{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "hide_input": true,
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    },
    "tags": [
     "remove-cell"
    ]
   },
   "outputs": [],
   "source": [
    "# for QR codes use inline\n",
    "# %matplotlib inline\n",
    "# qr_setting = 'url'\n",
    "# qrviz_setting = 'show'\n",
    "#\n",
    "# for lecture use notebook\n",
    "%matplotlib inline\n",
    "qr_setting = None\n",
    "#\n",
    "%config InlineBackend.figure_format='retina'\n",
    "# import libraries\n",
    "import numpy as np\n",
    "import matplotlib as mp\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import laUtilities as ut\n",
    "import slideUtilities as sl\n",
    "import demoUtilities as dm\n",
    "import pandas as pd\n",
    "from importlib import reload\n",
    "from datetime import datetime\n",
    "from IPython.display import Image\n",
    "from IPython.display import display_html\n",
    "from IPython.display import display\n",
    "from IPython.display import Math\n",
    "from IPython.display import Latex\n",
    "from IPython.display import HTML;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "slide_type": "subslide"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Linear Transformations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    },
    "tags": [
     "remove-cell"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/jpeg": "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",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {
      "image/jpeg": {
       "width": 650
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(Image(\"images/L7 F3.jpg\", width=650))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "slide_type": "subslide"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "So far we've been treating the matrix equation\n",
    "\n",
    "$$ A{\\bf x} = {\\bf b}$$\n",
    "\n",
    "as simply another way of writing the vector equation\n",
    "\n",
    "$$ x_1{\\bf a_1} + \\dots + x_n{\\bf a_n} = {\\bf b}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_number": 4
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "However, we'll now think of the matrix equation in a new way: \n",
    "\n",
    "We will think of $A$ as <font color=blue>\"acting on\" the vector ${\\bf x}$</font> to create a <font color=blue>new vector ${\\bf b}$.</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 5
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "For example, let's let $A = \\left[\\begin{array}{rrr}2&1&1\\\\3&1&-1\\end{array}\\right].$  Then we find:\n",
    "\n",
    "$$ A \\left[\\begin{array}{r}1\\\\-4\\\\-3\\end{array}\\right] = \\left[\\begin{array}{r}-5\\\\2\\end{array}\\right] $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 6
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "In other words, if ${\\bf x} = \\left[\\begin{array}{r}1\\\\-4\\\\-3\\end{array}\\right]$ and ${\\bf b} = \\left[\\begin{array}{r}-5\\\\2\\end{array}\\right]$, then $A$ _transforms_ ${\\bf x}$ into ${\\bf b}$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": [
     "remove-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/jpeg": "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",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {
      "image/jpeg": {
       "width": 650
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(Image(\"images/L7 F1.jpg\", width=650))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Notice what $A$ has done: it took a vector in $\\mathbb{R}^3$ and transformed it into a vector in $\\mathbb{R}^2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "How does this fact relate to the shape of $A$?   \n",
    "\n",
    "$A$ is $2 \\times 3$ --- that is, $A \\in \\mathbb{R}^{2\\times 3}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 8
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "This gives a __new__ way of thinking about solving $A{\\bf x} = {\\bf b}$.  \n",
    "\n",
    "To solve $A{\\bf x} = {\\bf b}$, we much \"search for\" the vector(s) ${\\bf x}$ in $\\mathbb{R}^3$ that are transformed into ${\\bf b}$ in $\\mathbb{R}^2$ under the \"action\" of $A$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "For a different $A$, the mapping might be from $\\mathbb{R}^1$ to $\\mathbb{R}^3$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": [
     "remove-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/jpeg": "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",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {
      "image/jpeg": {
       "width": 650
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(Image(\"images/L7 F2.jpg\", width=650))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "What would the shape of $A$ be in the above case?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Since $A$ maps from $\\mathbb{R}^1$ to $\\mathbb{R}^3$, $A \\in \\mathbb{R}^{3\\times 1}$.  \n",
    "\n",
    "That is, $A$ has 3 rows and 1 column."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "In another case, $A$ could be a square $2\\times 2$ matrix.  \n",
    "\n",
    "Then, it would map from $\\mathbb{R}^2$ to $\\mathbb{R}^2$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": [
     "remove-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/jpeg": "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",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {
      "image/jpeg": {
       "width": 650
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(Image(\"images/L7 F3.jpg\", width=650))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 9,
     "slide_helper": "subslide_end"
    },
    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We have moved out of the familiar world of functions of one variable:  we are now thinking about <font color = blue>functions that transform a vector into a vector.</font>\n",
    "\n",
    "Or, put another way, functions that transform multiple variables into multiple variables."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Transformations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Some terminology:\n",
    "\n",
    "A __transformation__ (or __function__ or __mapping__) $T$ from $\\mathbb{R}^n$ to $\\mathbb{R}^m$ is a rule that assigns to each vector ${\\bf x}$ in $\\mathbb{R}^n$ a vector $T({\\bf x})$ in $\\mathbb{R}^m$.   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 9,
     "slide_type": "subslide"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The set $\\mathbb{R}^n$ is called the __domain__ of $T$, and $\\mathbb{R}^m$ is called the __codomain__ of $T$.  \n",
    "\n",
    "The notation:\n",
    "\n",
    "$$ T: \\mathbb{R}^n \\rightarrow \\mathbb{R}^m$$ \n",
    "\n",
    "indicates that the domain of $T$ is $\\mathbb{R}^n$ and the codomain is $\\mathbb{R}^m$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 11
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "For $\\bf x$ in $\\mathbb{R}^n,$ the vector $T({\\bf x})$ is called the __image__ of $\\bf x$ (under $T$).  \n",
    "\n",
    "The set of all images $T({\\bf x})$ is called the __range__ of $T$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "hide_input": true,
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 12,
     "slide_helper": "subslide_end"
    },
    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": [
     "remove-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/jpeg": "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",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {
      "image/jpeg": {
       "width": 650
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# image credit: Lay, 4th edition\n",
    "display(Image(\"images/L7 F4.jpg\", width=650))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, the green plane is the set of all points that are possible outputs of $T$ for some input $\\mathbf{x}$.\n",
    "\n",
    "So in this example:\n",
    "   * The __domain__ of $T$ is $\\mathbb{R}^2$\n",
    "   * The __codomain__ of $T$ is $\\mathbb{R}^3$\n",
    "   * The __range__ of $T$ is the green plane."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 12,
     "slide_helper": "subslide_end",
     "slide_type": "subslide"
    },
    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": [
     "remove-cell"
    ]
   },
   "source": [
    "``` {toggle}\n",
    "Question Time! Q7.1\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 12,
     "slide_type": "subslide"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Let's do an example.   Let's say I have these points in $\\mathbb{R}^2$:\n",
    "\n",
    "$$ \\left[\\begin{array}{r}0\\\\1\\end{array}\\right],\\left[\\begin{array}{r}1\\\\1\\end{array}\\right],\\left[\\begin{array}{r}1\\\\0\\end{array}\\right],\\left[\\begin{array}{r}0\\\\0\\end{array}\\right]$$\n",
    "\n",
    "Where are these points located?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 15
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0 1 1 0]\n",
