuP9Mu\n9d9XbDYb27Zto7q6mldffZWCgoIx577zzjs8/vjjnD59GoAvfvGLdHZ28sYbb/h8PZfLhcFg4NKl\nS351+Z5ueDwe2trauHSphbY2iV6fQ3p6ls/JK7ruLlJOHiSh5jzutEyt5KtiShia62m/8U7cpuwp\npforxkal/k+BlpYW7rnnHr7yla9w//33893vfpePPvpoYPy1114bpq3rdDpuvPHGgeOqqqoJ28eN\n1AcPHjwIwJw5cwLwG0QXvmilvb29VFXVUlZ2kiNHOrHZCsjNXUpmZq5fmYieFCNdq2/CsuGvQXow\nNNWByzkF6wNLNGroUoBMSAp4qr/S0P1nQg1dCFEA/A7IAzzAb6SUPx8xZz3wOlDZ99RrUsofBtjW\nkJGTk8POnTs5fvw4NTU1PPvss8PGf/azn7F582Yef/xxQJNaLl26xFNPPcXly5dZsWIF3/ve9/y6\n5h//+EcAFbc7BCklFouF6upmGhp60emySUsrJj196l/nnbn5tN90F4mfniL11CGkIR5XenbM9TUN\nBUJqree6Os1TSvVXTJ0JJRchxAxghpSyXAiRChwF7pBSVgyZsx54XEp5+wTnihrJZSJ+//vfc++9\n9wbsfHPnzqWqqmpaVVkcC2+bnGlpGUH7sNNZLaSe+AsJdZW40rPwJKlwO59xOYnr7sT8+W00N5/j\nuutmkO4lqUsxNXyVXCZcoUspG4HGvp+tQoizQD5QMWLqtFnaNDc3YzKZJp7oB1VVVV5rzEwnrFYr\n9fUtVFdbcLszSEtbQG5u8AtkeVLT6bz2ZgyNNRiP7cPQXK9Fw8RNiyCwKSFcTtzJxoCk+iumjl8a\nuhCiCCiBTJD7AAAgAElEQVQFvIV8XCuEKBdCvCWEWBIA2yKW3bt3c9ttt03pHN70welWwwW0Tc6d\nO3fy8cdn2L+/mpqaFEym5eTmziExMYTVDoXAOXMO7ZvupmfpKgxtTcR1hL6vabRp6DqnXftA7DRT\nVBTYVH+lofuPz0uQPrnlVeBRKaV1xPBRoFBK2SOEuAXYCSzydp5t27YNRHGYTCZKS0sHwpP6/wMj\n/bhfagnU+fpjz2fPnj2sIFGk/L7BOO7t7WXnzl1cvtyJ05lERkYBNTXHgBZW9rUxO3JEmx+OY3vB\nfE69+l/oz5ezquQ6ZGLygLPtDy0MxvGZ+sqgnj/Qx3Gd7ayYcwUeTxtnz9ZSVfVpwN4v5eXlU3p9\nNB+XlZWxY8cOAL+i3nwKWxRC6IE3gbellE/6MP8ScLWU0jzi+ZjR0APJ2bNnWbJkCS6XK6Y3lLxv\ncmZjMERoezIpib9cReqxD9A57Dgzw9/XNNIwNNfRsnw1zrnxXHvt0nCbE7METEPv4xngzFjOXAiR\nJ6Vs6vv5M2gfFGZvcxWj2blzJ0DMOvPxMjkjGiFw5M+lPWcWSeeOk1xxDE9iMu70rHBbFkEIrK5e\nFs1Rdc8jgQk1dCHEGuCLwEYhxHEhxDEhxM1CiO1CiAf7pt0phDglhDgO/G/g7iDaHBMM1QdjtYaL\nr5mc/VJHpCLjE+hZvpr2TVtwGzMwNNYg7L1BuVa0aegScMf3kp0d+A85paH7jy9RLgeAcZeOUsqn\ngKcCZdR048iRI+Tk5ITbjIDgLZPTZCqMiW8f7vRMLOs+T3x9Jcbj+6HTjCsrDyK0FG8osNttZBek\nBSTVXzF1VOp/BCCEYPv27fzyl78MtymTpre3l8bGFiorzdjtqSQn55CaGrshbMLeS9LZoySfL8eT\nYsRtnF417Pvp/fQ4M/7pAbJnzgy3KTFNoDV0RZCJxpDFYGZyRjoyIZGe0jU45iwi9dg+4ptqcWbk\nIuOnTwckt8OBSPCQkZsbblMUfahaLmGiXx9sbGwEiKqyuU6nk8uXGzhw4BQffdREW1s22dnLycqa\nNamIlUjX0MfDlZFDxw1foHPVZ4mzWtC3NYDHM+nzRZOGbu8ykz47O2hymtLQ/Uet0MPMm2++CUCC\nnx3rw0G4MjkjHp0Oe9FiHDNmk3LmCIkXT+JOTceTGtsp8NLeTsacheE2QzEEpaGHmVtvvZW33347\nYmu4TLVc7XRE39ZE6rF96DtacGXmISM1zn4KuFxOaDnB4i98DnHtteE2J+bxVUNXDj3MCCEQQuCZ\nwtf0YDDdNjkDjttNQvU5Uj85AELgMuWALnYUzs5OM4X6y2R/4Q5YEtOVPiICVQ89whmqD0bKhuhA\nT85PzlNWdp7z5+NITi4mN3d+UJ15NGvoYxIXh33eEtpv3oo9fx7xzXXoujsnfFm0aOhSdmFMTYYg\nNoJWGrr/KA09Agi3Q4/aTM4owJOUgnXVRnrnFsdMX1OHo5eUFEFCYiIkJobbHMUQlOQSRqxWK0aj\nEbPZHLAuL/5ef/gmZ25oKxxON1wuEi+dIeXEQYjT4crIjcqGGhZLC0VFcWTZe+Gee0CVzA06Kg49\nCnjvvfcAQurMYzmTM+LR6+ldWIJj1tyo7WuqLcispKUVQLNNrdAjDKWhh4mysrKQ1nAJVE/OYBCT\nGvo4jNfXNNI19N7ebkwmAwadTqs8GcSUf6Wh+49aoYeRYDv06ZzJGQ1462tKhEuSDkcXc+akgdMJ\nqtVcxKE09DAihGDdunV88MEHAT1vqHtyKqZONPQ1dbvd9PRUU1Iyh7jubsjMhFtuCbdZ0wKloUcJ\ngYxwUZmc0Us09DW12axkZydrEp3DoTZDIxCloYeJ999/H4A77rhjSufxeDy0tLSEvyfnFJhuGvqY\nCMFH9ZfC3td0LNzuLjIzjdqBwwEBbpQ+EqWh+0/kfPxPM86cOQPAnDlzJvX60ZmcBeTmqhVTLCAN\n8fQUX429YD4p5R8S31CNy5SNTEwOm00ul5P4eCfJyX02SAnJ4bNH4R3l0MPEpUuXAPzStWN1k7O/\nSbNi+L1wG010Xv9Xg31NuzrC1te0p6eL/PzUwferEJAU3G+A/c2TFb6jHHqY8CfCRWVyTmMipK+p\nlF2kp+cNf1LFoEccSkMPE9XV1RQXF487x9eenNGO0tAHGetehLKv6Uj6U/0ThzpwKYO+Qlcauv9M\nuEIXQhQAvwPyAA/wGynlz73M+zlwC9ANbJNSlgfY1pjDW4SLyuRUjEc4+prabF0UFQ3JZvV4tMqR\nqo9oxDFhHLoQYgYwQ0pZLoRIBY4Cd0gpK4bMuQV4REr5V0KIa4AnpZSrvZxLxaGjaeE6nY4jR45w\n9dVXA6pcrcJ/QtHXVEpJZ2cVJSUFGAx9BcXsdi3KZevWgF9P4Z2AxaFLKRuBxr6frUKIs0A+UDFk\n2h1oq3iklIeEEOlCiDwpZdOkrI9xKiq0W7dixQo6OjpibpNTERpC0dd0INXfMKQ6pIpBj1j80tCF\nEEVAKXBoxFA+UDvkuL7vOYUXdu7cCcDBg2cD0pMz2lEa+iCTuReB7ms6FIeji5yctJFPhiTtX2no\n/uNzlEuf3PIq8KiU0jrZC27bto2ioiIATCYTpaWlA+FJ/f+BsX585513cuJEBfv3HyIuLp01a25D\nCDHwx9wfujZdjvuZ6vleeukY779fwLlzuTidUFLSRFycJD19Bk1N4HSauf32S9x559UR9fsPPT53\nrnxyr9fpONDagMjNZ31iMokXT3Kg5TIyKYVrFiwHBgt/+Xr8l/OfYO9tpLT0RgDKTmrjG7KzIS0t\n6H8v5eXlQT1/JB+XlZWxY8cOgAF/6Qs+1XIRQuiBN4G3pZRPehn/JbBXSvlS33EFsH6k5KI09OH0\n9PTQ2mqmpqadrq44dLoM0tIyiQ/gV+bpyB13wBVXwL/92/Dn//M/4aWX4MUXoaAgPLaFikD0NbVa\nLWRl2Zg9e8bwgfp62LQJ5s0LkLWKiQh0C7pngDPenHkfu4D7+i68GuhQ+vnEJCcnU1hYwPXXL2ft\n2kIWLnRit1fQ3FyB2dysNeJV+EVDA1y+DFddNXps5UptP2/fvtDbFWpcWXl0bNxM19UbiOs0ozc3\n+S3DDEv1H4oQKgY9QpnQoQsh1gBfBDYKIY4LIY4JIW4WQmwXQjwIIKX8E3BJCHER+BXwUFCtjgFG\n6oOpqanMm1fI+vUlrFkzi6KiHqzW0zQ3n6ejoxWXyxUeQ0NAIDX0jz/W/E1f8NAwLl3SxsLQHMpn\nArqfMMm+puAl1X8kQY5BB6WhTwZfolwOABMGuUopHwmIRdMcIQRpaWmkpaUxf76Hzs5OGhrM1NXV\n4XKlkpCQidFoQhdDHeQDyZEjYDTCggWjx956S5NaNm4MvV3hZDJ9TUel+g9FSrVCj1BUPfQowe12\nY7FYuHzZzOXLVtzuNJKSMklNTY+prNGpcuutsGQJ/Pu/Dz5ntcLPfgYXL8JPfwozZoz9+pjHx76m\nFks1S5fmDc8OBc2ZNzTAAw9oyUWKkOCrhq4cehTicrlob2+nvr6dxsYepDSRnJxJSopxWjv36mq4\n805Nblm+XPM9PT2adHzttbB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W09ycyaFDZvbsOcHZs5V0dHSMH+Pe2xvWeuIB09ClhI4O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\nxhiTICyhG2NMgvg/HtxDK78yajgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1058b3410>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "H = np.array([\n",
    "        [ 0.5, -0.2, 0.2, -0.1],\n",
    "        [ 0.4,  0.4, 1.5, 0.6]\n",
    "    ])\n",
    "P_moved = P + H\n",
    "\n",
    "plt.gca().add_artist(Polygon(P.T, alpha=0.2))\n",
    "plt.gca().add_artist(Polygon(P_moved.T, alpha=0.3, color=\"r\"))\n",
    "for vector, origin in zip(H.T, P.T):\n",
    "    plot_vector2d(vector, origin=origin)\n",
    "\n",
    "plt.text(2.2, 1.8, \"$P$\", color=\"b\", fontsize=18)\n",
    "plt.text(2.0, 3.2, \"$P+H$\", color=\"r\", fontsize=18)\n",
    "plt.text(2.5, 0.5, \"$H_{*,1}$\", color=\"k\", fontsize=18)\n",
    "plt.text(4.1, 3.5, \"$H_{*,2}$\", color=\"k\", fontsize=18)\n",
