Notebooks/MemberNBS/LinearAlgebra/Matrix as a Function & Plotting Vectors.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Matrix as a Function & Plotting Vectors\n",
    "** October 2017 **\n",
    "\n",
    "** Andrew Riberio @ [AndrewRib.com](http://www.andrewrib.com) **"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One of the fundamental concepts that is key to understanding and working with linear algebra is the notion that a matrix represents a linear transformation aka a linear function. This notebook will show you why that is.\n",
    "\n",
    "**Questions explored by this notebook**\n",
    "* Why is linear algebra called 'linear' algebra.\n",
    "* How does a matrix represent a linear function?\n",
    "* What is the geometric interpretation of linear funcitons?\n",
    "* How do we plot vectors using matplotlib?\n",
    "\n",
    "** Resources **\n",
    "* https://matplotlib.org/devdocs/api/_as_gen/matplotlib.axes.Axes.quiver.html"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Libraries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 214,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrices as Linear Functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A matrix defines a linear function. A matrix multiplied by a vector is equivalent of applying a linear function. A matrix multiplied by another matrix is equivalent to function composition. Below we have a matrix *a* and it's equivalent function defintion *fn*. We show here how they are equivalent. See the notebook *Linear Vs. Non-Linear Functions* for why we can define a linear function as a matrix. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 215,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAyBAMAAAAOzY77AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhAimburRN3N\ndmbBWFV7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABKklEQVQ4EWOQ//+JAQtg+v9fgEHYxRWLFAOr\ni7MAgwg2GZAYC0QuvaIFUwVEjnUCg/4CZElmUbg+zgAGRgEkucjKr3A5xg0MnF+Q5Bh4EXI8X3HL\nAbVwg1TCAZI+oNh6BbgEkIEqtxNZClWOvQC33HUUKRR9zAUMaciyyPa5MzBcwyHHtkmpcgMOOZ7/\n//8jy7FP/DMPHg/IeuBsaBzB+ciMUTnk0ACx2co7HBDhaRiALB/JwPQLJsc6fT6K3FQGBkGYHAPD\nfRQ5aQaG+Q6wvIImdz4AtxzQ7v0BOPQxMHAB8zIs/lDtA2aBCbjlyoHG4tDHW4BbzpyBLQGHPpYC\nBl5cctlK6rPh9lXKn3wAtAEGzv///xsuBxNEoWHuRBGEcgZADl8ZiadsBQBT61L1iH8aXAAAAABJ\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 1\\\\1 & 2\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2  1⎤\n",
       "⎢    ⎥\n",
       "⎣1  2⎦"
      ]
     },
     "execution_count": 215,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.init_printing()\n",
    "def fn(x,y):\n",
    "    return sp.Matrix( [2*x + 1*y, (1*x+2*y)] )\n",
    "\n",
    "a = sp.Matrix([[2,1],[1,2]]) # 2 * 2\n",
    "b = sp.Matrix([2,2]) # 2 * 1\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 216,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAyBAMAAACqpzYoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhB2zUSZuyJm\n3auNZPx4AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAiklEQVQoFWOQ//+JAQT0/39kEHZxBbNDXAwZ\nRMAsEOEIYSslOcDY4Q2cBTD2SgauBVA2x2+gaiibSwDB5i9RmwETj69hYDSAqon/ysApCmXzX2Dg\n+QFl8xYw8PyBsvkEEOIsQPVAYyHuecQQCDOHgWnmE5j5YFthaiAcqPvBHDqxR1BYIacZpLQEACLR\nO/GFmwPNAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}6\\\\6\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡6⎤\n",