      " [1 1 0 0]]\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA+EAAAPfCAYAAACoycg7AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjguMCwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy81sbWrAAAACXBIWXMAAB7CAAAewgFu0HU+AAA8oklEQVR4nO3df7RXBZ3v/9fhl8QPRYEUBTUz0DCv+YO+pSb+wGbSxk6NmJqoy5lmYlrXWdm61rTqluO0mrpmjreanFLUGh1SoevgeK0UkaYJkqnU0MKfHCAFQkUU8XDO9w+uZ7T4zee8P5zD47GWq42f/dn7XW04nyd7f/Zu6ezs7AwAAADQ7fo0ewAAAADYVYhwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAi/Zo9AAD0dM8//3zuuOOOzJ8/Pz/72c+yZMmSLF++PC+99FKGDRuWt771rXnve9+biy66KMOHD2/2uABAE7V0dnZ2NnsIAOjJfvjDH2bSpElbXG/EiBH5zne+k/e85z0FUwEAOyNnwgGgAcaMGZMTTzwxRx11VMaMGZNRo0alo6MjbW1tueWWW3LbbbdlxYoV+ZM/+ZPMnz8/hx9+eLNHBgCawJlwANhB69evT9++fTe7zsyZM9Pa2pok+cAHPpBbb721YjQAYCcjwgGgyKGHHpqHH344I0aMyPLly5s9DgDQBO6ODgBFBg8enCRZu3ZtkycBAJpFhANAgYULF+bnP/95kuSQQw5p7jAAQNO4MRsAdJMXX3wxS5Ysye23354vfelLWb9+fZLk4osv3uZttbW1bfb19vb2LF++PKNGjco+++yTfv38iAeAnZGf0ADQQNOmTcuFF164ydc/8YlP5Nxzz93m7Y4ZM2ar1128eHFGjx69zfsAALqfCAeAAkcccUT+8R//Me94xzuaPQoA0ETujg4ADfTss892XTr+0ksv5dFHH8306dMzY8aMvPnNb85Xv/rVnH766du83S1djr5s2bJMmDAhiTPhALAzE+EAUODGG2/M+eefn5aWlnz729/OBRdc0NDtt7W1dV2yLsIBYOfl7ugAUOC8887LmWeemY6OjnzsYx/LqlWrmj0SANAEIhwAipxxxhlJkjVr1uTf/u3fmjwNANAMIhwAiowcObJr+cknn2ziJABAs4hwACiyZMmSruUhQ4Y0cRIAoFlEOAAU+d73vte1/La3va2JkwAAzSLCAWAHTZs2LWvXrt3sOldeeWXuuOOOJMmBBx6Y4447rmI0AGAn4xFlALCDDjzwwKxevTof/OAHc9xxx+XNb35zhgwZktWrV+eBBx7Id7/73fz4xz9OkgwYMCCzZs3KKaec0tAZPKIMAHoGEQ4AO+jAAw/cqhutjR49Otdee20mTZrU8BlEOAD0DP2aPQAA9HQ/+tGP8sMf/jD33HNPFi5cmKeffjorV67MwIEDs/fee+eII47I6aefnsmTJ2fQoEHNHhcAaCJnwgGgF3AmHAB6BjdmAwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIB4AGWLBgQb7whS/kj//4jzNmzJjstttuGTJkSMaOHZsLLrgg9913X7NHBAB2Ai2dnZ2dzR4CAHqyE044IXPmzNnieuedd16+9a1vZcCAAQ2foa2tLWPGjEmSLF68OKNHj274PgCAHdev2QMAQE+3ZMmSJMm+++6bM888M8cff3z233//rF+/Pj/5yU9yxRVXZMmSJbnxxhvT3t6ef/7nf27yxABAszgTDgA76PTTT8+UKVPywQ9+MH379v2D11esWJFjjz02v/71r5Mkc+bMyfHHH9/QGZwJB4CewXfCAWAH/eu//msmT5680QBPkhEjRuSKK67o+vUtt9xSNRoAsJMR4QBQYOLEiV3Ljz76aPMGAQCaSoQDQIF169Z1Lffp48cvAOyq3JgNAArce++9XcuHHHLINr+/ra1ts68vW7Zsm7cJANQT4QDQzTo6OvLFL36x69eTJ0/e5m28etM1AKBncz0cAHSzK6+8MvPmzUuStLa25uijj27yRABAs3hEGQB0o3vvvTennHJK2tvb88Y3vjG//OUvs/fee2/zdrbmcvQJEyYk8YgyANiZuRwdALrJQw89lNbW1rS3t2e33XbL9OnTtyvAk4hqAOglXI4OAN3g8ccfz6mnnppVq1alb9++uemmm3LCCSc0eywAoMlEOAA02NKlS3PKKadk6dKlaWlpybXXXpvW1tZmjwUA7AREOAA00IoVKzJp0qQ89thjSZKrr746U6ZMafJUAMDOQoQDQIM899xzec973pNf/epXSZIvfvGL+au/+qsmTwUA7ExEOAA0wIsvvpjTTjstCxYsSJJ8+tOfzqWXXtrkqQCAnY0IB4AdtG7durS2tubHP/5xkuTiiy/O5Zdf3uSpAICdkUeUAcAOOvvss3PXXXclSU466aRcdNFFefDBBze5/oABAzJ27Niq8QCAnUhLZ2dnZ7OHAICerKWlZZvWP+CAA/LEE080dIa2traMGTMmSbJ48WLPFQeAnZTL0QEAAKCIy9EBYAe5qAwA2FrOhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABTp1+wBAIAGWL36v5Z//etk9903/AMN9PzLz6ft+ba8sO6FDBkwJKN3H53dd3OcAWyLls7Ozs5mDwEAPd0zzzyTefPmZd68eZk/f37mz5+flStXJknOP//8TJs2rfE77exMZs9Ovva1tM2YkTEdHUmSxUlG9+2btLYmU6cmEycmLS2N3z+7hM7Ozsx+Yna+Nv9rmfnwzKzvXN/1Wt+Wvmk9tDVTj56aiQdOTIvjDGCLRDgANMDm4qNbInzBgmTKlOShh5IkbUnG/L+XFicZ/dp1x49PbrghOfLIxs5Ar7dg2YJMmTElDy1/aIvrjh85Pje03pAjRznOADbHd8IBoMHGjBmTU089tft28IMfJO9+d1eAb9FDD21Y/wc/6L6Z6HV+8OgP8u7r3r1VAZ4kDy1/KO++7t35waOOM4DNEeEA0ACf/exnc/vtt+e3v/1tnnrqqXzzm9/snh0tWLDhMvM1a7btfWvWbHjfggXdMxe9yoJlC9L6L61Z88q2HWdrXlmT1n9pzYJljjOATXFjNgBogM9//vPdv5POzg2XoG9rgL9qzZrk/POTX/7Sd8TZpM7OzkyZMWWbA/xVa15Zk/Nnnp9f/uUvfUccYCOcCQeAnmL27K2/BH1THnwwuffehoxD7zT7idlbfQn6pjz4zIO590nHGcDGOBMOAD3F17/ekM2svfLrWT1+YkO2Re9z5Y8bc5x9ff7XM/HAiQ3ZFkBvIsIBoAdYsnBh9p0xI424uLff/7ktb37j81kdz3fm9+z2fHLpjIZcK3nbwtvy/MvPe444wO9xOToA9ACT3vrWtKxfv+UVt0K/rM9+WdKQbdHL7N6W9GnMcba+c32WPO84A/h9IhwAeoAhDd7e0Kxu8BbpFQa80NDNrV7nOAP4fSIcAHqA//OjHzV0e6sztKHbo5dY19i/7hk6wHEG8Pt8JxwAeoB9jj466ds3acAl6R19++WG/7tf1g9uwGD0KmvaR+c9d/fN+s4dP8769emX/XbfrwFTAfQuIhwAeoLdd09aW5NbbtnhTa2Z1Jo9xrhZFn9or+yeU0a35v8u3vHjrPWQVjdlA9gIl6MDQE8xdWpDNvPsOY3ZDr3TOW9pzPEx9RjHGcDGiHAA6CkmTkzGj9+hTbw89rC8NOGExsxDr/SON07MW/bYsePssDcelhMOcJwBbIwIB4CeoqUlueGGZPD2fZm7Y9Dg/PaL12/YDmxCS0tL/v7/uyGD+m3fcTa4/+Bc//7r0+I4A9goEQ4APcmRRyYzZmxziHcMGpyl/3tGXh5/ZDcNRm8yfq8j87+Pm7HNIT64/+DMOGtGjhzlOAPYFBEOAD3NpEnJnDlbfWn6y2MPy+LvzMmLx07q5sHoTY4dNSnfOXnOVl+aftgbD8ucC+dk0psdZwCb4+7oANAAc+fOzaJFi7p+vWLFiq7lRYsWZdq0aa9b/4ILLtixHR55ZPLAA8m99+blr3wt7bfflqSj6+WOvv2yZlJrnj1n6obvgLs0mO0wfq8jc/sfP5B5z9yb6x76Wu5ZNiPp81+PL+vXp19aD2nN1GOm5oQDTnAJOsBWaOns7Oxs9hAA0NNdcMEFuf7667d6/Ub++F2+PHnTGxdmTd6aJHlz7s5NPzzKY8hoqN/9LnnnxOeToUuS3VYnLw/Noz/fLwft5zgD2BbOhANAL7AmQ7uWH81bsn6wMKIbvLz7hn/+n6EDmjgLQA/lO+EA0ADTpk1LZ2fnVv8DAOyaRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAECRfs0eAADYcQMHrs7atRuW99vv1+ns3D3J7k2diV5oUHsy8uXkDeuTl/pm9frdMtLHSYBt0tLZ2dnZ7CEAoDd56qmn8g//8A+ZNWtWnnrqqey22245+OCDM3ny5EydOjWDBg1qyH46Ozvz7LOz8/jjX8uvfz0jZ5/dkSSZPj0ZObJvBg5szeDBUzNgwMS0tLQ0ZJ/sejo7O/PTdc9m2qolueeVFUnf/3qtb5LWkSMzdd99M3HYMMcZwFYQ4QDQQLNmzcq5556b5557bqOvjxs3LnfccUcOOuigHdrP6tULsnDhlLz44kNJkuXLk8mTN7y2IcL/a91+/cZn2LAbMmDAkTu0T3Y9D61bnUufXZjftL+4xXXHDxqUGw49NEcOHVowGUDP5TvhANAgv/jFLzJ58uQ899xzGTJkSP7u7/4u//7v/54f/ehH+fM///MkySOPPJLTTjstL7zwwnbv53e/+0H+8z/f3RXgW9Le/lBWrnx31q79wXbvk13Pj9f+Lh9e+Z9bFeBJ8tCLL+bd//mf+cHvftfNkwH0bCIcABrkr//6r/Piiy+mX79+ueuuu/I3f/M3eec735mTTjop11xzTb70pS8lSR5++OF85Stf2a59rF69IA8+2JqOjjXb9L7OzjVZtao169Yt2K79smt5aN3qfGzVg3mxs2Ob3remoyOtDz6YBatXd9NkAD2fCAeABpg/f35mz56dJLnooovyzne+8w/WueSSS3LooYcmSb761a/mlVde2aZ9dHZ2ZuHCKdsc4P/1/jV59tnz45tobE5nZ2cufXbhNgf4q9Z0dOT8hx92nAFsgggHgAaYOXNm1/KFF1640XX69OmTKVOmJElWrVrVFe1b69lnZ2/1Jeib0t7+YNatu3eHtkHv9tN1z271Jeib8uCaNbn32WcbMxBAL+OZEgDQAPfdd1+SZPDgwTnqqKM2ud4JJ5zQtTx37txMmjRpq/exdOnXt3/A11i16usZMGBiQ7ZF7zNt3dKGbOfrS5dm4p57NmRbAL3JThvh7e3t+e1vf9vsMQBgqzz44INJkv3333+zP7+GvubO0ffff3/a2tq2avtLlvwmTz112yZfX7ly48sbs379rbnwwoVZu9ZdrPk9A9uT6371useQba9bly/PwsGDM7TfTvtxEyBJss8++6Rf4Z9VO+0jytra2jJmzJhmjwEAAEAvtnjx4owePbpsfzvtd8KXLVvW7BEAAADo5arbc6e9PmjkyJFdy/PmzcuoUaOaOA291bJlyzJhwoQkjjO6j+Os91u6dGne8Y53JEk+8IEP5Kqrrtrs+hMmTMiyZctywAEHZO7cuVu1jyef/HGWLv3QJl9fuTL56Ec3LH/jG8nw4Zvf3l//9d1ZuvQtW7VvdiH7vZhc+YuGbe7u//bf8pZBgxq2PXYNfm5S4bXH2Wvbs8JOG+GvvSZ/1KhRpZcHsGtynFHBcdY77bbbbl3L/fv33+L/x+vXr0+y4fvhW3s87LPPH2fu3L5J1m9x3eHDk81/nuiX++8/Kn377r5V+2bXsXp9e8Y+vHQrjrIt69fSkqMOOii7+044O8DPTSpUfh882YkjHAB6itfebO2FF17Y4vpr1mx4zveQIUO2eh/9+u2ekSNbs3z5Lds+4O8ZObI1++wjwPlDI9MvrStH5pbly3d4W60jRghwgI3Yab8TDgA9xcCBAzNixIgk2eLdzletWtUV4dt6A9J99526fQN203bonabuu+9OtR2A3kaEA0ADHHrooUmSRYsWpb29fZPrPfzww3/wnq01bNjEDBo0fvsG/H8GDz4sw4adsOUV2WVNHDYs43fwe9yHDR6cE4YNa8xAAL2MCAeABjjuuOOSbLjU/P7779/kevfee2/X8rHHHrtN+2hpacmhh96QPn0Gb9eMffoMziGHXJ+Wlpbtej+7hpaWltxw6KEZ3Gf7PiYO7tMn1x9yiOMMYBNEOAA0wPvf//6u5euuu26j63R0dOSGG25IkgwbNiwnnnjiNu9n6NAjc9hhM7Y5xPv0GZzDDpuRoUOP3OZ9sus5cujQzDjssG0O8cF9+mTGYYflyNfcJwGA1xPhANAAEyZMyPHHH58k+fa3v52f/OQnf7DOFVdckYULFyZJLr744vTv33+79rXXXpPy9rfP2epL0wcPPixvf/uc7LXXpO3aH7umSXvtlTlvf/tWX5p+2ODBmfP2t2fSXnt182QAPZtbVgJAg1x11VU59thj89JLL+XUU0/N3/zN3+TEE0/MSy+9lJtvvjnXXHNNkmTs2LG55JJLdmhfQ4cemWOOeSDPPntvli79WpYvvy1JR9frLS39MmJEa/bdd2qGDTvBpcFslyOHDs0DxxyTe599Nl9bujQzli9/3ePL+rW0pHXEiEzdd9+cMGyY4wxgK7R0dnZ2NnsIAOgtbr/99nz4wx/O888/v9HXx44dm1mzZuXggw9u6H6feGJh3vSmtyZJHnnk7hx00FHp189jyGis59vbs+Tll7N6/foM7ds3++22m8eQAWwjEQ4ADfbkk0/mqquuyqxZs9LW1pYBAwbk4IMPzplnnpmPfexjGbSDd57emLa2tq5Hni1evDijR49u+D4AgB0nwgGgFxDhANAzuDEbAAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABF+jV7gO01a9aszJ8/P/Pnz89jjz2W5cuX57nnnsuQIUNy0EEHZeLEifnIRz6ScePGNXtUerCnnnoqs2bNyuzZs/Pzn/88bW1tWb9+fUaMGJGjjjoqH/rQh3LmmWemX78e+1uJncALL7yQBQsWZN68eZk3b17mz5+fJ554IklywAEHdC3D5ixZsqRr+S1veUve8IY35OCDD87kyZMzderUDBo0qInT0ZM988wzr/vzaf78+Vm5cmWS5Pzzz8+0adOaOyC9woIFC3LnnXfmvvvuy4MPPphnnnkm/fv3z7777pt3vetdueiii3L88cc3e0x6sOeffz533HFH5s+fn5/97GdZsmRJli9fnpdeeinDhg3LW9/61rz3ve/NRRddlOHDh3frLC2dnZ2d3bqHbtDe3p7+/ftvcb3+/fvnsssuyyc/+cmCqehtPvvZz+byyy/Pln6LHH300bn11luz//77F01Gb3PiiSdm9uzZG31NhLM1Zs2albPPPjurV6/e6Ovjxo3LHXfckYMOOqh4MnqDlpaWTb4mwmmEE044IXPmzNnieuedd16+9a1vZcCAAQVT0dv88Ic/zKRJk7a43ogRI/Kd73wn73nPe7ptlh57+m6PPfbIxIkT8453vCMHHXRQRo0alUGDBmXp0qWZPXt2rr322jz33HP51Kc+lWHDhuUv//Ivmz0yPczSpUvT2dmZwYMHp7W1NSeffHLe8pa3ZODAgVm4cGH+4R/+oetv0k455ZQsWLAgQ4YMafbY9ECv/YuePffcM0cffXR+8pOf5IUXXmjiVPQUv/jFLzJ58uS8+OKLXf9u5syZGTp0aG6++eb80z/9Ux555JGcdtppmT9/vj+n2CFjxozJoYcemrvuuqvZo9CLvHolz7777pszzzwzxx9/fPbff/+sX78+P/nJT3LFFVdkyZIlufHGG9Pe3p5//ud/bvLE9FRjxozJiSeemKOOOipjxozJqFGj0tHRkba2ttxyyy257bbbsmLFivzJn/xJ5s+fn8MPP7xb5uiRZ8KTZP369enbt+8mX3/88cdz1FFHZdWqVRk5cmSWLVu22fXh91166aUZPnx4PvrRj2bo0KF/8Pr69etzzjnnZPr06UmSyy67LJ/5zGeqx6QXuOaaazJkyJBMmDAhBx98cJLkwAMPzJNPPulMOFv06pUUffv2zfr165MkixcvzujRo5MkX/7yl/M//sf/SJJ8/vOfz2c/+9mmzUrP9D//5//MMccck2OOOSZ77713nnjiibzpTW9K4kw4jXH66adnypQp+eAHP7jRz+srVqzIsccem1//+tdJkjlz5rg0nW22pX5MNvwldmtra5LkAx/4QG699dZumaXHRvjW+OhHP5p//Md/TJI8+OCDGT9+fJMnordZuXJl9t1336xbty6HH354fvGLXzR7JHoJEc7WmD9/fiZMmJAkOffcc/Pd7343yesjvKOjI4cddlgWLlyYPffcM08//fRWfaULNkWE0wz/+q//mve9731Jkv/+3/97rrrqqiZPRG916KGH5uGHH86IESOyfPnybtlHr747+uDBg7uW165d28RJ6K2GDx/edZnKo48+2uRpgF3NzJkzu5bPOuusja7Tp0+fTJkyJUmyatWqTd5/AGBnNnHixK5ln7noTq82ZHf2Y6+N8Jdeeinf//73k2z4ADJ27NgmT0Rv9fLLLyfZcJwBVLrvvvuSbPjA8La3vW2T651wwgldy3Pnzu32uQAabd26dV3LPnPRXRYuXJif//znSZJDDjmk2/bTq47gV155JU899VRuvvnmvOtd78qiRYuSJBdeeOFGv9MLO+qZZ57JwoULk3Tvb1SAjXn1z5+DDz54s49KfO2fT6++B6Anuffee7uWfeaikV588cX85je/yVe+8pWceOKJXfdXufjii7ttnz327uiveu33kjbmlFNOyRVXXFE4EbuSL3/5y2lvb0+STJ48ucnTALuStWvXZsWKFUnS9f3vTdlzzz0zePDgrFmzJosXL64YD6BhOjo68sUvfrHr1z5zsaOmTZuWCy+8cJOvf+ITn8i5557bbfvvVWfCX2v48OG56aabcuedd2aPPfZo9jj0Qj/96U/z1a9+NcmGD8BTp05t7kDALuW1zwTfmseOvfodN4++A3qaK6+8MvPmzUuStLa25uijj27yRPRWRxxxRP7jP/4jX/7yl9PS0tJt++nxZ8L322+/PPDAA0mS9vb2LFmyJHfeeWe+/e1vZ+rUqXnsscfyqU99qlv/R2TX8/TTT+dP//RP097enpaWllx//fUZNGhQs8cCdiGvvWHMgAEDtrj+brvtlmTDPVMAeop77703n/zkJ5Mkb3zjG/ONb3yjyRPRG7z//e/v+sucl156KY8++mimT5+eGTNm5Nxzz81Xv/rVnH766d22/249E/5qoOzoP5t79EX//v1z2GGH5bDDDssRRxyR0047LVdffXX+4z/+Iy0tLfn0pz+diy66qDv/a9JkFcfZa61evTqnnXZa2trakiRf+MIXctJJJ3Xjf0N2BtXHGWzJwIEDu5Zfe8OiTXn1JpJveMMbum0mgEZ66KGH0tramvb29uy2226ZPn169t5772aPRS8wbNiwroY85phj8qEPfSi33XZbbrjhhjz22GM544wzuvUzW6+9HP3www/P5ZdfniS57rrrctdddzV5InqDtWvX5owzzsj999+fJPn4xz/e9bezAJVee8PRrbnEfM2aNUm27tJ1gGZ7/PHHc+qpp2bVqlXp27dvbrrpptc96QG6w3nnnZczzzwzHR0d+djHPpZVq1Z1y3669XL0fv36NeQurKNGjdqu951xxhld39O95ZZbcuqpp+7wLOx8qo6z9vb2TJ48Offcc0+S5M/+7M/c9G8X0uw/z+D3DRw4MCNGjMiKFSu6rszZlFWrVnVF+JgxYyrGA9huS5cuzSmnnJKlS5empaUl1157bVpbW5s9FruIM844I9OnT8+aNWvyb//2bznnnHMavo9u/054Mx8hMHLkyK7lJ598smlz0P26+zjr6OjIeeedl9tvvz1JctZZZ+Wb3/xmt+6TnY9HorCzOfTQQ3Pfffdl0aJFXU9q2JiHH374de8B2FmtWLEikyZNymOPPZYkufrqqzNlypQmT8WupKIhe+3l6EmyZMmSrmWX37Ej/uIv/iI333xzkuT000/PjTfemD59evVvH6AHOO6445JsuNT81ZuUbsxrn6977LHHdvtcANvjueeey3ve85786le/SpJ88YtfzF/91V81eSp2NRUN2asr4nvf+17X8tve9rYmTkJP9vGPfzzf+ta3kiQnn3xybrnllvTv37/JUwFsuLvrq/7lX/5lo+t0dHTkhhtuSLLhRjQnnnhixWgA2+TFF1/MaaedlgULFiRJPv3pT+fSSy9t8lTsiioaskdG+MyZM7Ns2bLNrjNnzpxcdtllSTZ8l/Pss8+uGI1e5nOf+1yuvPLKJMm73vWufP/73+96zA9As02YMCHHH398knRdrfP7rrjiiq77GVx88cX+EhHY6axbty6tra358Y9/nGTDn1Wv3mAZGmXatGmve7znxlx55ZW54447kiQHHnhg1xVnjdYjnxM+c+bMnHXWWTnttNNy8sknZ/z48Rk2bFhefvnlPProo7n99tszffr0dHR0JEk+85nPZNy4cU2emp7m6quvzuc///kkG55H/6UvfSmPP/74Zt8zbtw4H3DZZosWLcrcuXNf9+9evdv1Cy+88AePyPijP/qj7LPPPlXjsZO76qqrcuyxx77u+d8LFizIb37zm9x888255pprkiRjx47NJZdc0qwx6cHmzp2bRYsWdf16xYoVXcuLFi36gz+jLrjggqLJ6C3OPvvsricZnXTSSbnooovy4IMPbnL9AQMGZOzYsVXj0Ut87nOfyyWXXJIPfvCDOe644/LmN785Q4YMyerVq/PAAw/ku9/9btdfBA0YMCD/9E//lH79uieXWzo7Ozu7Zcvd6IILLsj111+/xfXe8IY35G//9m996GC7TJw48XXfo9wajz/+eA488MDuGYhea9q0abnwwgu3ev177rknEydO7L6B6HFuv/32nHPOOZt8VNnYsWMza9asHHzwwcWT0Rts7eeuV/XAj5Y0WUtLyzatf8ABB+SJJ57onmHotQ488MCtutHa6NGjc+2112bSpEndNkuPPBP+v/7X/8p73/ve3H333VmwYEF++9vf5plnnkmfPn2y1157Zfz48TnppJMyZcoUjwMCoNd73/velx/84Ad55zvfmWTD48sGDhyYgw8+OGeeeWY+9rGPZdCgQU2eEgCa50c/+lF++MMf5p577snChQvz9NNPZ+XKlRk4cGD23nvvHHHEETn99NMzefLkbv+Z2SPPhAMAr9fW1tb1DPDFixdn9OjRTZ4IANiYHnljNgAAAOiJRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFOnX7AEAoDd44YUXsmDBgsybNy/z5s3L/Pnz88QTTyRJDjjggK5lAGDXJsIBoAHe9773Zfbs2c0eAwDYybkcHQAaoLOzs2t5zz33zKRJkzJkyJAmTgQA7IxEOAA0wDnnnJPvfve7+c1vfpPf/e53ueuuuzJ8+PBmjwUA7GRcjg4ADfCRj3yk2SMAAD2AM+EAAABQRIQDAABAEZejA0AP0NbWttnXly1bVjQJALAjRDgA9ABjxoxp9ggAQAO4HB0AAACKOBMOAD3A4sWLN/v6smXLMmHChKJpAIDtJcIB2GW0t7enf//+O7yd6667LhdccMGOD7QNRo8eXbo/AKB7uBwdAAAAijgTDsAuo1+/flm4cOEOb2fUqFENmAYA2BWJcAB2KYccckizRwAAdmEuRwcAAIAiIhwAAACKiHAAAAAo4jvhANAAixYtyty5c1/371544YWu/5w2bdrrXvujP/qj7LPPPlXjAQA7CREOAA0wd+7cXHjhhRt9beXKlX/w2j333CPCAWAX5HJ0AAAAKNLS2dnZ2ewhAIAd09bWljFjxiRJFi9enNGjRzd5IgBgY5wJBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIBwAAgCIiHAAAAIqIcAAAACgiwgEAAKCICAcAAIAiIhwAAACKiHAAAAAoIsIBAACgiAgHAACAIiIcAAAAiohwAAAAKCLCAQAAoIgIB4AGeOqpp/KNb3wjZ511VsaNG5fBgwdn4MCBGT16dM4444zcdNNNaW9vb/aYAECTtXR2dnY2ewgA6Mk++9nP5vLLL8+WfqQeffTRufXWW7P//vs3fIa2traMGTMmSbJ48eKMHj264fsAAHacM+EAsIOWLl2azs7ODB48OB/+8Idz3XXXZe7cufnZz36WG2+8Mcccc0yS5Gc/+1lOOeWUvPDCC02eGABoFmfCAWAHXXrppRk+fHg++tGPZujQoX/w+vr163POOedk+vTpSZLLLrssn/nMZxo6gzPhANAziHAAKLBy5crsu+++WbduXQ4//PD84he/aOj2RTgA9AwuRweAAsOHD8/hhx+eJHn00UebPA0A0CwiHACKvPzyy0mSPn38+AWAXVW/Zg8AALuCZ555JgsXLkySHHLIIdv8/ra2ts2+vmzZsu2aCwCoJcIBoMCXv/zlrueET548eZvf/+r3vQGAns2N2QCgm/30pz/Ncccdl/b29owePTqPPPJIBg0atE3baGlp2ep13ZgNAHZezoQDQDd6+umn86d/+qdpb29PS0tLrr/++m0O8GRDWG/OsmXLMmHChO0dEwAoIsIB2GW0t7enf//+O7yd6667LhdccMEW11u9enVOO+20ru9zf+ELX8hJJ520Xft0ZhsAege3ZwWAbrB27dqcccYZuf/++5MkH//4x/PJT36yyVMBAM3mTDgAu4x+/fp13aF8R4waNWqzr7e3t2fy5Mm55557kiR/9md/liuuuGKH9wsA9HwiHIBdyvY8HmxbdHR05Lzzzsvtt9+eJDnrrLPyzW9+s1v3CQD0HC5HB4AG+ou/+IvcfPPNSZLTTz89N954Y/r08eMWANjApwIAaJCPf/zj+da3vpUkOfnkk3PLLbc05EZwAEDvIcIBoAE+97nP5corr0ySvOtd78r3v//97Lbbbk2eCgDY2fhOOADsoKuvvjqf//znkyT77bdfvvSlL+Xxxx/f7HvGjRvnLDkA7IJEOADsoFtvvbVrecmSJTnuuOO2+J7HH388Bx54YDdOBQDsjFyODgAAAEWcCQeAHTR79uxmjwAA9BDOhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFCkX7MHAIDeYNasWZk/f37mz5+fxx57LMuXL89zzz2XIUOG5KCDDsrEiRPzkY98JOPGjWv2qABAE7V0dnZ2NnsIAOjJ2tvb079//y2u179//1x22WX55Cc/2fAZ2traMmbMmCTJ4sWLM3r06IbvAwDYcc6EA0AD7LHHHpk4cWLe8Y535KCDDsqoUaMyaNCgLF26NLNnz861116b5557Lp/61KcybNiw/OVf/mWzRwYAmsCZcABogPXr16dv376bfP3xxx/PUUcdlVWrVmXkyJFZtmzZZtffVs6EA0DP4MZsANAAWwrqN73pTTnrrLOSJMuXL8/DDz9cMRYAsJMR4QBQZPDgwV3La9eubeIkAECziHAAKPDSSy/l+9//fpKkT58+GTt2bJMnAgCawY3ZAKCbvPLKK1m2bFn+/d//PX//93+fRYsWJUkuvPDCDB06dJu21dbWttnXly1btt1zAgB1RDgANNATTzyRN73pTZt8/ZRTTskVV1yxzdt99aZrAEDP5nJ0ACgwfPjw3HTTTbnzzjuzxx57NHscAKBJPKIMABrolVdeySOPPJIkaW9vz5IlS3LnnXfm29/+dgYOHJhPfOIT+dSnPpWWlpZt2u6WLkdvb2/P8uXLM2rUqOyzzz7p18/FbgCwMxLhAOwy2tvb079//x3eznXXXZcLLrhgm97zy1/+MieeeGJ+97vf5cILL8y11167w3MAAD2Py9EBoMDhhx+eyy+/PMmGiL/rrruaPBEA0AzOhAOwS3n44Yd3eBujRo3aru91L126NPvtt1+S5M///M9zzTXX7PAsAEDP4gtjAOxSDjnkkKbte+TIkV3LTz75ZNPmAACax+XoAFBkyZIlXctDhgxp4iQAQLOIcAAo8r3vfa9r+W1ve1sTJwEAmkWEA8AOmjlzZpYtW7bZdebMmZPLLrssSdKvX7+cffbZFaMBADsZ3wkHgB00c+bMnHXWWTnttNNy8sknZ/z48Rk2bFhefvnlPProo7n99tszffr0dHR0JEk+85nPZNy4cU2eGgBoBhEOAA2wbt26zJgxIzNmzNjkOm94wxvyt3/7t7nkkksKJwMAdiYeUQYAO2jFihW5++67c/fdd2fBggX57W9/m2eeeSZ9+vTJXnvtlfHjx+ekk07KlClTMmrUqGaPCwA0kQgHAACAIm7MBgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQBERDgAAAEVEOAAAABQR4QAAAFBEhAMAAEAREQ4AAABFRDgAAAAUEeEAAABQRIQDAABAEREOAAAARUQ4AAAAFBHhAAAAUESEAwAAQJH/HznnREWzzM7RAAAAAElFTkSuQmCC",
      "text/plain": [
       "<Figure size 600x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 495,
       "width": 496
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "square = np.array([[0, 1, 1, 0],\n",
    "                   [1, 1, 0, 0]])\n",
    "dm.plotSetup()\n",
    "print(square)\n",
    "dm.plotSquare(square)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 16
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Now let's transform each of these points according to the following rule.   Let \n",
    "\n",
    "$$ A = \\left[\\begin{array}{rr}1&1.5\\\\0&1\\end{array}\\right]. $$\n",
    "\n",
    "We define $T({\\bf x}) = A{\\bf x}$.  Then we have \n",
    "\n",
    "$$ T: \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 17
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "What is the image of each of these points under $T$?\n",
    "\n",
    "$$ A\\left[\\begin{array}{r}0\\\\1\\end{array}\\right] = \\left[\\begin{array}{r}1.5\\\\1\\end{array}\\right]$$\n",
    "\n",
    "\n",
    "$$ A\\left[\\begin{array}{r}1\\\\1\\end{array}\\right] = \\left[\\begin{array}{r}2.5\\\\1\\end{array}\\right]$$\n",
    "\n",
    "\n",
    "$$ A\\left[\\begin{array}{r}1\\\\0\\end{array}\\right] = \\left[\\begin{array}{r}1\\\\0\\end{array}\\right]$$\n",
    "\n",
    "\n",
    "$$ A\\left[\\begin{array}{r}0\\\\0\\end{array}\\right] = \\left[\\begin{array}{r}0\\\\0\\end{array}\\right]$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 18
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "square = \n",
      "[[0 1 1 0]\n",
      " [1 1 0 0]]\n",
      "A matrix = \n",
      "[[1.  1.5]\n",
      " [0.  1. ]]\n",
      "transformed square = \n",
      "[[1.5 2.5 1.  0. ]\n",
      " [1.  1.  0.  0. ]]\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 600x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 495,
       "width": 496
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = dm.plotSetup()\n",
    "print(\"square = \"); print(square)\n",
    "dm.plotSquare(square)\n",
    "#\n",
    "# create the A matrix\n",
    "A = np.array([[1.0, 1.5],[0.0,1.0]])\n",
    "print(\"A matrix = \"); print(A)\n",
    "#\n",
    "# apply the shear matrix to the square\n",
    "ssquare = np.zeros(np.shape(square))\n",
    "for i in range(4):\n",
    "    ssquare[:,i] = dm.AxVS(A,square[:,i])\n",
    "print(\"transformed square = \"); print(ssquare)\n",
    "dm.plotSquare(ssquare,'r')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 18
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "This sort of transformation, where points are successively slid sideways, is called a __shear__ transformation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 20,
     "slide_type": "subslide"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Linear Transformations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 20
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "By the properties of matrix-vector multiplication, we know that the transformation ${\\bf x} \\mapsto A{\\bf x}$ has the properties that \n",
