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.*\n",
    "\n",
    "<table align=\"left\">\n",
    "  <td>\n",
    "    <a target=\"_blank\" href=\"https://colab.research.google.com/github/ageron/handson-ml/blob/master/math_linear_algebra.ipynb\"><img src=\"https://www.tensorflow.org/images/colab_logo_32px.png\" />Run in Google Colab</a>\n",
    "  </td>\n",
    "</table>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Warning**: this notebook accompanies the 1st edition of the book. Please visit https://github.com/ageron/handson-ml2 for the 2nd edition code, with up-to-date notebooks using the latest library versions."
   ]
  },
  {
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": 6,
   "metadata": {},
   "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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4O3ATk1XJRxzSknTybNkzEyVJ2+OEn5lY8ckwEfGRiHgwIg72blkRERdExOcj4q6IuDMi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\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e352e72e8>"
      ]
     },
     "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": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWoAAAEACAYAAACatzzfAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\np0ItIpKcCrWISHIq1CIiyf0/qw4ATRntajAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32fc3c50>"
      ]
     },
     "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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JE273K2BhpPxXb1sdrc0mvZCwA8pLM3SfWfdIsVjEww8/jL1796KzsxM7duzA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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e33054240>"
      ]
     },
     "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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SdXV1slMhEokgl8sVdBij6IddirLZdFrisi8otajRzM6xTkLNe6pW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\nJEk6COATQRCG/+uf/h3A9jJOyYOL8f8Bx+4YbfPZ9uIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32f04e48>"
      ]
     },
     "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": {},
   "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": {},
   "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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usJvNSq6iH85z0fjUU4w3NUEkEjqKm2RsbMzkgHYum7xznkLFiROsuP12Tr32GuOrVoWO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\nzhn0/wHrBhfDSiWHRwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32e93400>"
      ]
     },
     "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": {},
   "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": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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no0xra/xn40n9IpWxSWMZftXwOj/ekZNHOHD8QOnX/uP7KdbFnCw6Web2A8cP\ncKLg9DzoS8+9lO7nd2fu+rllHm921myUUtxz2T3+BdmxA7p0gbZt6/w7CVEfhXWhzsjIsG6UaVV9\nsskrJzN62WhGXD2CMT3HuLKufmn9OHfiuaVfv5r4K7bnbmfe+nkVbp+4YmKZ+97zn/ew6+gulmw9\nPQxp7vq5dD63M11bd3UeQmuIizvdfnLI1n6ijbkkkzM2ZnIqrE/FVTrCNMijTP01LXMawxYPY+gV\nQ5nQu/pdB6vrQxcWl53PPOmGSRzKP1TmtoHvDSShVQKPX/V46W1ZWVncftntZZYb8B8DeOyTx5jz\n7Rx6X9Sbz3/+nK2HtjLx+rIFvUZvvAHffSeHiwtRF1prV77MQ1nGbM95naJas7Nma55BD35/sKPl\nX/nqFR0xNkKv3b22zO35Bfm6wdgGutebvaq9f/zf4vW9C+91tK7fp/1eN3u+mc47laf/9P6fdMNx\nDfXuo7sd3VdrrXVRkda33KL1unXO7yNEPeKrmzXW1/BtfXg4ytSp+RvnM2jhIJI7JzOz70xH9+lw\nTge01mVaEgCTVk6iWLs7n3nQZYM4XnCcf3z7DxZ8t4AbLr7B/5MTPPKI6U8LIWotfAt1ySjTrn70\nU4OgpE+Wvjmd5AXJ9Gnfh7Q70hzfv/dFvekU24kxy8YwcslIpmVO4+5/3c2srFm1Pg1XVb27W9rf\nQssmLRmxZARHTh5h0GWDnD9oybD/a6+t1b7rtvYTbcwlmZyxMZNT4Vmo9+wxl5YeKr74x8X0TetL\nz7Y9+XDAh37dN0JF8EH/D0iKT2Lq11N5YukTFBYXsjxlOU0bNa1xX2rl+8+JyAaR9O/Sn6OnjtKi\ncQv6dfTj6MgNG+CVV5wvL4SoUnjO+mjb1kzJsyXPGZbnLCdpdhLdWnVj9f2rvY4TOCNGQPv28Kc/\neZ1ECGs5nfURfoU6Lw+aNTOjTC2bkrdqxyoSZybSvmV7tgzd4nWcwMrLg+hozw/ZF8Jm9Xco0xmj\nTG3qSa3dvZbEmYmcveds64q068/TyJGwfXudirRNr92ZbMwlmZyxMZNT4VWoCwshI8O87bbIhl82\n0G16N6Ijo3kv+T2v4wTWtm2wcSO0aeN1EiHCRni1PgYPNlPyioutecv9/YHv6TC1AwDFY4pdGZxk\ntY0b4auvzGshhKhW/etRaw0REWaU6fz53uU4Q87hHNpNaQfUkyJ9/Lj5I9msmddJhAgJ9a9HXcko\nUy97Ujs5slqwAAATTElEQVSO7Ki0SNvYJ3Mt06uvunaouI3PE9iZSzI5Y2Mmp8Jn1odFo0z3HttL\nm8mmR1s0pij8t6QBTp6Ef/wDPv3U6yRChJ3waH3MnWum5B0+7PmUvAPHDxA70RwhWDi60L+5zaGs\nqAh+/BE6dPA6iRAho371qJUyo0wPHvRm/T65+bnEvBADwKmnThHZINLTPEFz4gSkpcG993qdRIiQ\nUn961CV9p0pGmQazJ3Xs1LHSIp3/ZH6VRdrGPlmdM336Kax29yhLG58nsDOXZHLGxkxO1ViolVJx\nSqlPlVKblFLrlVIPBSOYY716mcsLL/QswomCEzRPNaeZyhuVR+OGjT3L4only6GfO2dJF0JUVGPr\nQynVCmiltc5SSjUDVgP9tNbZ5ZYLfutj0ybo3NmMMvVoSt7JwpNEPRcFwJGRR2je2Pl5AcOC1ma3\nvKZNvU4iRMhxrfWhtd6jtc7yfX8M+A64oO4RXdC5s7n0qEgXFBWUFulDIw7VzyLdrx/s3+91EiHC\nml89aqVUPJAAfBWIMH4pGWX60UdVLhLInlRRcRGNxptdAfc9vo+YqBhH97OxT1brTF99BTExATlc\n3MbnCezMJZmcsTGTU473o/a1PRYAD/u2rCtISUkhPj4egJiYGBISEkpP0V7yJLl2/bLLzPWbb65y\n+aysrICsv1gX03Cweep2T91NbHSs4/uXcP358OL6zz+TdO+9EBHh+uNn+T4ctur3PYMteWy9buPr\nF6h64M/1ku9zcnLwh6Pd85RSDYF/A/+ntZ5SxTLB61GXjDKdNg3uuy846/TRWhMxzrwR2f7oduLO\nigvq+q2xZw9ERsI553idRIiQ5fbueW8Am6oq0kFXMsrUwyK99aGt9bdIgzl7S5rzU4gJIWrPye55\nVwMDgWuVUmuVUmuUUjcFPloV/BhlWv7tal2cWaSzH8im3dntavU4bmZyi9+ZDh+Gt98O6B9KG58n\nsDOXZHLGxkxO1dij1lp/CdhzHHRJcUhNDepqW0wwh6av+/M6OsZ2DOq6rdOkiTnAJbKeHHkphMdC\n6xByj0aZxk2KY+fRnWQOyaT7+d2Dtl4r7doF8+bB8OFeJxEi5IXnIeSVjDINtEumXsLOoztZMXiF\nFGmA9HTTfhJCBE1oFWo/R5nWtSfV4+89yD6QzbJBy0hsk1inx3IrUyD4lWn7dvj97wOWpYSNzxPY\nmUsyOWNjJqdCZx713LnmcunSoKzuutnXkbkrk0UDF5EUnxSUdVqvoMD8sYyK8jqJEPVK6PSogzjK\ntO+8vqRvSWdh8kL6dZJhQ4BpdyQkwGefQcuWXqcRIiyEV4+6mlGmbhvw7gDSt6ST9oc0KdJnmj8f\nbrpJirQQHgiNQl3LUab+9qSGpA9h3oZ5zOo3i+QuyX7dN1CZgsFRpquugiFDAp6lhI3PE9iZSzI5\nY2Mmp+wv1Js2mcs1awK6mkcWPcKMNTN4tc+rpCSkBHRdIWf9erNbZMd6vv+4EB6xv0ddcmLYAPa/\nRy0dReoXqbx4/YsMv0r2D67g/vuhb1+45RavkwgRVsKjR+1glGldjf9sPKlfpDI2aawU6crs2GHe\nzfTp43USIeotuwv1lVeaS98oU3/V1JOavHIyo5eNZsTVIxjTc0yt1uF2Ji9Um6llS1i16vQ7myCx\n8XkCO3NJJmdszOSUvYU6Lw+2bTOjTANgWuY0hi0extArhjKh94SArCPkZWbC5MnQwJ5RL0LUR/b2\nqHv1MrvlBaA3PefbOQxaOIjBXQczs+9M1x8/bDz1FFx6KQwY4HUSIcJSaPeo/Rhl6q/5G+czaOEg\nkjsnS5GuSbNm0psWwgJ2FmqXRpmW70mlb04neUEyfdr3Ie0Ob4be29gnqzTTwYNmQl6Ms3NBus3G\n5wnszCWZnLExk1P2FWqtYdYsM8rUxQ+wFv+4mL5pfenZticfDvjQtccNS3l50KMHFBV5nUQIgY09\n6ueeM73RkycdT8mryfKc5STNTqJbq26svn+1K48Z1saPNycFCEDrSQhxmtMetX2FWikzyjQzs+6P\nBazasYrEmYm0b9meLUO3uPKYYW/fPvM6xMZ6nUSIsBaaHyaWjDJdssSVh/v7u38ncWYirZq1sqZI\n29gnK5Ppo48gN9fzIm3j8wR25pJMztiYySm7CvVdd5lRpi58gLXxl43cl34f0ZHR7B6+24Vw9UBB\ngfl8ID/f6yRCiDPY0/rIyDD7Tv/8s99T8sr7/sD3dJjaAYDiMcWoIB9VF7I2bTKnOwviqc6EqM9C\nr0ft0vClnMM5tJvSDpAi7be8PGja1OsUQtQbodWjdmmU6Y4jO8oU6eXLl9c1mets7JNlZGTAO+/A\niy96HaWUjc8T2JlLMjljYyanaizUSqmZSqm9Sql1AUvRubO57Nq11g+x99he2kxuA0DRmCLZkvbX\nhx+ePkGDEMIqNbY+lFLXAMeAOVrr/6xmudq1Pvbsgdatzd4GtZySd+D4AWInmr0UCkcX0iBChgj5\npbgYZs+Gu++GhqFzvmMhQp2rPWqlVFsgPSCFum1bMyWvlr3p3PxcYl4we4mceuoUkQ0ia/U49dp3\n35mzt0TY0QkTor5wWqi93Xyq4yjTY6eOlRbp/CfzKxTpjIwMkpKS6prSVcHING4c/PSTs2Wb5+/j\nlpW/58YfNwU0k79sfO3AzlySyRkbMznlaqFOSUkhPj4egJiYGBISEkqfmJJGfpnrjzxCEsB991X+\n82quf7zkY2566yZoB3mj8lj5xcoKy2dlZTl+vGBdLxGox//tb5OYMgUOHixZX1LJGiu93rJlEp2f\ne52Mzz8PSJ7aXs/ynXHeljzBev3C5bqNr58N9aDk+5ycHPzhXeujsPD0PIkJ/g3uP1l4kqjnogA4\nMvIIzRs39+v+4W7kSDPv/9Spqpdp0gT++Ef429/grLOCl00IcZrbrQ/l+3JPLUeZFhQVlBbpQyMO\nSZEu5/Bh2Lq16iLdpAmccw7MmwfXXBPcbEKI2nGye94/gRVAB6XUNqXUvXVeay1HmRYVF9FovJmo\nt+/xfcREVX+oefm3qzZwO5PWsHSp2cNRKXME/jvvVL5skyYwdCh8/33ZIl0fnie32JhLMjljYyan\naizUWusBWuvztdaNtdYXaq1n1Xmtzz9vLv04VLlYF9PwWfMGYPfw3cRG19/JbocPw2OPmcIcEQG9\ne5tjhm6/HX74wRTvefPMCVoAoqPNGbW++gpeeAGiorzNL4TwjzeHkPs5ylRrTcQ48zdl+6PbiTsr\nrrYxQ5LW8Omn8NBDpw/iBFNwX34ZBg+ueP7Z/HwzAE9rGDsWHn1UzlErhG3s3T3Pz1GmZxbpHx/6\nsd4U6cOHzfz+l14qe/vtt8P//i9cfHH194+Kgi++gBYtoF27wOUUQgRe8I9w8GOU6ZlFOvuBbC46\n+yK/VmVjT6qqTJX1ml96yRTc6dPNTjJaw7vv1lykSyQkOCvSofQ8ec3GXJLJGRszORXcQl3yRPn2\nsaxJiwktAFj353