       "⎢ ⎥\n",
       "⎣6⎦"
      ]
     },
     "execution_count": 216,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a * b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 217,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAyBAMAAACqpzYoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhB2zUSZuyJm\n3auNZPx4AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAiklEQVQoFWOQ//+JAQT0/39kEHZxBbNDXAwZ\nRMAsEOEIYSslOcDY4Q2cBTD2SgauBVA2x2+gaiibSwDB5i9RmwETj69hYDSAqon/ysApCmXzX2Dg\n+QFl8xYw8PyBsvkEEOIsQPVAYyHuecQQCDOHgWnmE5j5YFthaiAcqPvBHDqxR1BYIacZpLQEACLR\nO/GFmwPNAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}6\\\\6\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡6⎤\n",
       "⎢ ⎥\n",
       "⎣6⎦"
      ]
     },
     "execution_count": 217,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fn(2,2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting Vectors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Due to linear algebra having such a strong geometric interpretation, it's helpful to have the tools to visualize what's going on. Here we will discuss how to plot vectors using matplotlib. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAD8CAYAAACMwORRAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1ceb648bd68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# [start_loc_x, start_loc_y, x_magintute, y_magintude]\n",
    "vects = np.array([[0, 2, 2, 0], \n",
    "                  [0, 2, 0, 2],\n",
    "                  [0, 4, 2, 0],\n",
    "                  [2, 2, 0, 2]])\n",
    "\n",
    "# Unpack vects into their respective tupules.\n",
    "X_LOC, Y_LOC, X_MAG, Y_MAG = zip(*vects)\n",
    "\n",
    "plt.figure()\n",
    "\n",
    "# GCA = Get current axes.\n",
    "ax = plt.gca()\n",
    "ax.quiver(X_LOC, Y_LOC, X_MAG, Y_MAG, angles='xy', scale_units='xy', scale=1)\n",
    "\n",
    "# Set the scale of the tics on the x and y axes. \n",
    "ax.set_xlim([-1, 10])\n",
    "ax.set_ylim([-1, 10])\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear Transformations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this section we will define a rectagle by a series of points, then we will define a linear transformation, a matrix, that transforms these points, thereby transforming the defined rectangle. First we will start by defining a function, *plotRect*, which plots a rectangle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 219,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plotRect(lb,lt,rt,rb):\n",
    "    \n",
    "    plt.figure(figsize=(8, 8))\n",
    "    ax = plt.gca()\n",
    "    \n",
    "    # Setup grid and axes. \n",
    "    ax.axhline(linewidth=2.0, color=\"black\")\n",
    "    ax.axvline(linewidth=2.0, color=\"black\")\n",
    "    ax.grid(color='b', linestyle='-', linewidth=1)\n",
    "    \n",
    "    ax.set_yticks(range(-10,11))\n",
    "    ax.set_xticks(range(-10,11))\n",
    "\n",
    "    # Set the scale of the tics on the x and y axes. \n",
    "    ax.set_xlim([-10, 10])\n",
    "    ax.set_ylim([-10, 10])\n",
    "    \n",
    "    # Plot square based on points. \n",
    "    # [x1,x2],[y1,y2]\n",
    "    plt.plot([lb[0],lt[0]], [lb[1],lt[1]], 'k-', lw=2)\n",
    "    plt.plot([lt[0],rt[0]], [lt[1],rt[1]], 'k-', lw=2)\n",