    "\n",
    "$$ A({\\bf u} + {\\bf v}) = A{\\bf u} + A{\\bf v} \\;\\;\\;\\mbox{and}\\;\\;\\; A(c{\\bf u}) = cA{\\bf u}$$\n",
    "\n",
    "for all $\\bf u, v$ in $\\mathbb{R}^n$ and all scalars $c$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 23
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We are now ready to define one of the most fundamental concepts in the course: \n",
    "the concept of a\n",
    "<font color=blue>__linear transformation.__</font>  \n",
    "\n",
    "(You are now finding out why the subject is called linear algebra!)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 24,
     "slide_helper": "subslide_end"
    },
    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "__Definition.__  A transformation $T$ is __linear__ if:\n",
    "1. $T({\\bf u} + {\\bf v}) = T({\\bf u}) + T({\\bf v}) \\;\\;\\;$ for all $\\bf u, v$ in the domain of $T$; and\n",
    "2. $T(c{\\bf u}) = cT({\\bf u}) \\;\\;\\;$ for all scalars $c$ and all $\\bf u$ in the domain of $T$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 24,
     "slide_type": "subslide"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "To fully grasp the significance of what a linear transformation is, don't think of just matrix-vector multiplication.  Think of $T$ as a function in more general terms.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 27,
     "slide_helper": "subslide_end"
    },
    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The definition above captures a _lot_ of transformations that are not matrix-vector multiplication.  For example, think of:\n",
    "\n",
    "$$ T(f) = \\int_0^1 f(t) \\,dt $$\n",
    "\n",
    "Is $T$ a linear transformation?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Checking the conditions of our definition:\n",
    "    \n",
    "$$ T(f + g) = T(f) + T(g) $$\n",
    "\n",
    "in other words:\n",
    "    \n",
    "$$  \\int_0^1 f(t) + g(t) \\,dt = \\int_0^1 f(t) \\,dt + \\int_0^1 g(t) \\,dt$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "and also:\n",
    "    \n",
    "$$ T(c \\cdot f) = c \\cdot T(f) $$\n",
    "\n",
    "(check that yourself)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "What about:\n",
    "    \n",
    "$$ T(f) = \\frac{d f(t)}{dt} $$\n",
    "\n",
    "Is $T$ a linear transformation?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "What about:\n",
    "    \n",
    "$$ T(x) = e^x $$\n",
    "\n",
    "Is $T$ a linear transformation?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Properties of Linear Transformations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "A key aspect of a linear transformation is that it __preserves the operations of vector addition and scalar multiplication.__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "For example: for vectors $\\mathbf{u}$ and $\\mathbf{v}$, one can either:\n",
    "1. Transform them both according to $T()$, then add them, or:\n",
    "2. Add them, and then transform the result according to $T()$.\n",
    "\n",
    "One gets the same result either way.  The transformation does not affect the addition."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "This leads to two important facts.\n",
    "\n",
    "If $T$ is a linear transformation, then\n",
    "\n",
    "$$ T({\\mathbf 0}) = {\\mathbf 0} $$\n",
    "\n",
    "and\n",
    "\n",
    "$$ T(c\\mathbf{u} + d\\mathbf{v}) = cT(\\mathbf{u}) + dT(\\mathbf{v}) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "In fact, if a transformation satisfies the second equation for all $\\mathbf{u}, \\mathbf{v}$ and $c, d,$ then it must be a linear transformation.   \n",
    "\n",
    "Both of the rules defining a linear transformation derive from this single equation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "__Example.__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given a scalar $r$, define $T: \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2$ by $T(\\mathbf{x}) = r\\mathbf{x}$.  \n",
    "\n",
    "($T$ is called a __contraction__ when $0\\leq r \\leq 1$ and a __dilation__ when $r > 1$.)  \n",
    "\n",
    "Let $r = 3$, and show that $T$ is a linear transformation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "__Solution.__\n",
    "\n",
    "Let $\\mathbf{u}, \\mathbf{v}$ be in $\\mathbb{R}^2$ and let $c, d$ be scalars.   Then"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "$$\n",