V0jO0YoFDecdJrPnEChgyRtoUQ9Vlwe9R+jDKNmxTHzqM7yRySSffzu7uQ0Hu1\n6TULIcKXfWNO/RhlesnUS9h5dCcrBq8I+SItW81CiLoKXqF2OMq0x997kH0gm2WDlpHYJrFOq/Si\nJ1VTr3nJkgy/e82BZmPvzsZMYGcuyeSMjZmcCk6h3rPHXH70UbWLXTf7OjJ3ZbJo4CKS4pMCn8sl\nstUshAik4PSoHYwy7TuvL+lb0lmYvJB+nfq5kilQpNcshHCDPT1qB6NM+7/bn/Qt6aT9Ic3aIl3V\nVvNtt8lWsxAisAJfqG+91VyWDGEqZ0j6ENI2pPFG3zdI7pLs6qrr0pNyul/ze+/512u2sU8mmZyz\nMZdkcsbGTE4FtlAXFpp9p0eMqPTHjyx6hBlrZvBqn1e5t2vdZz3VlVtbzYsWmftPnVr5zxMT4bzz\noKgoML+HECLMaK1d+TIPVc6992oNWhcXV/jRE0ue0DyDfvHLFyveL0iKi7VeskTrzp1NzJKvqCit\np0/XurCwdo9bVKT1+edr3aNHxZ99/73WSmn96KN1yy6ECH2+ulljfQ3cFnU1o0zHfzae1C9SGZs0\nluFXDQ9YhMpUttW8caO7veaICBg4EFavhuzssj+bPdus+5576v67CCHqCSfV3MkX5beox483m6cn\nT5a5edKKSZpn0CM+GeHy36aKli1bFrCt5pps2GC2nJ94ouztrVsf1//5n4FZZ20tW7bM6wgV2JhJ\naztzSSZnbMyE51vUTz1lRpk2alR607TMaQxbPIyhVwxlQm//Tr/lj5Kt5l69ArfVXJPOnc2xPSXD\nAgGWL4c9e6JISQnMOoUQYcpJNXfyxZlb1G+9ZTZbDx0qvWl21mzNM+jB7w92/a+SV1vNNZkyReuI\nCK2XLjXXBw/WOjJS6z17vMkjhLALDreoA3PAS8n+bAcPAjB/43ySFyST3DmZtDvSXFlfVfOab7sN\nJk604/Ds/fvhggugf394/XVo1cqcAuvf//Y6mRDCBt4d8FJulOkHmz8geUEyfdr3qVORLtmvuUsX\n5/s1e73fZGws3HyzyTR3Lhw5ApdfvtHTTJXx+nmqjI2ZwM5ckskZGzM55X6h7tXLXF54IYt/XEy/\ntH789sLf8uGAD/1+qGDsoRFogwbBsWMwfLg5V8JVV+33OpIQIsS42/rYuNF8irZmDcvPPkLS7CS6\nterG6vtXO3qMkhkaDz9sCnKJUJ6hUVAA559vukBDhpgWiBBCgPPWh7uF2vf9qu0rSZyZSPuW7dky\ndEu19wuFXrMQQgSCqz1qpdRNSqlspdQWpVTlx4P7rH3nFRJnJtKqWatKi3Rtes21ZWNPSjI5Y2Mm\nsDOXZHLGxkxO1ViolVIRwFTgRqAz0F8p1amyZTeeC902DiU6Mprdw3eX3u5VrznL4bkZg0kyOWNj\nJrAzl2RyxsZMTjV0sMwVwPda658BlFJpQD8gu/yCXR4wl0dHHmPpUu97zYcPHw78SvwkmZyxMRPY\nmUsyOWNjJqecFOoLgO1nXN+BKd6Ve6aYBs+cbrlIr1kIIerGSaGurNFd+SeQzxQTFaWs2UMjJyfH\n2wCVkEzO2JgJ7MwlmZyxMZNTNe71oZT6DfCM1vom3/WRmMMeXyi3nDu7jwghRD3iyu55SqkGwGbg\nOmA38DXQX2v9nRshhRBCVK/G1ofWukgp9SCwGLOXyEwp0kIIETyuHfAihBAiMOo868Ofg2GCRSk1\nUym1Vym1zussJZRScUqpT5VSm5RS65VSD1mQqbFS6iul1Fpfpqe9zlRCKRWhlFqjlPrA6ywASqkc\npdS3vufqa6/zACilWiil3lFKfaeU2qiUutKCTB18z9Ea32WuJf/WH1VKbVBKrVNKzVVKNar5XgHP\n9LDv/7ua64GTWahVfWEK/Q9AWyASyAI61eUx3fgCrgESgHVeZzkjUysgwfd9M0zf34bnKtp32QBY\nBVzhdSZfnkeBt4APvM7iy7MVONvrHOUyvQnc6/u+IXCW15nK5YsAdgFtPM5xvu/1a+S7/jZwj8eZ\nOgPrgMa+//c+AS6uavm6blGXHgyjtS4ASg6G8ZTW+gvgkNc5zqS13qO1zvJ9fwz4DrOPuqe01sd9\n3zbG/M/ueS9MKRUH9AFmeJ3lDIpATJusJaVUc+C/tNazALTWhVrrIx7HKq838KPWenuNSwZeA6Cp\nUqohEI35A+KlS4BVWuuTWusiYDlwW1UL1/UfXmUHw3hefGynlIrHbPF/5W2S0hbDWmAP8InW+huv\nMwGTgcex4I/GGTTwsVLqG6XUEK/DABcB+5VSs3xthulKqSZehyonGZjndQit9S7gJWAbsBM4rLVe\n4m0qNgC/VUqdrZSKxmyYtKlq4boWaucHwwgAlFLNgAXAw74ta09prYu11l2BOOBKpdSlXuZRSt0C\n7PW9+1BU/m/MC1dprS/H/A/1gFLqGo/zNAS6Aa9qrbsBx4GR3kY6TSkVCfQF3rEgSwzmnX5bTBuk\nmVJqgJeZtNbZwAvAEuAjTNu4sKrl61qodwAXnnE9Du/fUljL97ZrAfAPrfX7Xuc5k+9tcwZwk8dR\nrgb6KqW2YrbGeiml5nicCa31Ht/lPuBfVDdGITh2ANu11pm+6wswhdsWNwOrfc+X13oDW7XWB31t\nhveAqzzOhNZ6lta6u9Y6CdOq/b6qZetaqL8Bfq2Uauv7FPWPgBWf0mPX1liJN4BNWuspXgcBUErF\nKqVa+L5vgvkHXWHYVjBprUdprS/UWl+E+ff0qdb6Hi8zKaWife+EUEo1BW7AvHX1jNZ6L7BdKdXB\nd9N1wCYPI5XXHwvaHj7bgN8opaKUUgrzXHl+LIhS6lzf5YWY/nSVz5eTWR9V0pYeDKOU+ieQBJyj\nlNoGPF3yoYuHma4GBgLrfT1hDYzSWi/yMFZrYLZvlG0E8LbW+iMP89jqPOBfvjEJDYG5WuvFHmcC\neAiY62szbAXu9TgPUOaP/n1eZwHQWn+tlFoArAUKfJfTvU0FwLtKqZaYTH/VWudWtaAc8CKEEJaz\nZncjIYQQlZNCLYQQlpNCLYQQlpNCLYQQlpNCLYQQlpNCLYQQlpNCLYQQlpNCLYQQlvv/mmaH7nrq\nBksAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32e64c88>"
      ]
     },
     "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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9+nTp9U1J8WSx8NSsLB5MTfXuBB5VJ17dGDcUkuviK1eCh7cQPSStUTvoUDWy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\ni45/ywAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32e17a58>"
      ]
     },
     "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, subtracting 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": {},
   "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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r6Z8bNrDZ72d51670b9KE+a2uofeS8ex+7i249dYK35uiFBPatkVFOiNAJZrs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\n40ihFgRB8DhSqAVBEDyOFGpBEASP8//J8JL42Q1Z3wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32d3f0f0>"
      ]
     },
     "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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+f0LODsrvm6/8sRneW7Zs8R0hLcrvT8jZQfl985U/NsNbRCSfxGZ4NzQ0+I6Q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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32f73e80>"
      ]
     },
     "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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 $\\theta = \\dfrac{π}{2}$. In other words, if the dot product of two non-null vectors is zero, 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": {},
   "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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iXZ/3ZxCU019RyelFMIeyK1bQCfjPxYsD+fj6YtkyuO46V1jLXH45PPEEtG17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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32c344e0>"
      ]
     },
     "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": {},
   "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": {},
   "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": {},
   "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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 $\\lambda M$, 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",
    "A more concise way of writing this is:\n",
    "\n",
    "$(\\lambda M)_{i,j} = \\lambda (M)_{i,j}$\n",
    "\n",
    "In NumPy, simply use the `*` operator to multiply a matrix by a scalar. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "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 also defined on the right hand side, and gives the same result: $M \\lambda = \\lambda M$. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "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": [
    "This makes scalar multiplication **commutative**.\n",
    "\n",
    "It is also **associative**, meaning that $\\alpha (\\beta M) = (\\alpha \\times \\beta) M$, where $\\alpha$ and $\\beta$ are scalars. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "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": {},
   "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 $\\lambda (Q + R) = \\lambda Q + \\lambda R$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "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": {},
   "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 multiplication\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 $n \\times q$. It is noted simply $QR$ without multiplication sign or dot. 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",
    "$E = AD = \\begin{bmatrix}\n",
    "  10 & 20 & 30 \\\\\n",
    "  40 & 50 & 60\n",
    "\\end{bmatrix} \n",
    "\\begin{bmatrix}\n",
    "  2 & 3 & 5 & 7 \\\\\n",
    "  11 & 13 & 17 & 19 \\\\\n",
    "  23 & 29 & 31 & 37\n",
    "\\end{bmatrix} = \n",
    "\\begin{bmatrix}\n",
    "  930 & 1160 & 1320 & 1560 \\\\\n",
    "  2010 & 2510 & 2910 & 3450\n",
    "\\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "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": {},
   "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": {},
   "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": {},
   "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 $QR ≠ RQ$\n",
    "\n",
    "In fact, $QR$ and $RQ$ 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": {},
   "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": {},