    "    plt.plot([rt[0],rb[0]], [rt[1],rb[1]], 'k-', lw=2)\n",
    "    plt.plot([rb[0],lb[0]], [rb[1],lb[1]], 'k-', lw=2)\n",
    "\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we will define a rectangle by a series of points, (lb,lt,rt,br), and plot it using our above function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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2nGb1FOCd0+VLgR/fZM7BFLC2J3834FMzZpMkwJOAC2aK/DngpVX1VYCqunGm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\nb0mSBvP/AWdoT5oVmMbJAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1ceb64ac7f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Left bottom point\n",
    "lb = sp.Matrix([0,0])\n",
    "# Left top point\n",
    "lt = sp.Matrix([0,4])\n",
    "# Right top point\n",
    "rt = sp.Matrix([4,4])\n",
    "# Bottom right point\n",
    "\n",
    "rb = sp.Matrix([4,0])\n",
    "\n",
    "plotRect(lb,lt,rt,rb)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's definine a funciton that applies a 2 * 2 matrix to four points that we will use to transform the rectangle defined by the points above. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 221,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def transform(m,lb,lt,rt,rb):\n",
    "    lb_o = m * lb\n",
    "    lt_o = m * lt\n",
    "    rt_o = m * rt\n",
    "    rb_o = m * rb\n",
    "    \n",
    "    return [lb_o,lt_o,rt_o,rb_o]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will now define a matrix which we we will use to transform the rectangle. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 222,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAyBAMAAAAOzY77AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhCrmXbNZt0i\nu0TKQvJ9AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABKElEQVQ4EWOQ//+JAQtg+v9fgEHYxRWLFAOr\ni7MAgwg2GZAYC1Y51qJFDgg5wwCQShjgNGDdA5NjTc9HkVvIwKAJk2Ng6EeRu8bAYB8Atw9V7hsD\nw3sD7HKs34FyDdjleICB4a+AQw6oD5ccPjMZgG6xx+EWhrsMDOtx+QHo90pcfuc2YAXGADSsy+Sn\nLYAFJpBm1WhzgMshiSOY2OMIIj8qxxCupYIIKxgLGi6TGaIPwIRANLMokIDIcQkwME1Akgsr+wqX\n401gYAbx4IADIcefwMD3GS4BZCDJxRcw8P3AIXdegYEFJfMi6TtfgFsOn5lAtzDjcgvvBQYuXH4A\n+p0d2e/IfmDYzBDogMMPDEEdKchS7Ik/s2DhiSyOYA+y9ImvjMRTtgIAz1JTjnSjBSoAAAAASUVO\nRK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0\\\\0 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  0⎤\n",
       "⎢    ⎥\n",
       "⎣0  1⎦"
      ]
     },
     "execution_count": 222,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# M is an identity matrix. \n",
    "m = sp.Matrix([[1,0],[0,1]])\n",
    "m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 223,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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jeUuSNBjLW5KkwbSVd5InJrkqydeTrHSNI0nSsunc894P/BjwzsYxJElaOm0fCVpVVwMk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\nb0mSBvP/AWdoT5oVmMbJAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1ceb6657c18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lb_t,lt_t,rt_t,rb_t = transform(m,lb,lt,rt,rb)\n",
