{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# EE16A Discussion 2A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualizing Matrices as Operations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This exercise is going to help you visualize matrices as operations. For example, when we multiply a vector by a rotation matrix, we will see it \"rotate\" in the true sense here. Similarly, when we multiply a matrix by a scalar matrix, we will see it \"scale\". The way we will see this is by applying the operation to all the vertices of a polygon and seeing how the polygon changes.\n",
    "Let's first do the necessary imports and define some useful functions to do this."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "\"\"\"Function that takes the vertices of a polygon and applies a matrix \"transformation\" to each of them, effectively\n",
    "\"transforming\" the polygon.\"\"\"\n",
    "def transform_the_polygon(polygon, T):\n",
    "    \n",
    "    transformed_polygon = []\n",
    "    for point in polygon:\n",
    "        transformed_point = np.dot(T, point)\n",
    "        transformed_polygon.append(transformed_point)\n",
    "    return transformed_polygon\n",
    "\n",
    "\"\"\"Function that plots a polygon in the x-y plane, given its vertices as x-y coordinates. The plot is defined in terms\n",
    "of line segments connecting all adjacent vertices of the polygon.\"\"\"\n",
    "def plot_the_polygon(polygon):\n",
    "    fig = plt.figure(figsize=(5,5))\n",
    "    ax = fig.add_subplot(111, xlim = [-4, 4], ylim = [-4, 4])\n",
    "    for i in range(len(polygon) - 1):\n",
    "        ax.plot([polygon[i][0], polygon[i+1][0]],\n",
    "                [polygon[i][1], polygon[i+1][1]])\n",
    "    ax.plot([polygon[i+1][0], polygon[0][0]], [polygon[i+1][1], polygon[0][1]])\n",
    "    ax.grid(True)\n",
    "    ax.axhline(y=0, color='k', linestyle = '--', linewidth = 2)\n",
    "    ax.axvline(x=0, color='k', linestyle = '--', linewidth = 2)\n",
    "    #plt.show()\n",
    "\n",
    "Re_y = np.array([[-1, 0], [0,1]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we define our starting polygon, a square whose side is of length 1. Let's see what the square looks like."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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UNbOz3P3HvbtXSno+y3jASs1mU81mM3QMlFDW1/i+bGZnSzos6WVJf5M9EtDVv/Q8kLdM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\nD0ByKD4Ayfk/LeqTVXy/QK8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1569ff9ca20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "unit_square = [np.array([0,0]), np.array([1,0]), np.array([1,1]), np.array([0,1])]\n",
    "plot_the_polygon(unit_square)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part 1: Rotation Matrices as Rotations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's rotate the square by some arbitrary angle, say, 60 degrees."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x156a1fdfc18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# R is a \"rotation matrix.\"\n",
    "angle = np.pi/3 # 60 degrees in radians\n",
    "R = np.array([[np.cos(angle), -np.sin(angle)],\n",
    "              [np.sin(angle), np.cos(angle)]])\n",
    "rotated_square = transform_the_polygon(unit_square, R)\n",
    "plot_the_polygon(rotated_square)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And finally, we can \"reflect\" the square about the y-axis:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ODsq01TWz09z9h727V0h6Kct8wFLNZlPNZjN0DJRQ1tf4vmRmp0s6IOlVSX+TPRLQ1X/r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\nPgDJofgAJIfiA5Cc/weDn5Nh0/Kq3AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1569ff9c710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "reflected_square = transform_the_polygon(unit_square, Re_y)\n",
    "plot_the_polygon(reflected_square)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part 2: Commutativity of Operations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "The next natural question to ask is the following: Does the *order* in which you apply these operations matter?\n",
    "\n",
    "a) Let's see what happens to the unit square when we rotate the matrix by 60 degrees, and then reflect it along the y-axis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# As the name indicates, R_60 rotates the matrix by 60 degrees.\n",
    "rotation_angle = np.pi/3 # 60 degrees in radians\n",
    "R_60 = np.array([[np.cos(rotation_angle), -np.sin(rotation_angle)],\n",
    "                 [np.sin(rotation_angle), np.cos(rotation_angle)]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1569ff9c198>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "reflected_after_rotated_square = transform_the_polygon(unit_square, R_60)\n",
    "reflected_after_rotated_square = transform_the_polygon(reflected_after_rotated_square, Re_y)\n",
    "plot_the_polygon(reflected_after_rotated_square)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b) Now, let's reflect *before* rotating."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x156a23e7400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rotated_after_reflected_square = transform_the_polygon(unit_square, Re_y)\n",
    "rotated_after_reflected_square = transform_the_polygon(rotated_after_reflected_square, R_60)\n",
    "plot_the_polygon(rotated_after_reflected_square)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also scale the square: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x156a2518fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# S is a \"scalar matrix.\"\n",
    "S = np.array([[2, 0],\n",
    "              [0, 2]])\n",
    "scaled_square = transform_the_polygon(unit_square, S)\n",
    "plot_the_polygon(scaled_square)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}