    "T(c\\mathbf{u} + d\\mathbf{v}) = 3(c\\mathbf{u} + d\\mathbf{v})\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$ = 3c\\mathbf{u} + 3d\\mathbf{v} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$ = c(3\\mathbf{u}) + d(3\\mathbf{v}) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$ = cT(\\mathbf{u}) + dT(\\mathbf{v}) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus $T$ is a linear transformation because it satisfies the rule $T(c\\mathbf{u} + d\\mathbf{v}) = cT(\\mathbf{u}) + dT(\\mathbf{v})$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "__Example.__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Let $T(\\mathbf{x}) = \\mathbf{x} + \\mathbf{b}$ for some $\\mathbf{b} \\neq 0$.   \n",
    "\n",
    "What sort of operation does $T$ implement?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Answer: __translation.__\n",
    "\n",
    "Is $T$ a linear transformation?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "__Solution.__\n",
    "\n",
    "We only need to compare \n",
    "\n",
    "$$T(\\mathbf{u} + \\mathbf{v})$$ \n",
    "\n",
    "to \n",
    "\n",
    "$$T(\\mathbf{u}) + T(\\mathbf{v}).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "So:\n",
    "\n",
    "$$T(\\mathbf{u} + \\mathbf{v}) = \\mathbf{u} + \\mathbf{v} + \\mathbf{b}$$ \n",
    "\n",
    "and\n",
    "\n",
    "$$T(\\mathbf{u}) + T(\\mathbf{v}) = (\\mathbf{u} + \\mathbf{b}) + (\\mathbf{v} + \\mathbf{b})$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "If $\\mathbf{b} \\neq 0$, then the above two expressions are not equal.\n",
    "\n",
    "So $T$ is __not__ a linear transformation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "frag_helper": "fragment_end",
     "frag_number": 24,
     "slide_helper": "subslide_end",
     "slide_type": "subslide"
    },
    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": [
     "remove-cell"
    ]
   },
   "source": [
    "``` {toggle}\n",
    "Question Time! Q7.2\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### A Non-Geometric Example: Manufacturing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "A company manufactures two products, B and C.  To do so, it requires materials, labor, and overhead.\n",
    "\n",
    "For one dollar's worth of product B, it spends 45 cents on materials, 25 cents on labor, and 15 cents on overhead.\n",
    "\n",
    "For one dollar's worth of product C, it spends 40 cents on materials, 30 cents on labor, and 15 cents on overhead."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Let us construct a \"unit cost\" matrix:\n",
    "\n",
    "$$U = \\begin{array}{r}\n",
    "\\begin{array}{rrr}\\mbox{B}&\\;\\;\\;\\;\\mbox{C}\\;&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\end{array}\\\\\n",
    "\\left[\\begin{array}{rr}.45&.40\\\\.25&.30\\\\.15&.15\\end{array}\\right]\n",
    "\\begin{array}{r}\\mbox{Materials}\\\\\\mbox{Labor}\\\\\\mbox{Overhead}\\end{array}\\\\\n",
    "\\end{array}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Let $\\mathbf{x} = \\left[\\begin{array}{r}x_1\\\\x_2\\end{array}\\right]$ be a production vector, corresponding to $x_1$ dollars of product B and $x_2$ dollars of product C.\n",
    "\n",
    "Then define $T: \\mathbb{R}^2 \\rightarrow \\mathbb{R}^3$ by\n",
    "\n",
    "$$T(\\mathbf{x}) = U\\mathbf{x} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$ = x_1 \\left[\\begin{array}{r}.45\\\\.25\\\\.15\\end{array}\\right] + x_2 \\left[\\begin{array}{r}.40\\\\.30\\\\.15\\end{array}\\right]$$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$ = \\left[\\begin{array}{r}\\mbox{Total cost of materials}\\\\\\mbox{Total cost of labor}\\\\\\mbox{Total cost of overhead}\\end{array}\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The mapping $T$ transforms a list of production quantities into a list of total costs."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The linearity of this mapping is reflected in two ways:\n",
    "\n",
    "1. If production is increased by a factor of, say, 4, ie, from $\\mathbf{x}$ to $4\\mathbf{x}$, then the costs increase by the same factor, from $T(\\mathbf{x})$ to $4T(\\mathbf{x})$.\n",
    "2. If $\\mathbf{x}$ and $\\mathbf{y}$ are production vectors, then the total cost vector associated with combined production of $\\mathbf{x} + \\mathbf{y}$ is precisely the sum of the cost vectors $T(\\mathbf{x})$ and $T(\\mathbf{y})$."
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.9.18"
  },
  "rise": {
   "scroll": true,
   "theme": "beige",
   "transition": "fade"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}