    "plotRect(lb_t,lt_t,rt_t,rb_t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we see, the identity matrix preserves the original size of the square. Let's now play around with different matricies to see how they spatially transform the rectangle. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 224,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAyBAMAAAAOzY77AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nPRwyQlPKL984+P0/dj77/SP7/Ft/ABNMXXM7mV0CAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 0\\\\0 & 2\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2  0⎤\n",
       "⎢    ⎥\n",
       "⎣0  2⎦"
      ]
     },
     "execution_count": 224,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = sp.Matrix([[2,0],[0,2]])\n",
    "m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 225,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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WJGkwlrckSYOxvCVJGozlLUnSYNrKO8kTk1yZ5BtJVrrGkSRp2XTueR8Afgx4Z+MYkiQt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\npMFY3pIkDcbyliRpMP8fPfFRNjn+sLwAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1ceb65d0b70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lb_t,lt_t,rt_t,rb_t = transform(m,lb,lt,rt,rb)\n",
    "plotRect(lb_t,lt_t,rt_t,rb_t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we see, changing the values on the diagonal scales the rectangle. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 226,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAAyBAMAAAATs7BgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhCrmXbNZt0i\nu0TKQvJ9AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABRUlEQVQ4EWOQ//+JAR/Q//+RQdjFFZ8ShhAX\nQwYRvCpAko641LAWLXIAKQACqBrDAAgXQXIasO6B8sBqWNPzMdQsZGDQRFbDwNCPoeYaA4M9VBBq\nF6aabwwM7w0gBuFSw/odqKYBvxoeYMD6KxBQAzSHkBpi7GIAutmegJsZ7jIwrCfkd2AYVkKcDIsL\nzPDhNmCFxTYkfMrkpy2AaoJRrBptDlA2NAxhMljpUTVYgwUuSN3wCddSgZuMyYDYNZkh+gCmHAMD\nsyhIFKyGS4CBaQIWNWFlX+FqeBMYmME8dHUcCDX8CQx8n9HlQXwkNfEFDHw/CKg5r8DAgrUwQzLn\nfAFhNeh2Re4GgSko7gG6mZmQm3kvMHAR8jswDNmxhSGy3xk2MwQ6EPA7Q1BHCjYl7Ik/s4Di1E0/\n2CxCiNHXLmLqJiLqOABOc2CwB6TLmwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0\\\\0 & -1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  0 ⎤\n",
       "⎢     ⎥\n",
       "⎣0  -1⎦"
      ]
     },
     "execution_count": 226,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = sp.Matrix([[1,0],[0,-1]])\n",
    "m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 227,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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m9TTgndPly4Af32TOoRSwtid/F+BTM2aTJMATgQtnivw54CVV9VWAqrpxptxu\nrwSezWJ9z6KqvrRu8di5sqvq0qq6eVp8N4tzHcziMI+XzXgQ8JGq+lhV3QS8nsXjbsuq6p3A382R\ndVDup6vqvdPlvweuZoYzONbCl6fF201fs/w+JDkJ+BHgNXPkdUtyFxZF81qAqrqpqr4w8zBnAx+t\nqo/PlLcLuGOSXSyKdq7t8HcDV1TVV6bH9DtYfKbHht3CY+IcFn8oMf37b44k6zZR3ocx52lWr+Kf\nNkpP4JtPIrNVzwJenuSTwCuA82bMBjgL+GxVfXimvNOAs5JckeQdSR44U+6ap0+Hiy844kNAtyLJ\nOcANVfX+OfIOyn7x9H/374HnzZ3P4kjPXzTkzmHoUxknOQX4PhZ7yXPkHTU9PXUjcFlVzZIL/CaL\nPzy/PlPeegVcmmRfFmesnMOpwOeA35kO9b8mybEzZa85l5l2SKrqBhbb3k8Anwa+WFWXzpHNYq/7\nrCR3T3IMi6Orc/bHvarq09PlzwD3OpI7bfVTxY5Ymk+zemtjsNiA/laS57I4icxNM2afDfxiVb0p\nyRNZ/LV6RGemO8L18iQ2+Et+K/PdBdwNeDDwQOANSb6zpuM2W8z+H8CLWGxQXsTicP/TZsj9FRaH\nojfs1tZxVZ0PnJ/kPODpwPPnyJ1ucz5wM/D7c855I1k7VZJvB94EPOugIyibVovzUZw5vUbhoiRn\nVNWWnrNP8qPAjVW1L8nD55jnQR5aVTckOR64LMk10x7eVuxicXj3GVV1RZJXsTik+9ytThYgydHA\n45hpR2faSTiHxR8dXwD+KMmTq+p1W82uqquTvJTF06P/AOwHWs5bUlWV5MiO9szxnMBcX/zz57zP\nA85bt3wJ8JAZxjkNeM+M8/4i//S2uwBfmjF7F/BZ4KQZM98KPGLd8