    "plt.text(0.4, 2.6, \"$H_{*,3}$\", color=\"k\", fontsize=18)\n",
    "plt.text(4.4, 0.2, \"$H_{*,4}$\", color=\"k\", fontsize=18)\n",
    "\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we add a matrix full of identical vectors, we get a simple geometric translation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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8lHPNaMllNDLV1iORCJ2dPRw54iUadVBeXoPDkfkElIhGMHe3Yz22j7KuNgCS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\nLgc/c+ziTaeLAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10587ce50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "H2 = np.array([\n",
    "        [-0.5, -0.5, -0.5, -0.5],\n",
    "        [ 0.4,  0.4,  0.4,  0.4]\n",
    "    ])\n",
    "P_translated = P + H2\n",
    "\n",
    "plt.gca().add_artist(Polygon(P.T, alpha=0.2))\n",
    "plt.gca().add_artist(Polygon(P_translated.T, alpha=0.3, color=\"r\"))\n",
    "for vector, origin in zip(H2.T, P.T):\n",
    "    plot_vector2d(vector, origin=origin)\n",
    "\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Although matrices can only be added together if they have the same size, NumPy allows adding a row vector or a column vector to a matrix: this is called *broadcasting* and is explained in further details in the [NumPy tutorial](tools_numpy.ipynb). We could have obtained the same result as above with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.5,  3.5,  0.5,  4.1],\n",
       "       [ 0.6,  3.9,  2.4,  0.9]])"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P + [[-0.5], [0.4]]  # same as P + H2, thanks to NumPy broadcasting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Scalar multiplication\n",
    "Multiplying a matrix by a scalar results in all its vectors being multiplied by that scalar, so unsurprisingly, the geometric result is a rescaling of the entire figure. For example, let's rescale our polygon by a factor of 60% (zooming out, centered on the origin):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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F3/5W6eZOJ/T2Dv+elBQ1j3fVVeGx0V88Hg9tbW3U11uprbXT25uGyaRi3Acn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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105809f90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_transformation(P_before, P_after, text_before, text_after, axis = [0, 5, 0, 4], arrows=False):\n",
    "    if arrows:\n",
    "        for vector_before, vector_after in zip(P_before.T, P_after.T):\n",
    "            plot_vector2d(vector_before, color=\"blue\", linestyle=\"--\")\n",
    "            plot_vector2d(vector_after, color=\"red\", linestyle=\"-\")\n",
    "    plt.gca().add_artist(Polygon(P_before.T, alpha=0.2))\n",
    "    plt.gca().add_artist(Polygon(P_after.T, alpha=0.3, color=\"r\"))\n",
    "    plt.text(P_before[0].mean(), P_before[1].mean(), text_before, fontsize=18, color=\"blue\")\n",
    "    plt.text(P_after[0].mean(), P_after[1].mean(), text_after, fontsize=18, color=\"red\")\n",
    "    plt.axis(axis)\n",
    "    plt.grid()\n",
    "\n",
    "P_rescaled = 0.60 * P\n",
    "plot_transformation(P, P_rescaled, \"$P$\", \"$0.6 × P$\", arrows=True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dot product – Projection onto an axis\n",
    "The dot product is more complex to visualize, but it is also the most powerful tool in the box.\n",
    "\n",
    "Let's start simple, by defining a $1 \\times 2$ matrix $U = \\begin{bmatrix} 1 & 0 \\end{bmatrix}$. This row vector is just the horizontal unit vector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "U = np.array([[1, 0]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's look at the dot product $P \\cdot U$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 3. ,  4. ,  1. ,  4.6]])"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.dot(P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These are the horizontal coordinates of the vectors in $P$. In other words, we just projected $P$ onto the horizontal axis:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ps0xMTDg2j3su2Fwu2cv7d7De3l66uroACAQC9PT0cDTZua18mG27tLZXeCUe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\nq1ocHZ4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1054add90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_projection(U, P):\n",
    "    U_P = U.dot(P)\n",
    "    \n",
    "    axis_end = 100 * U\n",
    "    plot_vector2d(axis_end[0], color=\"black\")\n",
    "\n",
    "    plt.gca().add_artist(Polygon(P.T, alpha=0.2))\n",
    "    for vector, proj_coordinate in zip(P.T, U_P.T):\n",
    "        proj_point = proj_coordinate * U\n",
    "        plt.plot(proj_point[0][0], proj_point[0][1], \"ro\")\n",
    "        plt.plot([vector[0], proj_point[0][0]], [vector[1], proj_point[0][1]], \"r--\")\n",
    "\n",
    "    plt.axis([0, 5, 0, 4])\n",
    "    plt.grid()\n",
    "    plt.show()\n",
    "\n",
    "plot_projection(U, P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can actually project on any other axis by just replacing $U$ with any other unit vector. For example, let's project on the axis that is at a 30° angle above the horizontal axis:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Yw9///ndeesmZRuvZsxezZ88mLS3Nspi01D/wacpF+UVtbS1lZc7OvbY2ishI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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1059ec750>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "angle30 = 30 * np.pi / 180  # angle in radians\n",