   "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 $Q(RS) = (QR)S$. Let's create a $4 \\times 5$ matrix $G$ to illustrate this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "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)     # (AB)G"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "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(BG)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is also ***distributive* over addition** of matrices, meaning that $(Q + R)S = QS + RS$. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "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": {},
   "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 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",
    "$M I_n = I_m M = M$\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": {},
   "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": {},
   "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 matrix multiplication:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "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 matrix multiplication"
   ]
  },
  {
   "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": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python version: 3.5.3\n",
      "Numpy version: 1.12.1\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 an $m \\times n$ matrix, then $M^T$ is an $n \\times m$ 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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 5])"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32d8af60>"
      ]
     },
     "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ZuBDuvz8sQ7BsWbiQeuONYXmCaqGLogmnXmmWcpHVVS5mzAj3Da2FYp7vcVFf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\n4TjnLfdIPwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32d74d30>"
      ]
     },
     "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ttxWTlpZGenq6a/qjoqLC0eM7ue3xeCgrKwOI6LHHYU1bFJFTgJdbyKE/Bbyu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\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e330592b0>"
      ]
     },
     "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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J8OMfe7Zr8mR4/PGhY+Mkkmu5SCnp6OigodaMpbyRmus3kpKSicHg3a9Fp9NJW9tx1q4t\nICYmJvLqHQUQfy6KKoKMlJL77/8ONtujDHTmACv49NOPGfGLMSlJizC49VYtU6+yUkv0iCTq6zUZ\nqTuhYwAFBVqZghGSQSgr0x5joalJkwwKC4eOVVdrdi1aNLZzRyFOp5OmJjOnTpVx5kwLFnM8CTnz\nehc5vaW9vZm8vJRh5RjF6CiHHiJGmnls2bKV4uIK3O6HPYxOx+HQceHChdEvMnmyVu1uxQpNqqir\n6yusFUZ4nJF+9pk2w/bk0CsqtLFALQD3XHvx4qFjxcXaWIAceiTNzq1WK5WVdRw7Vk5pqROYTGpq\nHkmxcbiTff+/cTjMTJnSJ7eo2bnvKA09zOjq6uLBBx+ho+N3gKfmxAKdbgV79+5l5syZo59Qr9ec\nz4wZsG8fnD2r1R5PTva36f7l6FHtl0Z3EtoAdu7U1gyuHJLq4B+OHdNKIkydOnTsyBEtemP27MBc\nO8xxu920t7dTV9dKezvodCkkJWWi0/XNqnUOOy4fHbrdbiMhoYuUMJcHwx3l0EPEcPrgL3/5JO3t\ni4ChxYR6sFiu5IMP9vL1r3/d+wsajXDttTB/PhQVaTJMTk5YdLT3qBsfPw4LBiWWdHbCH/+oLfT+\n9KeBi+Q5dkw791NPDXzd7dYcemFhwEoshKuGbrfbaW5upbbWgtMZT1xcFqmpnuPMhdOBK8k3x9zW\nZmbu3PQBZXWVhu47yqGHEZWVlfznf/4Gq9VT7bP+rKCo6E9ju0huLtxxh+Yw9+/XHFdmZnjVgKmq\n0uqwd3TAn/+syURdXZpDXb4cvvMd784zFnmptlYL/Xz4Ye0LsD/nz2tfhmHocANB7yJnQytmsx2d\nLoWEhDySkkaeBEjpRiYk+XQtt7uJrKwZ4zFXgXLoIcPTzOPhh7+Pw/EQMJqUUkhNTSktLS2YTENK\n5oyOwaDpw7NmaSGOJSVaIaUk3z6E/mLIjLRHw77nHpgzZ/QT/OY3Wsu7HqTUFjaFgD17Br6emgqP\nPDL8uXquPfjXAWgz9wDq5xAeGrrT6aS1tY26ujasVj16fSopKclet28TQoc71vta5lZrByaTIGnQ\n+0/Nzn1HOfQwYffu3bz//h6czj96sbeBhISlfPLJJ1x//fVjv2hKClx3nVaKd9cubWackxP6uuhH\nj/qmU//rvw59bccO7V9fu9QcO6aFenpqUHzihCZdzYjOmaTVaqWpqY2Ghg7c7mQSEiaTmjq2OkHu\nOO+rEVosTSxapHrK+gMV5RIi+tepcLlc3HffP9PZ+TPAu1myxbKCXbs+Hr8hQsD06XDnnbBsmRYJ\n09g4/vP6wJD6JSdOaLLGeJKixhrNc+yY59m5lFr9+QDPoINdy8XtdtPa2srZs+WcPNlAQ0McSUnT\nSU3NJjZ2jEXfpETGeTdD70v1H5pMFOhaLtGImqGHAVarla4uKzEx9xEX930MhjyknEpXVx52uwAu\nAS4CpgJTAANu9wq2b/8VP/+5n4yIjYUlSzQZZs8eLYY7M1NrfhFMSkqgrQ0uuSS41wXtl0pLi2eH\nXlKiafphIIn4A18WOX3C7ULGxCC9jD+3WFrJyYnXahMpxo3KFA0jXC4XtbW1VFZWUlFRQWVlJY90\n673z519BbW0lra21xMVlotNlANW0tTV6rW16jZRQWgoffaQ1KsjJCXw/0fJyePVV+PxzbWFyzhwt\ntX+U9mDD8uGH2q8PbySXykr4y180p11To2WnzpkD3/xmn109Y7NmwcUXj92uEOJ5kTMFvd5/kU7C\n1gVS0rLuDq/2r6//nGXLUsnMzPSbDdGIt5miyqGHOUIIXnrpJdavXw9oC1Z1dXVUVFQQGxvLpZcO\nKYTpP2w2LUzv8GFtpj6GDjQhwxeHHuV4WuRMTPR+kdMXdJY2nKnptK8YfW1ncKq/YnhU6n+Y440+\naOmO3Oi/8KnX68nNzWX58uWBdeagNc5Yvhz+8R+1RJvyci180M8ERDe+9FLPmZ5hjj/vxbCZnEnG\ngDhzAOG04/YyBn20VH+lofuO0tDDmA8++ACAtLS00BqSng5f+pImO+zercWIB0OGGQ+hvmchwptM\nzkCic9hxGb0LpdVS/YcprqYYE0pyCWPuvvtuXnjhhZELcQWbri44dEiraWI0TljHGW4MXeQ0ER8f\n/EbGhoYq2pZ/EXvuyKGddrsNm+00q1cvCtivhWhCaehRQHJyMh0dHeHl0HtoaFB9TUNMMBY5fcXQ\nUE3L6ptxZo48825srGHuXCf5+R7q5SiGoDT0MMcbfbCjo4NVq1YF3pix4Me+plHTR9MPeHMvBper\nbWtLISUlH6MxI6TOHECCVzHoWqr/yMlESkP3nVE1dCFEHLALiO3e/29SyscH7RML/BlYAjQCd0op\ny/1v7sTj1ltvDbUJw6PTqb6mQcSfmZyBQiBHzRIdLtVfMX68klyEEIlSyk4hRAywF/i2lPLTfuPf\nAgqklA8JIe4EbpFS/qOH8yjJxUucTicGg4ELFy6Q76mEbDii+pr6Hc+LnMagLXL6hNuN3lxH0y0b\nRyz21tBQzqJFBqZMmRxE4yIbv/YUlVJ2dj+N6z5msFe+GfhJ9/O/AU97aadiGPZ3d+OZPn16iC3x\ngWjqaxpiApbJGUCE04E7MXlEZ96X6j8veIZNILz6pAkhdEKII0At8L6UcnB911ygAkBK6QJahBCq\n2s4IjKYPvvHGGwCRFwEQE6PVXF+/Xsu4rKzUUvlHQGnoGlJK3v30Uy5cqOLYsSqqq3XExeWRmjo5\nJBErviIc9lHroPuS6q80dN/xdobuBhYLIVKAzUKI+VLKk/12Gex1BENn8QBs2LChV0IwmUwUFhb2\nlsns+Q9U22t6HXr/Iv/hZN+o20lJFOl0MGkSaxwOqKykqKEB9PreErGDHXnP9uDxaN++at48Wlvb\n2PrxIY6WVzPlqnmkpCTz6efHgWoun63tv/+8tn/Ybp87iiN7Cj2VcA4eLAJg6dI1vdstLVV84xtf\n1P7+Ud5PxcXFI45H83ZRURGbNm0C8Ely9TlsUQjxY8AipXyy32vvAo9JKfd36+w1UspsD8cqDd1L\nhBDMmzePkydPjr5zuON0apUK9+3T5Jfs7PBqqBEihi5ypo69wmEYoG+soWPh5XTN8VxYTaX6jx2/\naehCiEzAIaVsFUIkAF8A/nPQbluBe4D9wFeAHb6brBhMWEe4+EKk9jUNAKHO5Awkwu3GnWgcdny0\nVH/F+PFGQ58M7BRCFKM57G1SyneEEI8LIW7s3udPQKYQ4hzwL8C/Bcbc6GEkfbDnV8wtt9wSJGuC\nRE9f0y9/WYtZr6wEh2NCaOh2u526ugaOHSvj/PlO7PYsUlOnYTSaBjjzHhkjEpFCjBiDrqX6e1/g\nTWnovjPqDF1KeQwYUgVKSvmTfs9tgHf1MhWjcubMGQAKCwtDbEmAGNzXtKVFK9kbZTLMWHtyRiqC\n4TsV2e02EhK6SEnxrXm0wjdUca4QMVK/xM2bNwNE90/Tfn1N14RBX1N/Mp6enD0LjJGJHLaXaFub\nmblz032K2lI9RX1HOfQwpCfCZUIQrn1Nx0AkZHIGDCmRCOQwi7paqn909mINJ1TGR4gYSR88cOAA\nWVlZwTMmxBR99FHI+5qOFX/35IxUDb03qchDEtlYU/2Vhu47kTcNmiBETYSLL4RLX1MviMRMzkAi\nHPZhI1wsliYWLVJ5hsFAlc8NQ4QQbNu2jXXr1oXalNARir6mo5oUfuVqw4WYNjO2SdPp6E4i6kFK\nSWPjUdaunacaQY8Dv9ZyUQSP2tpaAFavXh1iS0KMEFrc+pQpIe9rOp5FzomCcNhxGYdW2fQl1V8x\nfiaMhn7w4EHuuOMOTCYTer2eG2+8kR/84AcA7Nu3jzvuuIPU1FT0ej033HADjz76aEDtGU4ffOut\ntwCIm0ANI0bUSoPU19QToejJGbEausuF9CC5WK1NTJ06ti9hpaH7zoSZoS9dupTXXnuNuXPnsmLF\nil7HCbB8+XJee+01Zs+ezfLly3nnnXdCZueEinDxhSD1NY3mTM5AMzgG3el0ote3k5aWHxqDJiAT\nxqEDVFdXc+7cOR566KEhY+fPn6ekpISNGzcGxZbhYmzffffdCfdT3ut4Y50OZs+GvDy/9zUNl0XO\niI1DF2JIDPp4U/1VHLrvTCiH/sEHHyCE8LjYuH37doQQrF27NgSWDWRCRrj4Qnw8rFgBc+ZosesV\nFWPqazrRMjkDipS4B5X41VL9R+4tqvAvE0ZDB9i5cyeTJ09m3ryhxfW3b99OSkoKS5cuDYotI+mD\nE82hj1krHWNf03DuyRmRGrqUSAGy3wzdH6n+SkP3nQk1Q9+5cycJCQncd999SCl7pQ2Xy9UbJhhK\nucNisQBw/fXXh8yGiMOHvqYTOpMzkLicyLiEAUlFY0n1V4yfCePQS0pKKC8v59lnn+W+++4bMHbo\n0CFeeOEFrr766jGff+vWrbS1tVFZWUlNTQ1PPvkkuhFar3nSBz/44AMA0vygCUcSftFKExNh1SrN\nuffIMDk5uPX6iFrkjEQNXeew4Uoe+AXqj1R/paH7zoRx6B9++CFCCFatWjVkrKioaFz6eVtbG7fe\neitnz55lxowZFBYW8uKLL3L33Xf7dB4V4eIHuvua2o4epf297TQ0dtCZPIW4+ExSU8Mv4zQaEA4H\nrvSc3u2xpvorxs+E0dB37txJTk4Os2fPHjK2a9cuMjIyWLRo0ZjOnZKSwsGDB5kxQ5uRuN1urFbr\niMd40gcnqkP3l1YqpaSlpYXPjn/OzroYPltwG+4ZK8jqcpHocvrlGoEmEjV04bTjSjb1blssTUyf\nPv5Uf6Wh+86EmaHv3LnT4084KSW7d+/m2muvHTJ2+PBhDh065FUo4yWXaG23SktLcTqdfPWrX/XZ\nxo6ODpUhOgYcDgcNDY2UlDTS3h5LXFwWmZlpCCHoyp2N86JFGA99hKGuEkdGDqhUff/idOJO0pKK\ntNIezWRmDg08UASeCTFDP3HiBHV1daxcuXLIWHFxMS0tLQP086amJm677TZqa2txuVzce++97N+/\nf9TrvPXWWzz66KP8z//8D8mjtFcbTh+Mui5FXjBWrdRisXDmzAV27jzBZ5/ZEWI22dlzSU0duBjn\nzJxM8zW3Yylcgb65Eb25TqsVE4ZEooYuRF9SkT9T/ZWG7jtRXZzr9OnTPPbYYxw5coTz589z6aWX\nctlll/HMM89w4sQJnnjiid6xJUuWcMUVV/DUU08BWkTEvffeS2lpKa+//jq5ubleXdPpdHLJJZfw\n61//2qdoFafTicFg4MKFCz51+Z5ouN1umpqauHChgaYmiV6fRWpqhtfJK7qOdpKO7SOu/CyulHSt\n5KtiXBjqq2j+wu24TJnU13/OsmWpZGZmhtqsqMLb4lxR7dDHSkNDAxs3buShhx6itLSUffv2sXHj\nRq644gqP+7/33ns88sgjnDp1CoD169fT3t7O1q1bh71GUVHRgBnInj17WLlyJW63e8KFeg2+F57o\n6uqitraBkhIzNlsyiYlZJCePPcbZUF9F8qEiYixtYSXD7D9/LOJm6fqGKppv+Dp2QxxtbcdZu7bA\nL922vHlfTBS8deijSi5CiDwhxA4hxEkhxDEhxLc97LNaCNEihDjc/fiPsRoeDmRlZbF582aysrLQ\n6XQ899xzA5z5G2+8QWVlZe+2TqcbkH1aVlbmcz/Qv//97wATzpmPRO8i52dnKSo6y9mzMSQmziM7\ne9a4nDmAIzuX5mvvwLJoOYbmevQtDWErw4Q7Qmqt58ab6q8YP6PO0IUQk4BJUspiIUQycAi4WUp5\nut8+q4F/lVLeNMq5ImKGPhorVqzgpptu6q3WCPD73/8ep9NJdXU1zc3NPPXUUxgM3s/6ZsyYQWlp\nKdFwf8aLp0XOlJS0gH3Z6SytJB/9mLjKEpypGbgTVLid1zgdxHS0Yf7SBurrz3DllZNI9ZDUpRgf\nfquHLqWsBWq7n1uEEKeAXOD0oF0nzNRy7969vPzyywNee/DBB8d1ztLSUo8lCSYSFouFqqoGyspa\ncbnSSEmZTXZ24AtkuZNTabviOgy15RgP78JQX6XJMDETJghszAinA1ei0S+p/orx41OUixAiHygE\nPIV8LBealPCIAAAgAElEQVRCHBFCvC2EmO8H28KW48ePM2fOnHGdw1OM7USr4QLaIufmzZv59NOT\n7N5dRnl5EiZTAdnZ04mPD2K1QyFwTJ5O87o76VywDENTHTEtwe9rGmlx6DqHTftCbDOTn+/fVH8V\nh+47Xk9BuuWWvwHfkVJaBg0fAqZLKTuFENcDmwGPHm/Dhg29URwmk4nCwsLehY+e/8Bw305ISODy\nyy/32/l6Ys+nTp06YCEoXP7eQGx3dXWxefMWqqvbcDgSSEvLo7z8MNDA0u42ZgcPavuHYtuWN4vj\nf/sf9GeLWbboSmR8Yq+z7Vm0DMT2yaqSgJ7f39sxbc1cMv1i3O4mTp2qoLT0c7+9X4qLi8d1fCRv\nFxUVsWnTJgCfot68inIRQuiBt4B3pZRPebH/BWCJlNI86PWo0ND9zalTp5g/fz5OpzOqF5SklLS2\ntlJWVk9NTRc6XSYpKZkYDGHankxKYqtLST78ETq7DUd66PuahhuG+koaCpbjmBHLFVcsCLU5UYu/\ne4o+B5wczpkLIXKklHXdzy9D+6Iwe9pXMZTNmzcDRK0zHymTM6wRAnvuDJqzppBw5giJpw/jjk/E\nlRr8vqbhi8Di7GLOdFX3PBzwJmxxBfBVYG23Rn5YCHGdEOJBIcQD3bvdLoQ4LoQ4AvwXcGcAbY4K\n+uuD0VrDxdtMzh6pI1yRsXF0Fiyned0/4jKmYagtR9gC09c00jR0Cbhiu8jM9P+XnNLQfcebKJe9\nwIhTRynlM8Az/jJqonHw4EGysrJCbYZf8JTJaTJNi4pfH67UdFpXfYnYqhKMR3ZDmxlnRg6EaSne\nYGCzWcnMS/FLqr9i/KhM0TBACMGDDz7I7373u1CbMmb8nckZ7ghbFwmnDpF4thh3khGXcWLVsO+h\n6/MjTPr3jWROnhxqU6Iaf2voigATiSGLnhc555GaGv2zNRkXT2fhCuzT55B8eBexdRU40rKRsROn\nA5LLbkfEuUnLzg61KYpuJkS1xXCkRx+sra0FiKiyuQ6Hg+rqGvbuPc4nn9TR1JRJZmYBGRlTxhSx\nEu4a+kg407JoufoW2pZdQ4ylFX2Td31NhyOSNHRbu5nUqZkBk9OUhu47aoYeYt566y0A4nzsWB8K\nQpXJGfbodNjy52KfNJWkkweJP38MV3Iq7uToToGXtmbSpl8UajMU/VAaeoi54YYbePfdd8O2hst4\ny9VORPRNdSQf3oW+pQFneg4yXOPsx4HT6YCGo8y95YuIYaqQKvyHKp8bIQghEELgHsfP9EAw0RY5\n/Y7LRVzZGZI/2wtC4DRlwQhNwyONtjYz0/TVZN5yM8yP6kofYYHfyucqAkN/fTBcFkQDWa52JCJZ\nQx+WmBhsM+fTfN16bLkzia2vRNfRNuphkaKhS9mOMTkRAtgIWmnovqM09DAg1A49YjM5IwB3QhKW\nZWvpmjEvavqa2u1dJCUJ4uLjIT4+1OYo+qEklxBisVgwGo2YzWbS0oIfxzx0kTM7uBUOJxpOJ/EX\nTpJ0dB/E6HCmZWsNOSOM1tYG8vNjyLB1wV13gSqZG3BUHHoE8MEHHwAE1ZlHcyZn2KPX03XRIuxT\nZkRsX1NtQmYhJSUP6q1qhh5mKA09RBQVFQW1hktXVxelpRUUFR3j4ME2rNY8srMXkJ6eHXJnHpUa\n+gi4k4y0L7+W1jVfBunGUFcJTgcQ/hp6V1cHJpMBg06nVZ4MYMq/0tB9R83QQ0igHfpEzuSMBHr6\nmsZ/fpzk4/u18MYwlyTt9namT08BhwNUq7mwQ2noIUQIwapVq/joo4/8et5g9+RUjJ9I6Gvqcrno\n7Cxj0aLpxHR0QHo6XH99qM2aECgNPULwZ4SLyuSMXCKhr6nVaiEzM1GT6Ox2tRgahigNPUR8+OGH\nANx8883jOo/b7aahoSH0PTnHwUTT0IdFCD6puhDyvqbD4XK1k55u1DbsdjCZAno9paH7Tvh8/U8w\nTp48CcD06dPHdPzQTM48srPVjCkakIZYOuctwZY3i6TiPcTWlOE0ZSLjE0Nmk9PpIDbWQWJitw1S\nQmLo7FF4Rjn0EHHhwgUAn3TtaF3k7GnSrBh4L1xGE21X/UNfX9P2lpD1Ne3sbCc3N7nv/SoEJAT2\nF2BP82SF9yiHHiJ8iXBRmZwTmDDpayplO6mpOQNfVDHoYYfS0ENEWVkZ8+bNG3Efb3tyRjpKQ+9j\nuHsRzL6mg+lJ9Y/v78ClDPgMXWnovjPqDF0IkQf8GZgEuIA/Sil/62G/3wLXAx3ABillsZ9tjTo8\nRbioTE7FSISir6nV2k5+fr9sVrdbqxyp+oiGHaPGoQshJgGTpJTFQohk4BBws5TydL99rgcellL+\ngxDicuApKeVyD+dScehoWrhOp+PgwYMsWbIEUOVqFb4TjL6mUkra2kpZtCgPg6G7oJjNpkW5rF/v\n9+spPOO3OHQpZS1Q2/3cIoQ4BeQCp/vtdjPaLB4p5X4hRKoQIkdKWTcm66Oc06e1W3fJJZfQ0tIS\ndYuciuAQjL6mvan+hn7VIVUMetjik4YuhMgHCoH9g4ZygYp+21Xdryk8sHnzZgD27Tvll56ckY7S\n0PsYy73wd1/T/tjt7WRlpQx+MShp/0pD9x2vo1y65Za/Ad+RUloGD3s4xKO2smHDBvLz8wEwmUwU\nFhb2hif1/AdG+/btt9/O0aOn2b17PzExqaxYcSNCiN4Pc0/o2kTZ7mG853v11cN8+GEeZ85k43DA\nokV1xMRIUlMnUVcHDoeZm266wO23Lwmrv7//9pkzxWM7Xqdjb2MNIjuX1fGJxJ8/xt6GamRCEpfP\nLgD6Cn95u/3x2c+wddVSWPgFAIqOaeNrMjMhJSXgn5fi4uKAnj+ct4uKiti0aRNAr7/0Bq9quQgh\n9MBbwLtSyqc8jP8O2CmlfLV7+zSwerDkojT0gXR2dtLYaKa8vJn29hh0ujRSUtKJ9eNP5onIzTfD\nxRfDL34x8PX//m949VV45RXIywuNbcHCH31NLZZWMjKsTJ06aeBAVRWsWwczZ/rJWsVo+LsF3XPA\nSU/OvJstwN3dF14OtCj9fHQSExOZNi2Pq64qYOXKaVx0kQOb7TT19acxm+u1RrwKn6ipgepquPTS\noWNLl2rrebt2Bd+uYOPMyKFl7a20L1lDTJsZvbnOZxlmQKp/f4RQMehhyqgOXQixAvgqsFYIcUQI\ncVgIcZ0Q4kEhxAMAUsp3gAtCiPPA74GHAmp1FDBYH0xOTmbmzGmsXr2IFSumkJ/ficVygvr6s7S0\nNOJ0OkNjaBDwp4b+6aeav+kOHhrAhQvaWAiaQ3mNX9cTxtjXFDyk+g8mwDHooDT0seBNlMteYNQg\nVynlw36xaIIjhCAlJYWUlBRmzXLT1tZGTY2ZyspKnM5k4uLSMRpN6KKog7w/OXgQjEaYPXvo2Ntv\nQ24urF0bfLtCyVj6mg5J9e+PlGqGHqaoeugRgsvlorW1lepqM9XVFlyuFBIS0klOTo2qrNHxcsMN\nMH8+/PrXfa9ZLPCb38D58/CrX8GkScMfH/V42de0tbWMBQtyBmaHgubMa2pg40YtuUgRFLzV0JVD\nj0CcTifNzc1UVTVTW9uJlCYSE9NJSjJOaOdeVga3367JLQUFmu/p7NSk4yuugNWrQ21h+KDraB+2\nr6nd3oVOV8/FF08beqDdDl1d8NWvBtFahWpwEeYUFRWNuZqcXq8nKyuLrKwsHA4HZrOZiooqGhrs\nQBrJyekkRlDj4YMHi/xScfHAAW2y+fDDsHDh+O0KBf66F6PR09e0a+Z8kg8VDZBhhqT698du1zSt\nIDCez8hERTn0CMdgMJCTk0NOTg42m42mJjPl5WXU17vR6dJJTk4jPoR1tIPJgQOQlKRJLgrvGNzX\n1K2PBdFOSspUzwfY7TB5cnCNVHiNklyiFKvVSmOjmbIyM+3tOnS69KiPcb/2Wigs1HRyhe/oLK3E\n7HsfU/NRphXO174dB1Nbq8V/Ll4cfAMnMEpymeAkJCQwdWouU6fm0tHRQUODmbKyM7S0GNDr0zEa\n06KqzMCZM9DSAsuWhdqSyMWdnEptQSF5k2fB8eNaAlFODuj7uQmXK2iSi8J31DJ1iAhmjG1SUhL5\n+VNZtaqAFStymTGjC6v1FPX1Z2hubgh5jPt4Yq8//xx++EPtIQS8997ACJdII5R1bZxOJ3qDhdRF\ni+DOO7Vvx7o6aBzU1zQIMeig4tDHgpqhTyAGxrhPo7W1lbq6ZioqqnA4koiN1WLcI6n2+qxZ8POf\nh9qK6KC9vZm8vBTt/z8mRgsXmjUL9uzRQogyM1WWaJijNHQFbreblpYWamrMVFVZcLmMxMdrMe4q\ngWniUF9/hiuvnETq4EqKUkJpKXz0kRbUf/fdqnxukFFx6Iox4XQ6aWlpoarKTG1tJ263icTENJKS\nUiZ0jHu0Y7fbsNlOs3r1ouH/n7u64OWXIS4Orr4apkwJrpETGH8X51L4mXDVB/V6PZmZmVxyyRzW\nrl3AkiWJJCfX0Nh4lIaGcjo7B1dOHj+qHnofoboXbW1m8vNH6VUrhFbdTAj4+99hxw7o6AiYTeH6\nGQlnlIauGBaDwUB2djbZ2dnYbDbM5mbKy8upr3d2x7inT5gY92jH7W4iK2vGyDtZrVq6v9GohTSW\nlGir0ldeqdUrjqC1l2hFSS4Kn7FarTQ1NVNWZqatDWJiNOceF6cWyyIRq7WDmJhSrrhiwcg71tbC\n5s1ahbMe7HYtEiYzU6utkJMTWGMnKEpDVwSFjo4OGhs1597RoScmRktgiqYY92inoaGcRYsMTJky\nSgZoaakWF5rrobtka6v2WLhQC3ccruyuYkwoDT3MiRZ9MCkpienT81i5soCrrprKrFm27hh375t0\nKA29j2DfC22C1UxmZsboO3d0eKzMCGg9RvPytAyvl1+G06fH3dc0Wj4jwURp6Aq/IITAaDRiNBqZ\nOXMabW1t1Naaqaysxm5PIjY2DaMxLaJi3CcCFksrOTnxxMZ68YuqtRVG2k+n0+q82Gzw4Ydw4gSs\nWgVZWf4zWDEiSnJRBBS3201ra2t3jHs7TqeKcQ8n6us/Z9myVDIzM0ff+d13wWz2Pga9uRna27UC\nO0uWqISkcaA0dEXY4XK5+sW4d+BypXbXcVcx7qHA6XTS1nactWsLvPvl9NprvmeKulxQX6/Vg1m5\nUss8VV/kPqM09DBnIuqDMTExZGRksGjRRaxdu5ClS5MxGmvZseNP1NeX0dHRzkT/wg+mhj4g1d8b\nWlvBMHzbOo/ExGgyjNEI27fD1q3Q1OTVoRPxMzJevGkS/SchRJ0Q4ugw46uFEC3dzaMPCyH+w/9m\nKqKNniYdl146l8WLp7N4cTwJCZU0NByjoaECqzVwCSsKDYfDzJQpXiyGghae6HKNPdY8Ph6mTdNK\nYr72Guzbp2ntCr8yquQihLgKsAB/llIu8jC+GvhXKeVNo15MSS6KUejq6upu0tFMS4tEp0vrTmAK\nToW/iYJXqf79aWvTolc8hSz6isulxa7HxWmLpjNmDB89owD8WA9dSrlHCDF9tOt5bZlCMQLx8fHk\n5k4hN3cKnZ2dNDaaKS8/T319DDpdWtQ36QgWbW1m5s4dJdW/P1ar/5xuTIxWB6azU1tonT4dVqyA\ntDT/nH8C4y8NfbkQ4ogQ4m0hhGoA5gVKH+xjuHuRmJjItGl5XHVVAStXTuOiixzYbKd9inGPNIKl\noWup/uneH9DVpVVd9CeJiZoM09gIr76q9RC023uH1WfEd/wRh34ImC6l7BRCXA9sBuYMt/OGDRvI\nz88HwGQyUVhY2NsItuc/UG1PrO0eRto/OTmZgwcPIqVkxYol1NWZ2bp1Cw5HHEuXXk9ysoni4j0A\nvU2We5xjJG2fOVMc8OstWLAMk0lw4MABwMv/r44Ois6eBbOZNQUF2vixY9q4P7adTor+8hd44w3W\nfPObMG0axcXF3tsXZdtFRUVs2rQJoNdfeoNXYYvdkstWTxq6h30vAEuklGYPY0pDV/gNt9tNW1sb\nNTVmKivbcDqTiYvTmnSoGPfh8TrVvz/79sHJk4FPEuro0KJgZs7Uin4Nrs0+QfF3T1HBMDq5ECJH\nSlnX/fwytC+JIc5cofA3Op0Ok8mEyWRizhwXra2tVFebqa4ux+VKISFBS2BSMe599KX6z/PtwNGy\nRP1FUpImxdTUwF/+ApdfrtWH8TVccoLiTdjiy8DHwBwhRLkQ4l4hxINCiAe6d7ldCHFcCHEE+C/g\nzgDaGzUofbAPf9yLmJgY0tPTWbhwdneMuxGTqZ7Gxs+ory/FYmmLiBj3QGvoPqX696etLXhOVQjI\nyqKovl77ZfDaa1BREZxrRzjeRLmsH2X8GeAZv1mkUIyTnhj3rKwsHA4HZrOZiooqGhrsgBYGmZiY\nHGozQ4LV2sTChV7GnvenvT34USh6vVbwy2KBLVtgzhxYvlxLUlJ4RKX+KyYMNputO8bdTHOzu7tJ\nR9qEadLhc6p/34Hw7LP+iUEfK1JCQ4MWw758Ocyfrzn8CYKq5aJQjIDVaqWx0UxZmZn2dh06XXrU\nx7g3NzeQm9vOvHkzfTuwvV1LKgqHHqIOh9ZoIy1Na6gRDjYFAVXLJcxRGnofobgXCQkJTJ2ay1VX\nFbBqVT5z5zqx289QX38Ks7kOh8M++kkCQCA1dJ9S/fvT1eV/Y7ygJ6xxAAYDTJ2qzdiD0Nc00pg4\nv1kUimFISkrqbdTR3t5OfX0zFRWnaG6Ox2BIx2hMQx/hP+/tdhsJCV2keFv6tj9Wq/+TisZLT1/T\nzz9XfU37oSQXhcIDUkpaW1upq2umoqIVhyOJ2Fgtxj0Sm3Q0NtYwd66T/Pypvh985ow2E87L879h\n/mAC9DX1dxy6QjGhEEL0xrhfdJGblpaW7iYdFbhckdekQ0v1nzG2g4MVgz5WYmM1Gaa1Ff72Nygo\ngKVLJ2Rf08h4N0YhSkPvI9zvhU6nIz09nQULtBj3ZctSSU9vwGw+2h3j3uq3GPdAaOhWawcmkyAp\nKWlsJwhmDHo/PGroI9HT1/T0ab/1NY001AxdofABvV5PZmYmmZmZzJ/voLm5mYqKGhoaSoE0kpLC\nL8bdYmli0SIfCnENpq1NK3UbCUzwvqZKQ1co/IDNZsNsbqa83IzZ7OyOcU8PeYy7lJLGxqOsXTvP\n9+zQHjZt0vqIRmL6fZT0NVVx6ApFiLBarTQ1NVNWZqatDWJiNOceFxd8Z9Le3kJqah2LF88d2wlc\nLvjjH0ObVDReoqCvqYpDD3PCXTcOJtF2LxISEsjLm8KKFQtZtWoGc+e6cTrPUl9/kqam2hFj3P2t\noVutTUydOobY8x5CFIMOY9DQh2McfU0jDaWhKxQBpCfGfdq0XCwWC/X1ZsrLT9HcHIden05KShp6\nfWCkDKfTiV7fTlpa/thP4s9ORaGmp6+p2awV/Fq8WHtEyvqAFyjJRaEIMlJK2traqK01U1nZit2e\nRGxsGkZjml9j3Mec6t+fykp4663Illw8EWF9TVUcukIRpgghSE1NJTU1lYsuctPa2tod416J0+m/\nGHct1X/S+Iy1Wsd3fLgSpX1NlYYeIqJNNx4PE/le6HQ60tLSmD9/FmvXFmC3Hycjo5Hm5qPU118Y\nc4z7uFL9+9PeHrJ0er9p6CMxSl/TSEPN0BWKMCEmJobU1FQWLbqI+fOd3THutdTXlyKlqTfG3ZsO\nTG1tZubOTR9/t6bW1qjSmIclI0MrE3zoEJw6pZUQmDYtrGUYTygNXaEIc+x2O2ZzMxUVZhobHQih\nNelISBg+87O+/jirVs0Ye3ZoD5s3a7JLcnglSwWUMOxrqsIWFYooITY2lkmTcli2bB5XXz2HgoIY\nYmJKqa8/TmNjFV1dA3Xucaf69yeSskT9RWIiGI3Yjhxh2WWr+eyzz0Jtkdcohx4iJrJuPBh1L/oY\n7V7Ex8eTmzuFK65YwOrVM5k3TyLleerrT9LYWIPdbsNiaWL69HGk+vfgdmuLhiEqHRwUDb0/TqfW\nPKOiAkwmjl9yCYfOnWT9+o24XK7g2jJGvGkS/SchRJ0Q4ugI+/xWCHFOCFEshCj0r4kKhcITiYmJ\nTJuWx1VXFbBy5TQuushBV9cp4uIacDqdOByO8V2gJ6kownRkn7FYtPDMxkZYuBDWr4ebbqK4tpaE\nhDsoK0viqaeeDrWVXjGqhi6EuAqwAH+WUi7yMH498LCU8h+EEJcDT0kplw9zLqWhKxQBpLm5mR07\nSkhMzECIFiZNSiQvLx2TyeR7kw6zGf761+hs8+Z2a3+f1Qrp6XDppVroYj956f77H+ZPf5oJ3Ehi\n4pWcOnWYadOmhcRcv8WhSyn3CCGmj7DLzcCfu/fdL4RIFULkSCnrvDdXoVD4g5oaMybTdEymTNxu\nNy0tbdTUmImJqWTKlGSmTNGcu1cx7tEYg26zaTNxKWHOHG1Gnp3t8VfI/v3FwK3AHGy2R7jnnofY\nsWPr+COHAog/NPRcoKLfdlX3a4oRULpxH+pe9DGee+F0OqmsbMdo1JJjdDodRqOJ7OyZmEwF1Nen\ns3+/mR07jnLqVAktLS0jx7h3dYW0nrjfNHQpoaVF08Y7OmD5crj7brjmGq27kQcH7Xa7OXfuKKAp\nyC7X9zhwoIxXX33NPzYFCH+sdnj6uhr2XbJhwwby8/MBMJlMFBYWsmbNGqDvzay2J9Z2D+FiTyi3\ni4uLx3z8W2+9xZkznXzhC5oT6in0tXTpGmJiYrodFBQWXkVVVTNbtmzFYLDxpS99gZycdA4dOoQQ\nou/8O3bA55+zprv1XI+DXVNQEJTt4pKS8Z2vuBhaW1kzYwZMnUpRbCxkZrKmsHDU+/n5558jRAJw\nFFgDxNLR8S3uv/8hvvjFdaSlpQX0/VBUVMSmTZsAev2lN3gVh94tuWwdRkP/HbBTSvlq9/ZpYLUn\nyUVp6ApF4Dh8+Azt7ZNITvY+btrpdNDaasblMpOQYGfatDSys9NJTk6Gjz6CCxe0pJtIwmLRZuR6\nvSapXHyxzyn9f/3rX/nGN16ivX3zgNfj4v6Z226z8tJLz/rT4lHxdy0XgeeZOMAW4J+AV4UQy4EW\npZ8rFMHFZrNRV9dFVpZvqf56vYGMjBwgB7vdxrlzZs6cKcNodDP3fAkpMYKIiEIfvMh5zTVDFjl9\n4cCBI1gsQwP2bLb/l82bF1JUVNQ7sw4nvAlbfBn4GJgjhCgXQtwrhHhQCPEAgJTyHeCCEOI88Hvg\noYBaHCWMRyuNNtS96GOs96Kx0YwQ40v1j42NIzNzMtnZCxBiNjVnLZw418Tp0+U0NZnHHwbpI15p\n6DYbVFVBdbWWqn/bbXDnndqC5zgSovbuLUbKxR5GUujsfJqvfe1BukJYK344vIlyWe/FPg/7xxyF\nQjEWSkubSE6e4bfzxcfFkxITizN9GnanndLSdqASY4qerEwjyclJGELVkk5KrcZMezskJWmLnBdd\npD33EydOHKFnQXQoN2E2v8CPf/xTfvnLn/rtmv5A1XJRKCKcjo4Odu0qJTt7gd/OKWxdZLy1CUdW\nX8CalBKbzYrNZkGIDlJTDWRmGklOTvZrHfdhcTq1kEOHA/LytD6hubl+rwZZW1tLfv58bLYmhlea\na0hIuIRPP93BwoUL/Xp9T6h66ArFBKGurgmdzg+p/v3Q2YbGoAshiI9PJD4+ESklnZ0dnD/fgU7X\nhMkUT0ZGMklJSf537v0XOQsKYO7cgNYtLy4uJi6uEJttJP85ma6un3LXXRspLt4TnC80L1C1XEKE\n0o37UPeiD1/vhZSSsrJmUlP9G4ki7F3IYWenmnNPSEgmNTWHpOR82tqMnDtn4ejRMsrLa7BYLLjH\nE8PudlO0Z48WOy6Etsh5992avBLgJhSHDxfT2elJPx+IlPdz4YKBp5/+n4Da4wtqhq5QRDCtra10\ndcWTkhLr1/PquqwIL+VRndCRlGQEjLhcLpqbO2hoaEWvryczMwmTKZnExETvFmz7Z3JmZ2uLnMNk\ncgaKPXuO4HTe6MWeOjo6/sC///tKbr31ZqZOnRpw20ZDaegKRQRz8uTn1NSkYjJl+vW88eeOknz0\nExxZY6/j4nI56ey04HZbMBgcZGcnk5qaTEJCwsAdBy9yXnKJ3xc5fWHKlLnU1LwOjKSNS6ABOI9O\n92Ouvz6Tt956JWA2KQ1doYhyelL9TaZ8v587xtKK2zC+CPSYGD1Gowkw4XQ6qKmxUFXVQFyci+xs\nI6lJCcS1t/ctcq5eHZBFTl+wWCw0NFQAcwE3WiWT88DnCPEXEhLc6PUtWK3nMRhiycubxZw5s1m/\n/paQ2dwf5dBDRLgmJoQCdS/68OVeNDc343KlBGRBLqajDenHsES93tBdYyYNZ6uZxuJq6oQN94KL\nyFy1mIzZM4mPjx9wTCjeF2fPnkVKN0bjJVitF0hOTmfq1FnMmzeb117bQWcnHDhwgFmzZpEWhg2l\nlUNXKCKUigoziYmTAnLuGEsrcpwz9AG43cS0mdF1WdGlptN59W04Jk+n0+WktqoZd8VZ0tP1TJuW\nTkZGOrGx/l0T8JbFixfz3ntvMWnSJGbOnEliYmLvWE3NeXbv3s3SpUtDYps3KA1doYhAbDYbO3ac\nJitrkf/LuUpJxt//iDM9B3TjC4QTdhsxLY0IJF1T59A1eyHO9KGLnFoYpIWODjNCtJCVFcfUqemk\npaWFLoFpEC+99BJf+9rXRq5QGSCUhq5QRDH+SPUfDuGwI9yusTtzKYmxtKLrbMcdn0THouXYp16E\ne4Sm1kIIkpKMJCUZkXIaFksbhw6Z0emqmTw5iSlT0khLSwtpvPett94KQGVlJXndFSjDDRWHHiJU\n7HUf6l704e290FL9/ZtM1IPOZkWKMbgGlxN9Uy2G+kqcxlRaV96I+Yav0TWncERnPhghBMnJqZSX\nl5GevoimpkwOHGjlww+PcfLk5zQ3N48vxn2M9ETnPP/880G/trcoh65QDMPBgwe54447etu33Xjj\njeVAN7sAABEjSURBVPzgBz8AYN++fdxxxx2kpqai1+u54YYbePTRR4NiV0dHB21tggQfnKQvCFuX\n1zHoADqrBUN9JfqWRqyzC2j+4l20rboJx6Rp445Y0el0pKSkkZ09C5OpgJoaE/v2NbJjx1HOnLlA\na2tr0CWQ3//+90G9ni8oDV2hGIW5c+cye/Zs3n777SFjs2fPZtasWWzbti1o9pSUlHPunIHMzMkB\nOX9sdSkpn7w3oI7LEPotcrpS0+m8+FLsk6cjY4NTbNfpdNLe3ozDYSY+voupU03k5Gh13APZIm7V\nqlXs3r076F8iSkNXKPxAdXU1586d46GHhlaFPn/+PCUlJWzcuDFo9vSl+s8L2DV01g6GK0rl7SJn\noNHr9aSlZQFZOBx2SkqaOXeuksREB9Onp5GVlU5SABKTHnzwQXbv3u338/oLJbmECKUb9xHO9+KD\nDz5ACMG6deuGjG3fvh0hBGvXrvXb9Ua7Fz2p/gZD4ML6tKSifueXkpj2Fgx1FeisHXQsWo75H+7G\ncvk1ODM89+T0Bz0t9EbDYIglIyOH7Ox5GAxzOHMmhl27Stm79zgVFVVY/djsuv/CaDiiZugKxQjs\n3LmTyZMnM2/e0Bnx9u3bSUlJCWpccnV1E7GxgW0Jp7O0IvWx2iJnSyPC6cCek4d1yWpNhgmTyoKe\niIuLJy5uCjCFrq5OTpwwI+V50tNjmDo1jYyMdOLG0fii/8Loj370Iz9Z7T+Uhq5QjEB+fj4Gg4GV\nK1cipezVZ10uF6+99hrr1q3jzTffDIotTqeTHTuOYzIVBCx8T0qJ4a0/k9bRijsuEevshdimz8WV\nEn5Zkb7QE+MuZTPZ2XHk5aWTnj62GHchBLm5uUGdpSsNXaEYJyUlJZSXl/Pss89y3333DRg7dOgQ\nL7zwAldfffW4r/HTn/6U5557btR9A5Xq73Q6OHToI95//0127txCbloGr/7mlaAucgaaxMRkEhOT\nkXIqFks7hw+bEaKaSZMSyctL741k8oZVq1axa9euAFs8NrzS0IUQ1wkhTgshzgohfuBh/B4hRL0Q\n4nD34z5P51H0Ec66cbAJ13vx4YcfIoRg1apVQ8aKiorGrZ8//fTTPPHEE5SWlg4473Boqf7+kVva\n21vYtu0VHnnkLlavzub73/8P3nxTR1tbLQ8+8jNs0+eE3Jl7q6H7ghbjnkJ2dj4ZGYtoacnmwIE2\nduw4zvHj5zGbzaPGuD/wwAN+t8tfjPqVJITQAU8D1wDVwAEhxJtSytODdn1FSvntANioUISEnTt3\nkpOTw+zZs4eM7dq1i4yMDBYtWjTm8z/88MN89NFHPP7446Pua7PZqKvrIisrZczXq6kpo6hoC++9\nt4WzZ/djMKyis/Nm4Elstkzi47/A+vWPsmLFdWO+RiSh0+kwGk0YjSZcLhf19a1UVJgxGMrJzU1h\n8uR0UlNTh4RBhnPGqDe/MS4DzkkpywCEEK8ANwODHXpw45YiHFVdsI9wvRc7d+70aJuUkt27d3Pt\ntdcOGTt8+DCHDh0acyjjcPdiLKn+UkpOnz7Mjh1b2LbtTRobqxDiRmy2h4DNOBx9YX16/f/DvHmJ\nfPOb4bPQt3TpmqBdKyYmhtTUdCAdp9NJVVUzpaX1xMWV9sa4G43G7k5N4bsw6o3kkgtU9Nuu7H5t\nMLcKIYqFEK8JIcLra0uh8JETJ05QV1fHypUrh4wVFxfT0tIyQD9vamritttuo7a2FpfLxb333sv+\n/fv9Zo+3qf52u41PPtnG//k//8Q110zjgQfu4oUXOqmu/m/s9lpstueBW4D+Mdp/w2h8nV/96kV0\n4yzGFQ30xLhnZ88hKWkBFy4ksHdvFR99dJSSknIsFgsAf/jDH0Js6VC8maF7mhIMDlXZArwspXQI\nIR4E/hdNolEMg6oB3kc43YvTp0/z2GOPceTIEYQQPP/885w4cYJnnnmGEydO8MQTT/SOPffcc5w6\ndYqnnnqKjIwMXnzxRe69915KS0t5/fXXyc0dIdNyGDzdi55U/+xsz4ky7e0t7NnzNu+++yYHD27H\nYJhPZ+fNSLkduJiRfzyfIS7uW/z2t+9hMgU2HNJXDh4sCuos3RN6vYGMjBwgh64uK8eP1+B2n2H5\n8qvYt29PSG3zhDcOvRKY1m87D01L70VK2dxv84/AL4Y72YYNG8jPzwfAZDJRWFjY+wbuWRBS2xNr\nu4dwseeVV17xON7Q0MA3v/lNj/Y3NDRwyy23cPPNN7N27Vp+9KMfsWTJEhYsWDDi9YqLiwf8/cXF\nxUP2nzZtJjpdeu8iYY+T69netu0Ntmz5Ky7X14DnsNtv7TkjUAes6bdNv+13MRi+xXe/+3PmzVsy\n7PlDtX3mTHFIrl9YuAKbrYsDBz7E5bKzaNFioIvjx/eRmGjg6qtX8cILmygq2jngC9if78eioiI2\nbdoE0OsvvWHUOHQhRAxwBm3GXQN8CtwlpTzVb59JUsra7ue3AN+TUl7p4VwqDl0R1Rw5coRDhw5x\n//33D3j9jTfe4LLLLhuyiFZUVPR/27v/2KjrM4Dj70dIe+1KSxRqlyE6MtgmSmggoEPtNmftOtMS\nM5x2CQQUBmhg/mFcXNxspmR0gfFjJjNd54DYsUwzmQYtCC3EroXyS+mKgziKY0XaAGVQvf589sdd\n72o5endw/X6vd88ruXC9fu57Tz9cn37u+Xw/3w+lpaVUV1eHPJ6qUlPzIWlp37zq6tDW1v8yf34e\n5879FNWnIoxUSU0tIS8vjZdeqhjW65/EI1Wlq6uTri4vnZ1eVL2A75aaKmRlecjM9DBmjAePx3dL\nSUlxrZ8iPQ89ooVFIlIArMdXc69Q1V+LSCnQoKpvi8gqoAjoBs4Dy1T1eIjjWEI3SWnOnDkUFRUF\nrtYIUF5ezrZt26ivr2fp0qUsWLCAyZMnf+F57e3t1NWdJTv760Mev6Wlmfnz87h48ReoPh42HpEN\n3HLLn6isrMXjSQvbfqTq7e0NJO3u7mDSFukkIyOFzEwPWVke0tODiTvS89GdFNOEHiuW0IPiqW7s\ntmTpi8rKSkpKSoZsM7gvmpo+5syZLMaOHRf2+J98coIFC77NpUu/AYZ6nVrS0x+msrKOCRMmRRa8\nC6KpoXd3d9HZ6aWry0tvbzBxjx7dS2amJ3BLS/Ml7dTU1BE1AWwrRY2JI42NjUyZMiWq5/T09HD6\n9CXGjr0tovYTJ06mrGwLy5bdD1wAngzR6iypqT/ixRcr4jqZh9LX1xcok3R1fbFMkpY2KjDazsjw\n4PGMDZRJkomN0I1xwL59+5g9e3ZUz2lra+PAgUtkZ0eWeF955QXKy32LlFJSMunq2gp8f0CLHjye\nB3jkkXtYseJXUcXipJ6e7sBou6cnmLRHjephzJjUQH174Gg71pdDOHDgAGVlZezYsYPLly9TUFDA\n1KlTWb16NfX19axdu5aqqio6OjrIz89n+vTprFq1KqYxDGQlF2NGuEOH/sWlSzlkZGQN2e7cubM8\n+GAOAA8//BOee+73HD1az/LlRXz++V8A3/nyo0c/y+23H6a8/B1X9+aEkTMpGS+bm1jJJc4lS904\nEtYXQf19EelS/4Gj8jff/DhQRrnzzrtYt+6vrFw5D6/3TeAsGRlbWbv2oKPJPNyk5E039U9KZuDx\njPvCpKTb74t429wkEpbQjYlD4Zb6hxqVDzZjRh5lZVt45pm5AKxb93ZEk6vXItyk5Lhx/WWSG0fM\npKTTm5vEgpVcjIlDtbWNqH415EbQVxuVX83+/bvp7Ozi3nuv76Jb0U1Kekb8pOTChQvZuXNnyOue\nz507lz179nD+/HlHykBWcjFmhLraUv9IRuWhzJoV3Sgy3KRkTk7/aDsLj+fmYZmUjAfV1dWkpaWx\naNGiKzY3qaqqIj8/P+4WZFlCd4nb9cF4Yn0RVFNTE1jqP1C0o/JwopuUzHRlUtLN94UTm5uArxa/\nfv16Nm7ceN3HAkvoxsQVVeXUqQtkZfn2ML3WUXm/65mUTGbDvblJv+effz5wOd5YsBq6MXFk4FL/\naEblib5S0mklJSXU1NTQ0tJyxfeKi4upq6ujtbX1ul7j3XffpbGxkaamprBbEFoN3ZgRqKXlHB0d\nvcyc6fvdHTgqt5WSzhnuzU0uXrxIZ2cn48ePj0W4AfYn2iWDLx2bzKwvfHp6eli58mnmzZsKwObN\n+3niiZ/T2nqC1tajXLjwAaNGnSQn5wJ33KHMmpXFfffdygMPTCMvbxq5uVOYNGki2dnZZGZmjvhk\n7tb7wonNTd544w2Ki4tjHruN0I2JE7t27eK9997ioYd+SGnpi/5JyRTXJiWTjVObmzQ0NJCbmzss\nP4PV0I2JE319fXi9XtLT090OxUShra2NxYsXs3z5cpqbm6mvr2fx4sXcfffdIdtv2LABr9eLqnLw\n4EFOnjzJkiVLhizV2LVcjDHGQdFubgKwadMm9uzZE7NJUauhu8TqxkHWF0HWF0EjrS9yc3OvSOYA\na9as4bXXXrvi8e3bt/Pqq6+yd+9eXn755ZjEYDV0Y4wZRrW1tVRWVl7xeGFhIYWFhTF9LSu5GGPM\nMGpsbMTr9TJz5sxrPoaVXIwxJg50dHRcVzKPRkQJXUQKROQjETkuIs+G+H6KiGwVkRMiUiciE2Mf\namIZafXB4WR9EWR9EZQofRHtTlXXI2xCF5EbgN8BDwJTgcdE5BuDmj0OnFfVycA6oCzWgSaaI0eO\nuB1C3LC+CLK+CLK+iF4kI/RZwAlVPaWq3cBWYPASp2Jgk//+68D9sQsxMbW3t7sdQtywvgiyvgiy\nvoheJAn9K8B/Bnx92v9YyDaq2gu0i8iNGGOMcUwkCT3UzOrgU1UGt5EQbcwAzc3NbocQN6wvgqwv\ngqwvohf2tEURuQt4QVUL/F//DFBVXT2gzTv+NvtEZBRwRlWzQxzLkrwxxlyDWF0+twH4mojcCpwB\nHgUeG9TmLWABsA+YB+y+1oCMMcZcm7AJXVV7ReQpYAe+Ek2Fqh4TkVKgQVXfBiqALSJyAjiHL+kb\nY4xxkKMrRY0xxgwfx1aKhluclCxEpEJEzorIh27H4jYRmSAiu0WkSUSOisgKt2Nyi4ikisg+ETns\n74tfuh2Tm0TkBhE5JCJ/dzsWt4lIs4h84H9v7B+yrRMjdP/ipOP4zk9vwVeXf1RVPxr2F48zInIP\ncBnYrKrT3I7HTSKSA+So6hERyQAOAsXJ+L4AEJF0Vf3Mf2JBLbBCVYf8BU5UIvI0MAPIVNUit+Nx\nk4j8G5ihqhfCtXVqhB7J4qSkoKrvA2H/Y5KBqn6qqkf89y8Dx7hyjUPSUNXP/HdT8c1vJWU9VEQm\nAIXAH9yOJU4IEeZqpxJ6JIuTTBITkduA6fjOlEpK/jLDYeBTYKeqNrgdk0t+CzxDkv5BC0GBKhFp\nEJEhd6B2KqFHsjjJJCl/ueV1YKV/pJ6UVLVPVXOBCcBsEbnd7ZicJiI/AM76P7kJoXNHsvmWqs7E\n96nlSX/ZNiSnEvppYOAVGCfgq6WbJCcio/El8y2qus3teOKBqv4PqAEKXA7FDXOAIn/d+M/Ad0Rk\ns8sxuUpVP/X/2wb8DV8JOySnEnpgcZKIpOA7Tz2ZZ69t5BH0R6BJVde7HYibRGSciGT576cB3wOS\nbnJYVZ9T1YmqOglfntitqvPdjsstIpLu/wSLiHwJyAcar9bekYTuv2BX/+KkfwJbVfWYE68db0Sk\nEvgHMEVEPhGRhW7H5BYRmQP8GPiu/5SsQyKSjKNSgC8D1SJyBN88QpWqbnc5JuO+m4H3/XMr9cBb\nqrrjao1tYZExxiQI24LOGGMShCV0Y4xJEJbQjTEmQVhCN8aYBGEJ3RhjEoQldGOMSRCW0I0xJkFY\nQjfGmATxf0KNOgvBmMdIAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e3302d4a8>"
      ]
     },
     "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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FEOJ0jfv8+Sn6+voGNuk4QTI5uEmHtbMFKVO5S+YARpOWvAEScWxH\n9lB+YAcpazmRcxYRnTWXpLsqqxp3zeo/N3cxKnSJ0tALhNIHB9F7XxgMBiorK1m8WKtxv/hiN5WV\nXXi9O4m9/SJhkylnNe5bDu068wmTmUTlNOLT6kjZ7NgONFPxlz9Q8fwjWA80Y/T3jnnNcLgfj0dg\nt+dnOeJ8off3hR5RI3SFIgtMJhPV1dVUV1ezqK6H8I4/02ly4PMfBRxYrQ4slvyUIsoyy+mySBEN\nY9+1BbHjdZLuKkLzlmg17nbnWa8LBntYtkzfC3EpcoPS0BWK8bJzJ7zxBsycSTweJxAI0tUVIBhM\nIoQTq9WZt6V+h2II92MM9IFMEa+aTuScxcSm1yOt5Ugp6e7eydq1C4vCHapIj6pDVyjyiZTwyCNg\nNoPtzBF5NBolEAjS2RkkHAaDwYHV6sSc5UYZ44nJEApi7PeBhNj0enpqZmGdZ6TxsjHX1VPoGFWH\nrnOUPjhIUfZFdzf4/WclcwCLxUJ1dRWLFs1h8eJpzJolSSZb8fmOEwj0jrqO+1kaejYIQcru1Grc\na2dhDPTheOMp5r32AmzaBMePQzxPa8jngaJ8XxQYpaErFOPh4EEYpTb9FFarFavVSk1NFeFIBF9f\ngK6uFvr7zRgMDsrLHRiNefgYCkHc5iBe66Z8bj20t8O772oxz58P550H06Zl9DsoigcluSgU2ZJI\nwEMPgcejSS5ZIqUkFArR2xukp6efRMKKyeTAZrNjyGH5YzDoo6oqTH39kPVlkkno7YVwGCwWOP98\nmDcPamvHXeOuyD+qDl2hyBft7RCNjiuZg/bhPOVOnTkzRSgUwusN0NPTTSplw2x2YrWVYxATS7DJ\nZIDKyooznzQaobpaexyPw759sGOHJh0tWgTnnKOd19k67orMUH+SC4TSBwcpur7Yty+tdj4eDAYD\nDoeD2bNnsGzZHNrjx3A6/fQHj+LzdRCJ9I+rxj2RiFNWFqe8fJSVJM1mbWReXw8ul1a18+ij8Otf\nw/bt4C3sPjVF977QAWqErlBkQySiadHTpuX80kajEbvdzty5M0kmkwSDQbq7e+nr6wTsWCxOLBZr\nRjswhUIBZs1yZL5bU1kZTB+QZiIR2LoV/vY3qKiAxYthzhwt6St0jdLQFYpsOHQIXngB6uom7Zbx\neJxgsJ+u7gABfwJwYLM5KSuzjvgan+8YixdPw2oduU1GhEKa5i6lNppfvFgb0ReZ67TYUXXoCkU+\nePxx6O8v2Gg1FosRCATo6uqnv18ihOZOHWpgisUiGAydnH/+7NzePBgEn7ZJB7NmaZp7XR1M9I+G\nYkxUHbrOUfrgIEXTF34/tLWB82x7fa5o2jV6HXpZWRlVVVWcf/5sliyZTn29BNrw+Y7j93tJJOKE\nwwFqavKwzZ3DoSXyWbMgENC+qWzcCM8/D0ePQizzNeQzoWjeFzpiTA1dCPEgcCPQIaVMazcTQnwf\nuA7oBzZIKZtzGqVCoQeOHtVK+3RSAWKxWKipsVBTU004HMbnC9LZeQKzOUUq5SaRSIy6jvu4EUL7\nhuJyQSoFnZ1w5IhWQXPuuVqN+/Tpqsa9AIwpuQghVgFB4KF0CV0IcR3weSnlDUKIS4HvSSkvG+Fa\nSnJRFCejWP31RDAYZOfOdiwWJ9BPRYWF6mondrs9//uIJpPQ16fp7mazVuN+7rma9q72MJ0QOatD\nl1L+VQgxZ5Qm64GHBtpuEUK4hRDTpJQdmYerUOicU1b/SZwMHQ9ebwC7vRa73UVKpujv1wxMBkM3\nlZVWKiu15G7Ih4nIaISqKu0nkdDctLt2aRr7woWagam6WhmY8kguenYWcGLIcevAc4pRUPrgIEXR\nFxla/SfKWBr6aCSTSXp6wthsWgWKQRiw2Ry43dOx2+fQ1+fkwIEAO3ce5cSJdoLBYM7WcT8Lkwlq\narSKGI8Hdu/Watwffhjefht6erRvPaNQFO8LnZGLd2i6rwEj/k9t2LCBhoYGADweD42NjaxZswYY\n/A9Ux6V1fAq9xHPW8apVsG8fTW1t0NXFmqVLtfMDyTeXx82HD4/79c9t3Upra5T3XHAOMLjQ16Xn\nLsVgMLK77SgAK+YuoqcnyLNvbsVojHH9JY14PE62HDqEECI/v9/06dpxVxdrolHYsoWm1laYO5c1\nt9wCbvdZ/d/c3JzZ/88UPG5qamLjxo0Ap/NlJmRUtjgguTw1gob+E+AlKeXvBo73AavTSS5KQ1cU\nJS0t8NRTupdb3n23hXC4Aqs18xrxZDJBKBQklQpQVpagpsaB2+3ANhnzBENr3KurtRr32bO1ahrF\nGeR6LRdB+pE4wJPA54DfCSEuA/qUfq6YUuTQ6p8v4vE4fX1xXK5RrP5pMBpNOJ0ewEMiEefkyQCt\nrV3YbClqapy4XA4sljxt0lFerv2AVuP+yitacp8xQ6txr6/Xfb/rjTE1dCHEI8DrwHlCiONCiLuF\nEPcIIT4NIKV8FjgihDgEPAB8Nq8RTxGUPjiIrvvilNXf45mU241XQ/f7A0AWVv80mExmXK5K3O7Z\nwAxOnIDdu9vYt+84PT1e4vlcS31ojXt/P7z4Ik1f/So8+ywcPqwthqYYk0yqXO7MoM3ncxOOQqEz\nWlq0Wmudl911dASw2XK3vkxZmWXAfVpFLBbh6NEA0ILTZaKm2onDYcc8ztUmR2VojXtPj/azaZNW\nGTNvHixYoNW45+PeUwBl/VcoRqPAVv9MiEQivPNO58DIOn9IKYlGw0SjQYTox+02U13txOFw5K/G\nPRbT1m7v79eSu93OAweP8XDzPj7ykZtYv/4mZs6cmZ976wi1lotCMVH8fm0p2VmzdOMOTUd7excn\nTxpxuSon7Z5SSiKRfmKxfgyGfjweK1VVjvEZmFIpTdo69TM0Rzgc2oRpTY228qPTyZY9e7hi7fsw\nmZYjxEEaGs7ljjtu4kMfWs+SJUsmJDvpFZXQdU5TU9PpcqVSR7d9sXMnvPEGTOIIsGnXrtNlf5kg\npWTXrqOUldVhMhVGhkjJFOFQP4lEAIMhQlWVjcpKJ+Xl5WcamOLxwaQdi2l/JKXU5JSKCi1x19Zq\n34YcDpreeos1731v2nv+8If/xb/8y38RCr0KvEVZ2ROYTE9gtxu4+eabuPXW9axatSo/slABUDsW\nKRQTQUrN5VhRMXbbAtLf3088bqa8vHCJyyAM2O1OwEkykcDX4aX3yAnKUgEq3Vacbjs2iwVhs2ku\n0nPO0f51OrUReHl5evfoKEauz33uM7z88hs8/fTniUQeIhZbSyz2XUKhXfz0p0/w61//M8nkYa65\n5jruvHM91157LS4dy2a5Qo3QFYp0dHVpzkad154fP95Gb68du32Sk1UyiSEWRkQjiHgUEAhAIkk5\nPCQ81UScFfgERM0xTB4Ts+dPo7a2EkeO6sxDoRBLl17GkSN/j5R/n6ZFK/AUTucTRKOvcdFFV/DR\nj67nppveT53O/1+HoyQXhWIivP46vPPO4C4+OiSZTLJz5zHs9jk53Vx6KCIeQ0TDGKJhRDKBFAYE\nEmk0k3BXkqioJeGpJmV3krQ5SNnsaSuCYrEogUAvyaQXuz3BnDmV1NRUjr5FXgYcPHiQ5ctX0t//\nFHDpKC0DwCbKy58klXqW+voGbr/9Jm6+eT3Lli3Tve6uErrO0a1uXAB01xeJBDz0kFZ7PskabDYa\nus/n49ChMG73BP/opFKIWARDLIIhqk1KSgFCQrLcQcJTTbKihoSz4nTilhbruCeKI5Ew/f1acne5\nYM6cSqqrK8/aXSnT98Xjjz/BRz7yBUKhbUBNBhEkgNcwm5/AbH6M+fPraW7+63h+lUlDaegKxXhp\nb9eMLDqfUOvuDmCxZKHxJ+IYolriFnFtUlIiQRhIuiqITZ9NwlND0uEmVe4gWe6APEy0Wq02rFYb\nMJNwuJ933ukllTpAZaWJ2bMrqaqqpKysLOPrfeAD6/nMZ/7GT35yJ6HQ88BY31ZMwGri8aWYzZu5\n9tqrJ/Db6As1QlcohvOXv2iGoqqqQkcyIvF4nJ07W3C5Gs6UC6RExKKn9W1SKW10h0SWWUl6qoh7\naki6q0iWO0mVO0hZR5iUnESklIRCQfr7vQjRR02Nhfr6SioqKjKqVEkkEqxceQ3bt19BPH5/Bnf0\nU17+Xu6++0p+8IP/VJLLeFAJXaF7IhH41a9g2jRdu0N7Ors4cTCE22IfcVIyXlFDyunRtO1yB7Is\nT2uy5BgpJf39fkIhLwaDjxkz7MycWUFFRcWoNe6dnZ0sXHgRXu+PgfePcod+ysuv49ZbF/OLX/xY\n98kclOSie3SnGxcQXfVFga3+Z2nop5yS4bCm7RsMICXeY52UueYTnTY7o0nJYkIIgcPhZt++7Vx4\n4VX09PhobfViNLZQV+dkxoxK3G73WZt01NbW8vTTv+c971lPOPwGMC/N1SOAg2uvvYMHH/xRUSTz\nbFAJXaEYyu