keBezb8f57CDM93At/DYk/o\nuunrZhZ/YZ8w83x3zzHfdXlPBd4FHDP3up3yv+nxssmMhwCXrFv+psfcbeV34BC5t5u2B7/UsW6n\nMZ7HDK/fAP4riyMa17HYs/oK8LqmOb9gpjmfAFy3bvks4M9mnOc5wKUz5j0BeO265Z8EfrtpHf8a\n8J+2cP9vekwA1wInTpdPBK49kpzb+mHz2U6zOv1VSpJvA34VePVss1w8t/Kw6fIjgbkOb8NiD/6a\nqrp+xsw/YfGiNZKcxuLFKLOccD/JiesWH8/ikNOWVNUHqur4qjqlqk5hsSG8f1V9ZqvZSe6zbvEc\n4JqtZk7r7jiaAAABwElEQVS5j2FxmPRxVfWVOTKb/A1wnySnTntD57J43N1mTa8BeS1wdVX9xoy5\n91x7V0CSOwKPZobfh6o6r6pOmn53zwX+qqqevNVcgCTHJrnT2mUWR6fmeMx9BvhkkrUPzDibxRkz\n57Lho4m34hPAg5McM/1+nM3itRCzWNcfu1k83/0Hc2WzeLw9Zbr8FODIjqx1/GWyib9EHs9ig/xV\nFnuZ6/cEzmexZ3gt0yu6NznGM1m8KvVDwEuY9pRnmv9DgX0sXql7BfCAGbN/F/jZmdf30cDrWDzI\n3ws8csbs/wV8ALhy+qU8seH35Trme7X5m6b1cCXwp8C9Z8r9CIvnkvdPX7O8in3KvsXHyybzHjs9\nLj7K4rD8XPO8kMXzj/84zfenZsp9KIunZa5ct34fO0Pu9wLvm3IPMNO7Ow4a4+HM+GpzFu8SeP/0\nddXM/39nAnun9fEnwF1nyj0W+D/AXWZety9k8cfWgWk7dPsZs/83iz9e3g+cvYWcf/aYAO4O/CWL\nnb7LgbsdSZZnWJMkaTC39cPmkiTpIJa3JEmDsbwlSRqM5S1J0mAsb0mSBmN5S5I0GMtbkqTBWN6S\nJA3m/wPciVDwYUOXlwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1ceb67839b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lb_t,lt_t,rt_t,rb_t = transform(m,lb,lt,rt,rb)\n",
    "plotRect(lb_t,lt_t,rt_t,rb_t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we see, changing the signs on the diagonal translates the rectangle over an axis. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 236,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAyBAMAAAAOzY77AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhCrmXbNZt0i\nu0TKQvJ9AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA30lEQVQ4EWOQ//+JAQtg+v9fgEHYxRWLFAOr\ni7MAgwg2GZAYC1Y51qJFDgg5wwCQShjgNGDdA5NjTc9HkVvIwKAJk2Ng6EeRu8bAYB8Atw9V7hsD\nw3sD7HKs34FyDdjleICB4a+AQw6oD5ccPjMZgG6xx+EWhrsMDOtx+QHo90pcfuc2YAXGADSsy+Sn\nLYAFJpBm1WhzgMshiSOY2OMIIj8qhwgnZBYsXFDTLgOzKFARRA497YaVfYXLoaddBo5ROXAZgpZ2\n2RN/ZsHCEzn0EWxYPCBEEKwBkMNXRuIpWwGwM0uvUPbbnQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0\\\\1 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  0⎤\n",
       "⎢    ⎥\n",
       "⎣1  1⎦"
      ]
     },
     "execution_count": 236,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = sp.Matrix([[1,0],[1,1]])\n",
    "m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 237,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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FL3yBKVOmsMkmm3DdddcxYcL4yqyLsrmNW89/mV2tdVEyu2q5teyY19UXcxc9bB4RgyNi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\n+Ekg6P3O1vkAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1ceb68edb00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lb_t,lt_t,rt_t,rb_t = transform(m,lb,lt,rt,rb)\n",
    "plotRect(lb_t,lt_t,rt_t,rb_t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When we start changing the values on the counterdiagonal, we start skewing the rectangle. By changing the values on the diagonal and counterdiagonal, we can perform any linear transformation to our square. If we apply successive linear transformations, affine transformations, we can do things like rotation. Play around with the values for the matrix to see how different values result in different linear transformations of the rectangle. "
   ]
  }
 ],
 "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": "3.5.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}