    "U_30 = np.array([[np.cos(angle30), np.sin(angle30)]])\n",
    "\n",
    "plot_projection(U_30, P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Good! Remember that the dot product of a unit vector and a matrix basically performs a projection on an axis and gives us the coordinates of the resulting points on that axis."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dot product – Rotation\n",
    "Now let's create a $2 \\times 2$ matrix $V$ containing two unit vectors that make 30° and 120° angles with the horizontal axis:\n",
    "\n",
    "$V = \\begin{bmatrix} \\cos(30°) & \\sin(30°) \\\\ \\cos(120°) & \\sin(120°) \\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.8660254,  0.5      ],\n",
       "       [-0.5      ,  0.8660254]])"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "angle120 = 120 * np.pi / 180\n",
    "V = np.array([\n",
    "        [np.cos(angle30), np.sin(angle30)],\n",
    "        [np.cos(angle120), np.sin(angle120)]\n",
    "    ])\n",
    "V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at the dot product $V \\cdot P$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.69807621,  5.21410162,  1.8660254 ,  4.23371686],\n",
       "       [-1.32679492,  1.03108891,  1.23205081, -1.8669873 ]])"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V.dot(P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first row is equal to $V_{1,*} \\cdot P$, which is the coordinates of the projection of $P$ onto the 30° axis, as we have seen above. The second row is $V_{2,*} \\cdot P$, which is the coordinates of the projection of $P$ onto the 120° axis. So basically we obtained the coordinates of $P$ after rotating the horizontal and vertical axes by 30° (or equivalently after rotating the polygon by -30° around the origin)! Let's plot $V \\cdot P$ to see this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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C6NH+t98PKJWauAJ0x5IihCgG8ISU8l0l9h1otJJ7qoSdZjMNQL7wAiUDKI2U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\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105083210>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P_rotated = V.dot(P)\n",
    "plot_transformation(P, P_rotated, \"$P$\", \"$V \\cdot P$\", [-2, 6, -2, 4], arrows=True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Matrix $V$ is called a **rotation matrix**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dot product – Other linear transformations\n",
    "More generally, any linear transformation $f$ that maps n-dimensional vectors to m-dimensional vectors can be represented as an $m \\times n$ matrix. For example, say $\\textbf{u}$ is a 3-dimensional vector:\n",
    "\n",
    "$\\textbf{u} = \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}$\n",
    "\n",
    "and $f$ is defined as:\n",
    "\n",
    "$f(\\textbf{u}) = \\begin{pmatrix}\n",
    "ax + by + cz \\\\\n",
    "dx + ey + fz\n",
    "\\end{pmatrix}$\n",
    "\n",
    "This transormation $f$ maps 3-dimensional vectors to 2-dimensional vectors in a linear way (ie. the resulting coordinates only involve sums of multiples of the original coordinates). We can represent this transformation as matrix $F$:\n",
    "\n",
    "$F = \\begin{bmatrix}\n",
    "a & b & c \\\\\n",
    "d & e & f\n",
    "\\end{bmatrix}$\n",
    "\n",
    "Now, to compute $f(\\textbf{u})$ we can simply do a dot product:\n",
    "\n",
    "$f(\\textbf{u}) = F \\cdot \\textbf{u}$\n",
    "\n",
    "If we have a matric $G = \\begin{bmatrix}\\textbf{u}_1 & \\textbf{u}_2 & \\cdots & \\textbf{u}_q \\end{bmatrix}$, where each $\\textbf{u}_i$ is a 3-dimensional column vector, then $F \\cdot G$ results in the linear transformation of all vectors $\\textbf{u}_i$ as defined by the matrix $F$:\n",
    "\n",
    "$F \\cdot G = \\begin{bmatrix}f(\\textbf{u}_1) & f(\\textbf{u}_2) & \\cdots & f(\\textbf{u}_q) \\end{bmatrix}$\n",
    "\n",
    "To summarize, the matrix on the left hand side of a dot product specifies what linear transormation to apply to the right hand side vectors. We have already shown that this can be used to perform projections and rotations, but any other linear transformation is possible. For example, here is a transformation known as a *shear mapping*:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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yNq11Sin1+wB+BolT/okFm4jIGgVbXENERObL+0IkF94IpdRmpdQx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\nTAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10550f3d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_shear = np.array([\n",
    "        [1, 1.5],\n",
    "        [0, 1]\n",
    "    ])\n",
    "plot_transformation(P, F_shear.dot(P), \"$P$\", \"$F_{shear} \\cdot P$\",\n",
    "                    axis=[0, 10, 0, 7])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at how this transformation affects the **unit square**: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Li/tebmoiuHQpVFRQ/N73wvjxrSO+4mmq9CVKubX9CWJPypQ3bIDjxym+/nq49lqCpaWw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\nxpy4YRhGEmNO3DAMI4kxJ24YhpHE/H/ODg6bXljFOQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105166f90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Square = np.array([\n",
    "        [0, 0, 1, 1],\n",
    "        [0, 1, 1, 0]\n",
    "    ])\n",
    "plot_transformation(Square, F_shear.dot(Square), \"$Square$\", \"$F_{shear} \\cdot Square$\",\n",