7dk2/zH+qU7O6G1tbBc2mckv1CcOjNdmqnLZncOAuAwWDA5arA5aogmUzS1tbHsWPd\nmM3HqK93M316JS6X63Rivvzyy/nGN/6Nr33tZkKhN4ChqzXGsNk+TDgMf/rTb/j5z9fw6U9/uiC/\nV75QkotCcYp8W/0zcUrW1IDbrSVyhyPtxOzhw8c5eNBMdfWM3MdYJCQSCQKBXuJxL1ZrhPp6D9Om\nDda4f/CDH2HTJguRyC/QVv5OYLPdycqVUZ555g/cdNN6Nm16HgC/34/T6SzcL5MBSkNXKLIlF1Z/\nKbUKmXBYS9yp1GDitlq1idaamsyckmkvL2lq2onNthCzOfNKkKlMPB7D79fKIMvL48yZU0F5uYVL\nL13LsWNfQMpPYrVuYMWKDl544YnT5ZFvv/02F110EQAPPPCArkfrKqHrHF3pxgVGF30hJTzyiDYi\nzmRThWRyMGlHo4Mjeim1+vVTo22PZ3C0ncFGEWP1RV9fH2+80UFt7YIMf7HiZdu2JlasWJPVa6LR\nCIGAl1Sql76+I3zyk3cTj69i2TIvr7zy3FlGJikl1113ve5H62pSVKHIhu5uTXIZbgkfYVISs1kb\nZc+erSXvU6Nte34nJU+e7KGsTL/llIXGYrFiscwEZuJyzeV//I9v8fzzj/Ef//Gf9PX5MBqNZ+zA\nJITg+eefOz1ad7lcuh+tj4YaoSsUQGjzZixvv43R6Txz+Van88xJyVOjbev4nZLjJZFIsHnzbjye\npflbf3yKcqrGXcpeamst1NVVUll5Zo378NG6z+fTzYJeSnJRKLLgozfczBMvvcT1V63hjluu533r\n1mGfNk1XbtGuri62bQtQW3tOoUMpWrQa9wChkGZgmj69nLq6SjweD6aB1R31qK1nmtAzmokRQqwT\nQuwTQhwQQvxLmvN3CSE6hRBvD/z83XiCLiV0vY/mJKOHvvif3/5fxJIxfr+pgo9/8bdUnbuI1e/9\nED/96c9ob2+ftDhG64sTJ7yUl5eO3LJtW1POr6nVuLuorW2gqmoZfX21bN3qZ/Pm3ezefQiv10tj\nYyOpVIprr13HPffcgxACv9+f81jyQSabRBuAHwLXAouBO4QQ56dp+lsp5YUDP7/IcZwKRV5ZvHgx\nv/zlzykvbyIQ+APR6HFeeeVOvvjFzTQ0LGTRosu5//7/YM+ePRTiW2Y0GqWjIzL5y+ROYQwGA06n\nh9rac/B4ltLZWcmWLV42b97Jvn1H+O1vf8O2bdsAcLvd/PSnPy1wxGMzpuQihLgMuFdKed3A8VcA\nKaX81pA2dwErpJT/MMa1lOSi0DX33POPPPzwYcLhJxgc78SAlykrexKT6QmczjI+/OH13HLLTaxc\nufL0V/V80traxq5dCWpq6vN+r1JnsMa9F4slRF2dm89+9tO8+OJfgMJo6znT0IUQNwPXSik/PXD8\nUeASKeUXhrS5C/gG0AUcAL4opWxJcy2V0BW6JhaLcfHFa9iz5wYSia+laSGBHRgMT2C3P0EqdZx1\n667njju0XXFytXnDcF57bTdSzsVms+fl+or0JBJxfD4vyaSXEyd2cM89dwCTr63nUkNPd5HhWflJ\noEFK2Qi8CPwqg+uWNHrQjfWCnvqirKyMZ5/9A3b7j4AX0rQQQCOp1L0EAm/T37+dxx67lLvv/inV\n1TNZtep6fvKTB2hraxvX/dP1RX9/P36/KLlkng8NPVtMJjNVVdOorV3I4sXrefzxQzQ2XsE999zD\nvffeV+jwziKT74otwOwhx3XAyaENpJS9Qw5/BnyLEdiwYQMNDQ0AeDweGhsbTxspTr2Z1XFpHZ9C\nL/GsWbOGP/3p16xbdzOx2E+AW09FOPDvmmHHnyMQ+BzwNK+99iZvvvlD/vu/H+P++7+a9f2bm5vP\nOj979jkYDJWnE9wps81UP96/v7kg929sXEk0GmHr1hdJJmMsW7YciLB7998oLzfzyCMP09p6hHff\nPXSGESyX77+mpiY2btwIcDpfZkImkosR2A+8B2gD3gTukFLuHdJmupSyfeDxB4EvSymvSHMtJbko\nioZ///dv841vPEYo9AqQ6dKzz+Jw3M0rrzzP8uXLJxyDsvrnByklsViUWCxCNBpBygjaSowRLBaB\n223F5bLidFqxWrWfsrKygq3OmNM6dCHEOuB7aBLNg1LKbwoh7gO2SimfFkJ8A7gJiANe4O+llAfS\nXEcldEXRIKVk3boP8fLLM4hGf5zBK17Ebr+DF198iksvHW1/y8wpJat/Pkgmk6eTdjw+mLSFiOJw\nlOFyWXG7rZSXDybuyZjkzhZlLNI5uli/RCfouS98Ph+LF19Ma+v/C3xslJavUV7+AZ599lFWr149\n7vsN74s9e96lrc2Nx1M97msWK9ms5RKPx4hGI8RiEZLJwcRtMiVxuaynf2w2LWlbLJaz1lPXM2ot\nF4UiB7jdbp577jEuu2wtodAFwLI0rd4EVvH1r/9/E0rmw0kkErS0BPB4GnJ2zWImlUqdlklisTNl\nEpvNeHq07XBYsVo9p2WSUkKN0BWKDHj44V/zmc98nVBoK+AZcmYnNts1hMMdANxww4089dSTOdFa\nS9Xqn0jET4+2E4nBpG00JnA6Laf17aGj7am+to2SXBSKHPOJT3yO3/ymlXD4j2jTSfuw2dby4IPf\n4Y47buPHP/4xn/vc5wBobm7mggsumND93n57P4HAdBwO98SD1xnFNilZaFRC1zl61o0nm2Lpi2g0\nyooVq9m79wMkk7dis63mRz+6n7vvvut0G5/Ph8ejjeDHM1o/1RfRaJTNm/dRU7OsqJPYRCYli+V9\nMRkoDV2hyDEWi4Vnn/0DS5ZcQiz2A7797X89I5mDprlLKU+P1g0Gw7hG693dXoSoLJpkPtakZHX1\nKWRvKocAAAVHSURBVJmksignJYsFNUJXKLLkzTe3cuTIMW677ZZR201ktK5Hq392k5LWkpyUzBdK\nclEodEK22np/fz+vvHKU2trFkxHeWahJSf2hErrOUfrgIKXQF5mO1puampg9+xwOHjRTXT0jb/EU\nw6RkKbwvMkVp6AqFjshUW5dScuxYL273wpzcd6xJyaqqU5OSDqzWat06JRWZoUboCsUkM9pofbxW\n/6nulCx11AhdodApo43WT57soaws/TZzyimpGAs1Qi8QSh8cpJT7Yuho/frrb2D16vWsWvUxEok4\nsdiZMonRmMDlKp1JyVJ+XwxHjdAViiJg+GjdaLRz8cWXDJuUdJW8U1KRGWqErlDohFgshpQSiyXT\ntdcVpYIqW1QoFIopQi73FFXkgeHbr5Uyqi8GUX0xiOqL7FEJXaFQKKYISnJRKBQKnaMkF4VCoSgx\nMkroQoh1Qoh9QogDQoh/SXO+TAjxWyHEQSHEG0KI2bkPdWqh9MFBVF8MovpiENUX2TNmQhdCGIAf\nAtcCi4E7hBDnD2v2CcArpZwPfBf4dq4DnWo0NzcXOgTdoPpiENUXg6i+yJ5MRuiXAAellMeklHHg\nt8D6YW3WA78aePwo8J7chTg16evrK3QIukH1xSCqLwZRfZE9mST0WcCJIcctA8+lbSOlTAJ9QojK\nnESoUCgUiozIJKGnm1kdXqoyvI1I00YxhKNHjxY6BN2g+mIQ1ReDqL7InjHLFoUQlwFfl1KuGzj+\nCiCllN8a0ua5gTZbhBBGoE1KWZvmWirJKxQKxTjI1eJcW4FzhRBzgDbgduCOYW2eAu4CtgAfBjaP\nNyCFQqFQjI8xE7qUMimE+DzwZzSJ5kEp5V4hxH3AVinl08CDwMNCiINAD1rSVygUCsUkMqlOUYVC\noVDkj0lzio5lTioVhBAPCiE6hBA7Cx1LoRFC1AkhNgsh9gghdgkhvlDomAqFEMIihNgihNg+0Bf3\nFjqmQiKEMAgh3hZCPFnoWAqNEOKoEGLHwHvjzVHbTsYIfcCcdACtPv0kmi5/u5RyX95vrjOEEKuA\nIPCQlHJZoeMpJEKI6cB0KWWzEMIBvAWsL8X3BYAQolxKGRooLHgN+IKUctQP8FRFCPH/ABcBLinl\nTYWOp5AIIQ4DF0kpe8dqO1kj9EzMSSWBlPKvwJj/MaWAlLJdStk88DgI7OVsj0PJIKUMDTy0oM1v\nlaQeKoSoA64Hfl7oWHSCIMNcPVkJPRNzkqKEEUI0AI1olVIlyYDMsB1oB16QUm4tdEwF4jvAlynR\nP2hpkMAmIcRWIcSnRms4WQk9E3OSokQZkFseBf5xYKRekkgpU1LK5UAdcKkQYlGhY5pshBA3AB0D\n39wE6XNHqXGFlHIF2reWzw3ItmmZrITeAgxdgbEOTUtXlDhCCBNaMn9YSvlEoePRA1JKP9AErCtw\nKIVgJXDTgG78G+BqIcRDBY6poEgp2wf+7QL+hCZhp2WyEvppc5IQogytTr2UZ6/VyGOQXwB7pJTf\nK3QghUQIUS2EcA88tgHvBUpuclhK+VUp5Wwp5TloeWKzlPLjhY6rUAghyge+wSKEsAPXALtHaj8p\nCX1gwa5T5qR3gN9KKfdOxr31hhDiEfj/27tjEwSCIIzC769CbOCaMLMNI2s8EDQxETESa7CRucBL\nL14Y3xdvsNFjYXZZnsCU5JvkPHpPoyQ5ACfguF7Jeif5x1MpwB64J/nwmyNcq+oyeE8abwc81tnK\nC5ir6ra12IdFktSEX9BJUhMGXZKaMOiS1IRBl6QmDLokNWHQJakJgy5JTRh0SWpiAf0VWu905yRw\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32faaba8>"
      ]
     },
     "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": {},
   "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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FFC4n3vRMPLkFI9zD2MVub2HChGxdez+O0Rr6CJESnn0W7rlH6eYuF/T0DP2e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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32cdd5c0>"
      ]
     },
     "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": [
    "### Matrix multiplication – Projection onto an axis\n",
    "Matrix multiplication 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 $U \\cdot P$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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FLBbj/PlpmptbgPgY9+ZmH+3tN+Lz3UEw6OfFF0McOfI6J0+eZmJiImfzuOeD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\n69CNMaZEWIdujDEl4v8D0e6TQPIUTz4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32cc1be0>"
      ]
     },
     "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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6ha48Vl9fz969FWzbZqO6GiIjnZ17bKx1f3yChTGGv//977z4ojON1rt3H159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\nQ1dKqRChHbpSSoUI7dCVUipE/H+SXvyWudtSKwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32d68828>"
      ]
     },
     "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": [
    "### Matrix multiplication – 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": {},
   "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 product $VP$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "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 $V_{1,*} P$, which is the coordinates of the projection of $P$ onto the 30° axis, as we have seen above. The second row is $V_{2,*} P$, 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 $VP$ to see this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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04O0PAEqlJq4AXbGkCiFKADwkpXxLiX0HG63kniphp8VCE5BPPUXJAEojJXDN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\nIvDoo7RdWUmO03GBMncuOeu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rCzhwANiwIaAtFPrU8dQQ7MwZbXPppeSYdu+mVY+uuoqc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\ngP8PoIfUTdnslWoAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32be3da0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P_rotated = V.dot(P)\n",
    "plot_transformation(P, P_rotated, \"$P$\", \"$VP$\", [-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": [
    "### Matrix multiplication – 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 matrix multiplication:\n",
    "\n",