    "                    axis=[0, 2.6, 0, 1.8])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's look at a **squeeze mapping**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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TpNPm8pM/ufHJSPE6qVGXyNIWu/aiuzdEZYyNDbF7d5Ldu3fYHYqx1sRfMJif+HO63DLx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\nDieJWgghHE4StRBCONz/B0B3dEep8wH8AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105a4ac10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_squeeze = np.array([\n",
    "        [1.4, 0],\n",
    "        [0, 1/1.4]\n",
    "    ])\n",
    "plot_transformation(P, F_squeeze.dot(P), \"$P$\", \"$F_{squeeze} \\cdot P$\",\n",
    "                    axis=[0, 7, 0, 5])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The effect on the unit square is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Ii4ikMnXyIjGmTr4D1MmHEvdO3symmtkqMyszszui3H6dmS2N/FlgZhM6E0ZE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\nQ15EJIVpyIuIpDANeRGRFKYhLyKSwv4/jieNHDELoFIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1058023d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_transformation(Square, F_squeeze.dot(Square), \"$Square$\", \"$F_{squeeze} \\cdot Square$\",\n",
    "                    axis=[0, 1.8, 0, 1.2])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's show a last one: reflection through the horizontal axis:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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274/HL17RsKPDJPm2Nrj+evPct7+9iCCt44rf60s4MWbpeYv5y2Sgt3eqhz0x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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105bd8dd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_reflect = np.array([\n",
    "        [1, 0],\n",
    "        [0, -1]\n",
    "    ])\n",
    "plot_transformation(P, F_reflect.dot(P), \"$P$\", \"$F_{reflect} \\cdot P$\",\n",
    "                    axis=[-2, 9, -4.5, 4.5])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrix inverse\n",
    "Now that we understand that a matrix can represent any linear transformation, a natural question is: can we find a transformation matrix that reverses the effect of a given transformation matrix $F$? The answer is yes… sometimes! When it exists, such a matrix is called the **inverse** of $F$, and it is noted $F^{-1}$.\n",
    "\n",
    "For example, the rotation, the shear mapping and the squeeze mapping above all have inverse transformations. Let's demonstrate this on the shear mapping:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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7IRSiXZdSOiOdJFz9cmy2bgWrFerrj9/0p7fKuPH6UPyfe+1asu6/n6VK8fbb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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1055438d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_inv_shear = np.array([\n",
    "    [1, -1.5],\n",
    "    [0, 1]\n",
    "])\n",
    "P_sheared = F_shear.dot(P)\n",
    "P_unsheared = F_inv_shear.dot(P_sheared)\n",
    "plot_transformation(P_sheared, P_unsheared, \"$P_{sheared}$\", \"$P_{unsheared}$\",\n",
    "                    axis=[0, 10, 0, 7])\n",
    "plt.plot(P[0], P[1], \"b--\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We applied a shear mapping on $P$, just like we did before, but then we applied a second transformation to the result, and *lo and behold* this had the effect of coming back to the original $P$ (we plotted the original $P$'s outline to double check). The second transformation is the inverse of the first one.\n",
    "\n",
    "We defined the inverse matrix $F_{shear}^{-1}$ manually this time, but NumPy provides an `inv` function to compute a matrix's inverse, so we could have written instead:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1. , -1.5],\n",
       "       [ 0. ,  1. ]])"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F_inv_shear = LA.inv(F_shear)\n",
    "F_inv_shear"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Only square matrices can be inversed. This makes sense when you think about it: if you have a transformation that reduces the number of dimensions, then some information is lost and there is no way that you can get it back. For example say you use a $2 \\times 3$ matrix to project a 3D object onto a plane. The result may look like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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T2riRefu3qy+2v+2zVfxV8+yQRaV8Bjo9hXJUXEyF9+l5SdMi4m3bcyQ9Iml6xTWhe/pi\n+9v+oKSHJC1s7NGXbsxDPiL+tNWyxoGXKXHsYqrXWrzG3sa//2P7YRV/7vdyyO+WNK3p/tTGY0P7\nfHyUPr1q1PVv/sBHxGO277Z9WkS83qUaq5Z5+4+qH7a/7QkqAn51RLznGiOV9Bmoerrm6MVU0ggX\nUzW+7dR0MdV/dKvAMbJJ0jm2z7I9UdK1Kt6LZmslXS9Jti+RNHh0aiuBUde/ee7R9gwVp/um+Q/e\nYLWed868/Y9quf59sv3/WdJ/RcQ/tFheymdgzPfkR3GnpAdt/7UaF1NJUvPFVCqmeh5u/NzB0Yup\n1ldVcBki4rDtWyWtV/FFuzIittq+uVgc90TEOtuX235J0n5JN1ZZc5naWX9JV9u+RdJBSQckLaiu\n4vLZvl9STdLptl+RtFjSRPXB9pdGX3/l3/6zJP2lpF/Z/qWKqeq/U3HGWamfAS6GAoDEqp6uAQCM\nIUIeABIj5AEgMUIeABIj5AEgMUIeABIj5AEgMUIeABL7PyTYAoze4eQfAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105775dd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot([0, 0, 1, 1, 0, 0.1, 0.1, 0, 0.1, 1.1, 1.0, 1.1, 1.1, 1.0, 1.1, 0.1],\n",
    "         [0, 1, 1, 0, 0, 0.1, 1.1, 1.0, 1.1, 1.1, 1.0, 1.1, 0.1, 0, 0.1, 0.1],\n",
    "         \"r-\")\n",
    "plt.axis([-0.5, 2.1, -0.5, 1.5])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looking at this image, it is impossible to tell whether this is the projection of a cube or the projection of a narrow rectangular object. Some information has been lost in the projection.\n",
    "\n",
    "Even square transformation matrices can lose information. For example, consider this transformation matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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RNXOfuJ7buVyfy7WB6ssWauYiIg5QZi4iEmLKzEVEsoiauU9cz+1crs/l2kD1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\ngcH+WokETwMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10500dcd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_project = np.array([\n",
    "        [1, 0],\n",
    "        [0, 0]\n",
    "    ])\n",
    "plot_transformation(P, F_project.dot(P), \"$P$\", \"$F_{project} \\cdot P$\",\n",
    "                    axis=[0, 6, -1, 4])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This transformation matrix performs a projection onto the horizontal axis. Our polygon gets entirely flattened out so some information is entirely lost and it is impossible to go back to the original polygon using a linear transformation. In other words, $F_{project}$ has no inverse. Such a square matrix that cannot be inversed is called a **singular matrix** (aka degenerate matrix). If we ask NumPy to calculate its inverse, it raises an exception:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "LinAlgError: Singular matrix\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    LA.inv(F_project)\n",
    "except LA.LinAlgError as e:\n",
    "    print(\"LinAlgError:\", e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is another example of a singular matrix. This one performs a projection onto the axis at a 30° angle above the horizontal axis:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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XXzxPefn8pN6uopj0psxcxGcHD77J228XUlQU3OZjauzpR5l5krme27lcXzy1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\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105a41550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "angle30 = 30 * np.pi / 180\n",
    "F_project_30 = np.array([\n",
    "               [np.cos(angle30)**2, np.sin(2*angle30)/2],\n",
    "               [np.sin(2*angle30)/2, np.sin(angle30)**2]\n",
    "         ])\n",
    "plot_transformation(P, F_project_30.dot(P), \"$P$\", \"$F_{project\\_30} \\cdot P$\",\n",
    "                    axis=[0, 6, -1, 4])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But this time, due to floating point rounding errors, NumPy manages to calculate an inverse (notice how large the elements are, though):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  1.20095990e+16,  -2.08012357e+16],\n",
       "       [ -2.08012357e+16,   3.60287970e+16]])"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.inv(F_project_30)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you might expect, the dot product of a matrix by its inverse results in the identity matrix:\n",
    "\n",
    "$M \\cdot M^{-1} = M^{-1} \\cdot M = I$\n",
    "\n",
    "This makes sense since doing a linear transformation followed by the inverse transformation results in no change at all."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.,  0.],\n",
       "       [ 0.,  1.]])"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F_shear.dot(LA.inv(F_shear))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another way to express this is that the inverse of the inverse of a matrix $M$ is $M$ itself:\n",
    "\n",
    "$((M)^{-1})^{-1} = M$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1. ,  1.5],\n",
       "       [ 0. ,  1. ]])"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.inv(LA.inv(F_shear))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Also, the inverse of scaling by a factor of $\\lambda$ is of course scaling by a factor or $\\frac{1}{\\lambda}$:\n",
    "\n",
    "$ (\\lambda \\times M)^{-1} = \\frac{1}{\\lambda} \\times M^{-1}$\n",
    "\n",
    "Once you understand the geometric interpretation of matrices as linear transformations, most of these properties seem fairly intuitive.\n",
    "\n",
    "A matrix that is its own inverse is called an **involution**. The simplest examples are reflection matrices, or a rotation by 180°, but there are also more complex involutions, for example imagine a transformation that squeezes horizontally, then  reflects over the vertical axis and finally rotates by 90° clockwise. Pick up a napkin and try doing that twice: you will end up in the original position. Here is the corresponding involutory matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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UNGpGkLz5pukxBoOeGz3iaX/8xyZxztRrHR42n2RaWkyPt6PD3Obt5Em46Sbz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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x104ff9fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_involution  = np.array([\n",
    "        [0, -2],\n",
    "        [-1/2, 0]\n",
    "    ])\n",
    "plot_transformation(P, F_involution.dot(P), \"$P$\", \"$F_{involution} \\cdot P$\",\n",
    "                    axis=[-8, 5, -4, 4])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, a square matrix $H$ whose inverse is its own transpose is an **orthogonal matrix**:\n",
    "\n",
    "$H^{-1} = H^T$\n",
    "\n",
    "Therefore:\n",
    "\n",
    "$H \\cdot H^T = H^T \\cdot H = I$\n",
    "\n",