    "$f(\\textbf{u}) = F \\textbf{u}$\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 $FG$ results in the linear transformation of all vectors $\\textbf{u}_i$ as defined by the matrix $F$:\n",
    "\n",
    "$FG = \\begin{bmatrix}f(\\textbf{u}_1) & f(\\textbf{u}_2) & \\cdots & f(\\textbf{u}_q) \\end{bmatrix}$\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": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ifbFrl3w8fejDD34gy8f/6Z8kbgmHjec4flwuzPp8xvzvn/5U+uTdd6WPdu+W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\nREQFhEWbiKiAsGgTERWQ/w+9kz6Op0gtkgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32bdf358>"
      ]
     },
     "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), \"$P$\", \"$F_{shear} P$\",\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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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5sqp2kb8mgASDNBdNMN3Tb0R32snL06Sd8ZZ1axhdkVSa+ICGOvLW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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32d62da0>"
      ]
     },
     "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), \"$Square$\", \"$F_{shear} Square$\",\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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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am46UXM50gDz2mN0ROop0fWxkmUyhbDI0ZH5xJiYKI++GBjOSW0fyrmSbXzw+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\nIYTD/X89xnDf72x6IwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32b74518>"
      ]
     },
     "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), \"$P$\", \"$F_{squeeze} P$\",\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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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hhvwgYEuj5a0ce4jfCPy9I6E601nDxiU6QlQ+5lKmcCYV+NfHF47z73ECKBwx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\nQ15EJIlpyIuIJDENeRGRJPa/BwEbCrfj2D8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32b40b38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_transformation(Square, F_squeeze.dot(Square), \"$Square$\", \"$F_{squeeze} Square$\",\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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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FwmWz0N8/3cNOJExyCoVMScKiadl5FQiYG5jr11/8fCJh6tIzvfvdZljet75l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\nkryFEKII/X/W+4jkVUmwMgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32f119e8>"
      ]
     },
     "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), \"$P$\", \"$F_{reflect} P$\",\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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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FaNMllMxKJwlXvxyf994Dq/WYwuxTb5Vy7YdC8X/uDRvI/u1vOWtwkLff3odS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\nYRaCAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32fc3f28>"
      ]
     },
     "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": {},
   "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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jPv4E720TEbskbW8c6M6gk/FL0gWNj6lLbB/Tm9JGjMHb6A9qvY0yO60xRfuI7SlVFzNc\nbP+tik81KwfdVMproNtTKNviYqqWWu2hDD4CPriNW7TpV52Mf5mk+yJiwPZsFZ9qMn2aaaeTbZTZ\nbyRNjogPbU+XtFTSFyquqXS2P6Xik/p1jT36/3dzi7sc8Gtg2EM+Iv5xqNsaB16Ojn0XU707xGO8\n0/j9e9t1SV+R1M8hv1FS84HUYyRtGtRmg4pTUzfZHivpsIho9/G2X7Qd/6Cx3qkhrq1IbKOK53+P\nVq+RtJoDLyIes32H7SMjYmuVdZXJ9jgVAX9vRHzsGiOV9Bqoerpmz8VU0n4uprI9vrG852Kq13pV\n4DBZJel425MbY5ulYls0e0jFNpGki1R8y2cWbcffeNPfY6b6/zlvxRp63nmZpMslyfZpkrbvmdpM\nZMjxN889256q4nTvNAHf8J+SXouIfx/i9lJeA8O+J9/GLZKWNE6Pe1tFmKn5YipJX5b0c9t7Lqb6\n14hYW1XBZYiIXbbnSHpCxZgWRcTrtudLWhURD0taJOle2+sk/VFFEKbQ4fivtX2epAFJW7VvZyAF\n2/dJqkmaYPttSXMljZcUEbEwIh61PcP2m5I+kPSt6qotX7vxS7rQ9jUqnv8/qTjDLA3b/yDpMklr\nbL+kYhrmRhVnnJX6GuBiKABIrOrpGgDAMCLkASAxQh4AEiPkASAxQh4AEiPkASAxQh4AEiPkASCx\n/wOE7vvJmGtThwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32e643c8>"
      ]
     },
     "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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m2bJ86uvnZneg4hQ18yxz/fwQY9XnSmPPxHlnrLW0tv6JtWsvo6ioyNd9T1YUj02XaDWL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\nIuIANXMREQeomYuIOEDNXETEAf8f7n8C1UvhnekAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32f36b38>"
      ]
     },
     "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": {},
   "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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85eXzkvq8imLSmzJzEZ8dOPA2775bSFFRcBcfU2NPP8rMk8z13M7l+uKprbOz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\nA9TMRUQc8P8BDQN3GjTsujoAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32fcf2b0>"
      ]
     },
     "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": {},
   "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": {},
   "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": {},
   "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}$\n",
    "\n",
    "Once you understand the geometric interpretation of matrices as linear transformations, most of these properties seem fairly intuitive.\n",
    "\n",
    "A matrix that is its own inverse is called an **involution**. The simplest examples are reflection matrices, or a rotation by 180°, but there are also more complex involutions, for example imagine a transformation that squeezes horizontally, then  reflects over the vertical axis and finally rotates by 90° clockwise. Pick up a napkin and try doing that twice: you will end up in the original position. Here is the corresponding involutory matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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mjfNtEBlDat7zFY2aESS//a3pMQaDnhs94ml/+qcm6UzXax0aMp9kWlpMj7ej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\nEEJkIEneQgiRgSR5CyFEBvr/4uU+QQV3U9gAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32c29470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_involution  = np.array([\n",
    "        [0, -2],\n",
    "        [-1/2, 0]\n",
    "    ])\n",
    "plot_transformation(P, F_involution.dot(P), \"$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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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KCwUVFWGUlYVRVlaIcLgG4XAYhYWFtl0LNRsm/2zOBZu5KU6e1A3byp5oTuzT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\nnIjIAP8fgrAarPBEMFgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32cd89b0>"
      ]
     },
     "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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WLH8XcVlZ1BEVh/L9IpFQOiiuSmXMAeX7RQpWSDpIZwJxNWkSrFrlvxn3xAed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\nIiIl7P8DykDSSJg1BVAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32faa860>"
      ]
     },
     "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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9PZdDR6xTG+dNOTEwf/XmfKYVtlOca22vptZEahtSOGfmcd8ytuUOdPj3rzid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\nEFBKqRj2/wG+fq6hrJDmBAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32ca8dd8>"
      ]
     },
     "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": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32d9c7f0>"
      ]
     },
     "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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": {},
   "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 Algebra. 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. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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