    "It corresponds to a transformation that preserves distances, such as rotations and reflections, and combinations of these, but not rescaling, shearing or squeezing.  Let's check that $F_{reflect}$ is indeed orthogonal:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0],\n",
       "       [0, 1]])"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F_reflect.dot(F_reflect.T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Determinant\n",
    "The determinant of a square matrix $M$, noted $\\det(M)$ or $\\det M$ or $|M|$ is a value that can be calculated from its elements $(M_{i,j})$ using various equivalent methods. One of the simplest methods is this recursive approach:\n",
    "\n",
    "$|M| = M_{1,1}\\times|M^{(1,1)}| - M_{2,1}\\times|M^{(2,1)}| + M_{3,1}\\times|M^{(3,1)}| - M_{4,1}\\times|M^{(4,1)}| + \\cdots ± M_{n,1}\\times|M^{(n,1)}|$\n",
    "\n",
    "* Where $M^{(i,j)}$ is the matrix $M$ without row $i$ and column $j$.\n",
    "\n",
    "For example, let's calculate the determinant of the following $3 \\times 3$ matrix:\n",
    "\n",
    "$M = \\begin{bmatrix}\n",
    "  1 & 2 & 3 \\\\\n",
    "  4 & 5 & 6 \\\\\n",
    "  7 & 8 & 0\n",
    "\\end{bmatrix}$\n",
    "\n",
    "Using the method above, we get:\n",
    "\n",
    "$|M| = 1 \\times \\left | \\begin{bmatrix} 5 & 6 \\\\ 8 & 0 \\end{bmatrix} \\right |\n",
    "     - 2 \\times \\left | \\begin{bmatrix} 4 & 6 \\\\ 7 & 0 \\end{bmatrix} \\right |\n",
    "     + 3 \\times \\left | \\begin{bmatrix} 4 & 5 \\\\ 7 & 8 \\end{bmatrix} \\right |$\n",
    "\n",
    "Now we need to compute the determinant of each of these $2 \\times 2$ matrices (these determinants are called **minors**):\n",
    "\n",
    "$\\left | \\begin{bmatrix} 5 & 6 \\\\ 8 & 0 \\end{bmatrix} \\right | = 5 \\times 0 - 6 \\times 8 = -48$\n",
    "\n",
    "$\\left | \\begin{bmatrix} 4 & 6 \\\\ 7 & 0 \\end{bmatrix} \\right | = 4 \\times 0 - 6 \\times 7 = -42$\n",
    "\n",
    "$\\left | \\begin{bmatrix} 4 & 5 \\\\ 7 & 8 \\end{bmatrix} \\right | = 4 \\times 8 - 5 \\times 7 = -3$\n",
    "\n",
    "Now we can calculate the final result:\n",
    "\n",
    "$|M| = 1 \\times (-48) - 2 \\times (-42) + 3 \\times (-3) = 27$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To get the determinant of a matrix, you can call NumPy's `det` function in the `numpy.linalg` module:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "27.0"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = np.array([\n",
    "        [1, 2, 3],\n",
    "        [4, 5, 6],\n",
    "        [7, 8, 0]\n",
    "    ])\n",
    "LA.det(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One of the main uses of the determinant is to *determine* whether a square matrix can be inversed or not: if the determinant is equal to 0, then the matrix *cannot* be inversed (it is a singular matrix), and if the determinant is not 0, then it *can* be inversed.\n",
    "\n",
    "For example, let's compute the determinant for the $F_{project}$, $F_{project\\_30}$ and $F_{shear}$ matrices that we defined earlier:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.det(F_project)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's right, $F_{project}$ is singular, as we saw earlier."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0816681711721642e-17"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.det(F_project_30)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This determinant is suspiciously close to 0: it really should be 0, but it's not due to tiny floating point errors. The matrix is actually singular."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.det(F_shear)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Perfect! This matrix *can* be inversed as we saw earlier. Wow, math really works!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The determinant can also be used to measure how much a linear transformation affects surface areas: for example, the projection matrices $F_{project}$ and $F_{project\\_30}$ completely flatten the polygon $P$, until its area is zero. This is why the determinant of these matrices is 0. The shear mapping modified the shape of the polygon, but it did not affect its surface area, which is why the determinant is 1. You can try computing the determinant of a rotation matrix, and you should also find 1. What about a scaling matrix? Let's see:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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/Hw2pFAYHB9HTE0MkEkYyWaQbe0EABYUBjNQtdnr1rrRvn87Ev/ENYOPGme+XSCRQUNCH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\ngg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x104edf8d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_scale = np.array([\n",
    "        [0.5, 0],\n",
    "        [0, 0.5]\n",
    "    ])\n",
    "plot_transformation(P, F_scale.dot(P), \"$P$\", \"$F_{scale} \\cdot P$\",\n",
    "                    axis=[0, 6, -1, 4])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We rescaled the polygon by a factor of 1/2 on both vertical and horizontal axes so the surface area of the resulting polygon is 1/4$^{th}$ of the original polygon. Let's compute the determinant and check that:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.25"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.det(F_scale)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Correct!\n",
    "\n",
    "The determinant can actually be negative, when the transformation results in a \"flipped over\" version of the original polygon (eg. a left hand glove becomes a right hand glove). For example, the determinant of the `F_reflect` matrix is -1 because the surface area is preserved but the polygon gets flipped over:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.0"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.det(F_reflect)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Composing linear transformations\n",
    "Several linear transformations can be chained simply by performing multiple dot products in a row. For example, to perform a squeeze mapping followed by a shear mapping, just write:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "P_squeezed_then_sheared = F_shear.dot(F_squeeze.dot(P))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since the dot product is associative, the following code is equivalent:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "P_squeezed_then_sheared = (F_shear.dot(F_squeeze)).dot(P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the order of the transformations is the reverse of the dot product order.\n",
    "\n",
    "If we are going to perform this composition of linear transformations more than once, we might as well save the composition matrix like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "F_squeeze_then_shear = F_shear.dot(F_squeeze)\n",
    "P_squeezed_then_sheared = F_squeeze_then_shear.dot(P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From now on we can perform both transformations in just one dot product, which can lead to a very significant performance boost."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What if you want to perform the inverse of this double transformation? Well, if you squeezed and then you sheared, and you want to undo what you have done, it should be obvious that you should unshear first and then unsqueeze. In more mathematical terms, given two invertible (aka nonsingular) matrices $Q$ and $R$:\n",
    "\n",
    "$(Q \\cdot R)^{-1} = R^{-1} \\cdot Q^{-1}$\n",
    "\n",
    "And in NumPy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ True,  True],\n",
       "       [ True,  True]], dtype=bool)"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.inv(F_shear.dot(F_squeeze)) == LA.inv(F_squeeze).dot(LA.inv(F_shear))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Singular Value Decomposition\n",
    "It turns out that any $m \\times n$ matrix $M$ can be decomposed into the dot product of three simple matrices:\n",
    "* a rotation matrix $U$ (an $m \\times m$ orthogonal matrix)\n",
    "* a scaling & projecting matrix $\\Sigma$ (an $m \\times n$ diagonal matrix)\n",
    "* and another rotation matrix $V^T$ (an $n \\times n$ orthogonal matrix)\n",
    "\n",
    "$M = U \\cdot \\Sigma \\cdot V^{T}$\n",
    "\n",
    "For example, let's decompose the shear transformation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.89442719, -0.4472136 ],\n",
       "       [ 0.4472136 ,  0.89442719]])"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U, S_diag, V_T = LA.svd(F_shear) # note: in python 3 you can rename S_diag to Σ_diag\n",
    "U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2. ,  0.5])"
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S_diag"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that this is just a 1D array containing the diagonal values of Σ. To get the actual matrix Σ, we can use NumPy's `diag` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2. ,  0. ],\n",
       "       [ 0. ,  0.5]])"
      ]
     },
     "execution_count": 126,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S = np.diag(S_diag)\n",
    "S"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's check that $U \\cdot \\Sigma \\cdot V^T$ is indeed equal to `F_shear`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1. ,  1.5],\n",
       "       [ 0. ,  1. ]])"
      ]
     },
     "execution_count": 127,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.dot(np.diag(S_diag)).dot(V_T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1. ,  1.5],\n",
       "       [ 0. ,  1. ]])"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F_shear"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It worked like a charm. Let's apply these transformations one by one (in reverse order) on the unit square to understand what's going on. First, let's apply the first rotation $V^T$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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EYb3hU5+C4a3POSAiuakmUAKSlu/vEDMYOBBeegku0ZwDIsWk/20BFK0mUOD1/a+uer0I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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10550fbd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_transformation(Square, V_T.dot(Square), \"$Square$\", \"$V^T \\cdot Square$\",\n",
    "                    axis=[-0.5, 3.5 , -1.5, 1.5])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's rescale along the vertical and horizontal axes using $\\Sigma$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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peW0UZFvHq6ElnurTSSye0ejbxp5sSi+0OvwH/ncaHV0xg27n/Rc0cu3Fc3ng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\nUiqK/R9kJsBsOB4OigAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105067d10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_transformation(V_T.dot(Square), S.dot(V_T).dot(Square), \"$V^T \\cdot Square$\", \"$\\Sigma \\cdot V^T \\cdot Square$\",\n",
    "                    axis=[-0.5, 3.5 , -1.5, 1.5])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we apply the second rotation $U$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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qVq58g4aGAey99xD69tUD4KPok5+EZ5/NzCOHhw/3NYG4xx/3PYXaPuPmkEP8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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1058a8190>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_transformation(S.dot(V_T).dot(Square), U.dot(S).dot(V_T).dot(Square),\"$\\Sigma \\cdot V^T \\cdot Square$\", \"$U \\cdot \\Sigma \\cdot V^T \\cdot Square$\",\n",
    "                    axis=[-0.5, 3.5 , -1.5, 1.5])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And we can see that the result is indeed a shear mapping of the original unit square."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Eigenvectors and eigenvalues\n",
    "An **eigenvector** of a square matrix $M$ (also called a **characteristic vector**) is a non-zero vector that remains on the same line after transformation by the linear transformation associated with $M$. A more formal definition is any vector $v$ such that:\n",
    "\n",
    "$M \\cdot v = \\lambda \\times v$\n",
    "\n",
    "Where $\\lambda$ is a scalar value called the **eigenvalue** associated to the vector $v$.\n",
    "\n",
    "For example, any horizontal vector remains horizontal after applying the shear mapping (as you can see on the image above), so it is an eigenvector of $M$. A vertical vector ends up tilted to the right, so vertical vectors are *NOT* eigenvectors of $M$.\n",
    "\n",
    "If we look at the squeeze mapping, we find that any horizontal or vertical vector keeps its direction (although its length changes), so all horizontal and vertical vectors are eigenvectors of $F_{squeeze}$.\n",
    "\n",
    "However, rotation matrices have no eigenvectors at all (except if the rotation angle is 0° or 180°, in which case all non-zero vectors are eigenvectors).\n",
    "\n",
    "NumPy's `eig` function returns the list of unit eigenvectors and their corresponding eigenvalues for any square matrix. Let's look at the eigenvectors and eigenvalues of the squeeze mapping matrix $F_{squeeze}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.4       ,  0.71428571])"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigenvalues, eigenvectors = LA.eig(F_squeeze)\n",
    "eigenvalues # [λ0, λ1, …]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.,  0.],\n",
       "       [ 0.,  1.]])"
      ]
     },
     "execution_count": 133,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigenvectors # [v0, v1, …]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Indeed the horizontal vectors are stretched by a factor of 1.4, and the vertical vectors are shrunk by a factor of 1/1.4=0.714…, so far so good. Let's look at the shear mapping matrix $F_{shear}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.,  1.])"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigenvalues2, eigenvectors2 = LA.eig(F_shear)\n",
    "eigenvalues2 # [λ0, λ1, …]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  1.00000000e+00,  -1.00000000e+00],\n",
       "       [  0.00000000e+00,   1.48029737e-16]])"
      ]
     },
     "execution_count": 135,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigenvectors2 # [v0, v1, …]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wait, what!? We expected just one unit eigenvector, not two. The second vector is almost equal to $\\begin{pmatrix}-1 \\\\ 0 \\end{pmatrix}$, which is on the same line as the first vector $\\begin{pmatrix}1 \\\\ 0 \\end{pmatrix}$. This is due to floating point errors. We can safely ignore vectors that are (almost) colinear (ie. on the same line)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Trace\n",
    "The trace of a square matrix $M$, noted $tr(M)$ is the sum of the values on its main diagonal. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "123"
      ]
     },
     "execution_count": 136,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = np.array([\n",
    "        [100, 200, 300],\n",
    "        [ 10,  20,  30],\n",
    "        [  1,   2,   3],\n",
    "    ])\n",
    "np.trace(D)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The trace does not have a simple geometric interpretation (in general), but it has a number of properties that make it useful in many areas:\n",
    "* $tr(A + B) = tr(A) + tr(B)$\n",
    "* $tr(A \\cdot B) = tr(B \\cdot A)$\n",
    "* $tr(A \\cdot B \\cdot \\cdots \\cdot Y \\cdot Z) = tr(Z \\cdot A \\cdot B \\cdot \\cdots \\cdot Y)$\n",
    "* $tr(A^T \\cdot B) = tr(A \\cdot B^T) = tr(B^T \\cdot A) = tr(B \\cdot A^T) = \\sum_{i,j}X_{i,j} \\times Y_{i,j}$\n",
    "* …\n",
    "\n",
    "It does, however, have a useful geometric interpretation in the case of projection matrices (such as $F_{project}$ that we discussed earlier): it corresponds to the number of dimensions after projection. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 137,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.trace(F_project)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# What next?\n",
    "This concludes this introduction to Linear Algeabra. Although these basics cover most of what you will need to know for Machine Learning, if you wish to go deeper into this topic there are many options available: Linear Algebra [books](http://linear.axler.net/), [Khan Academy](https://www.khanacademy.org/math/linear-algebra) lessons, or just [Wikipedia](https://en.wikipedia.org/wiki/Linear_algebra) pages. "
   ]
  }
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