MML Python, Ch.03 Analytic Geometry.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import numpy.linalg as npl\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib.pyplot import figure\n",
    "from sympy import *\n",
    "from sympy.abc import x\n",
    "from IPython.core.interactiveshell import InteractiveShell\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib import cm\n",
    "import random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "InteractiveShell.ast_node_interactivity = \"all\"\n",
    "plt.rcParams[ \"figure.figsize\" ] = (10,10)\n",
    "origin2D = np.vstack([0,0])\n",
    "origin3D = np.vstack([0,0,0])\n",
    "scale = 10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 03, Analytic Geometry\n",
    "## 3.1 Norms\n",
    "### Example 3.1 (Manhattan Norm), Example 3.2 (Euclidean Norm)\n",
    "#### Figure 3.3\n",
    "$$\n",
    "||x||_1 := \\Sigma_{i=1}^n|x_i| = 1, \\\\\n",
    "|x_2| + |x_1| = 1, x_2 = 1-|x_1|\\\\\n",
    "||x||_2 := \\sqrt{\\Sigma_{i=1}^{n}x^2_i} = \\sqrt{x^Tx} = 1, \\\\\n",
    "\\sqrt{x_1^2 + x_2^2} = 1, x_2 = \\sqrt{1-x_1^2}\n",
    "$$\n",
    "Where we state \"$= 1$\", to produce a unit shape. \n",
    "\n",
    "1. For the $l_1$ norm, we are selecting $x_0$ and $x_1$ such that the sum of their absolute values is 1.\n",
    "2. For the $l_2$ norm, we are selecting $x_0$ and $x_1$ such that the square root of the sum of their squared values is 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.0 & 0.0\\\\0.0 & -0.0201998950135421\\\\0.0 & -0.0399829930525427\\\\0.0 & -0.0593485110724477\\\\0.0 & -0.0782951385367167\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0.0,                 0.0],\n",
       "[0.0, -0.0201998950135421],\n",
       "[0.0, -0.0399829930525427],\n",
       "[0.0, -0.0593485110724477],\n",
       "[0.0, -0.0782951385367167]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# L1 and L2 Norm.\n",
    "# Building L1 and L2, (x,y) coordinates for 4 quadrants, where y in the first is of the form 1-x, and sqrt(1-x**2).\n",
    "xRight = np.linspace(0,1,50)\n",
    "xLeft = np.linspace(-1,0,50)\n",
    "l1normData = np.hstack([np.vstack(np.vstack([xRight,xLeft,xLeft,xRight]).ravel()),\n",
    "                       np.vstack(np.vstack([1-xRight, xLeft+1, -xLeft-1, xRight-1]).ravel())])\n",
    "l2normData = np.hstack([np.vstack(np.vstack([xRight,xLeft,xLeft,xRight]).ravel()),\n",
    "                       np.vstack(np.vstack([np.sqrt(1-xRight**2), np.sqrt(-xLeft**2+1), -np.sqrt(-xLeft**2+1), -np.sqrt(1-xRight**2)]).ravel())])\n",
    "# L1 and L2 Norm.\n",
    "\n",
    "Matrix(l1normData[:5]-l2normData[:5]) # The difference between the first 5 values of the L1 and L2 norm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Data and figure initialization.\n",
    "plt.axis([-scale/5,scale/5,-scale/5,scale/5])\n",
    "plt.grid(alpha=.1)\n",
    "plt.title(\"L1, Manhattan Norm, in red and L2, Euclidean Norm, in green.\")\n",
    "# Data and figure initialization.\n",
    "\n",
    "plt.scatter(l1normData[:,0],l1normData[:,1],s=.5,color=\"r\")\n",
    "plt.scatter(l2normData[:,0],l2normData[:,1],s=.5,color=\"g\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.2 Inner Products\n",
    "### Example 3.3 (Inner Product That Is Not the Dot Product)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1],\n",
       "[2]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3\\\\4\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[3],\n",
       "[4]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.vstack([1,2])\n",
    "y = np.vstack([3,4])\n",
    "Matrix(x)\n",
    "Matrix(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3 & 4\\\\6 & 8\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[3, 4],\n",
       "[6, 8]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}9\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([[9]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix(np.dot(x,y.T))\n",
    "Matrix(x[0]*y[0] - (x[0]*y[1]+x[1]*y[0])+2*x[1]*y[1] )\n",
    "# This can be called an inner product as it is symmetric and a bilinear mapping, that is it \n",
    "# does still combine elements from 2 symmetric spaces and outputs 1 element,\n",
    "# and it turns out, this has a different output from the dot product."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3.4 (Symmetric, Positive Definite Matrices)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}9 & 6\\\\6 & 5\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[9, 6],\n",
       "[6, 5]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}9 & 6\\\\6 & 3\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[9, 6],\n",
       "[6, 3]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A1 = np.vstack([[9,6],[6,5]])\n",
    "A2 = np.vstack([[9,6],[6,3]])\n",
    "Matrix(A1)\n",
    "Matrix(A2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = symbols('x')\n",
    "y = symbols('y')\n",
    "A1 = Matrix(A1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 9 x^{2} + 12 x y + 5 y^{2}$"
      ],
      "text/plain": [
       "9*x**2 + 12*x*y + 5*y**2"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A1expr = simplify(Matrix([x,y]).T*A1*Matrix([x,y]))[0]\n",
    "A1expr\n",
    "f = lambdify([x,y],A1expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\text{True}\\\\\\text{True}\\\\\\text{True}\\\\\\text{True}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[True],\n",
       "[True],\n",
       "[True],\n",
       "[True]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix(np.array([f(-1,-1),f(-1,1),f(1,-1),f(1,1)]) > 0) # Checking for positive definiteness."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above is positive definite because it is symmetric and the expression output is always greater than 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}9 & 6\\\\6 & 3\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[9, 6],\n",
       "[6, 3]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A2 = Matrix(A2)\n",
    "A2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 9 x^{2} + 12 x y + 3 y^{2}$"
      ],
      "text/plain": [
       "9*x**2 + 12*x*y + 3*y**2"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A2expr = simplify(Matrix([x,y]).T*A2*Matrix([x,y]))[0]\n",
    "A2expr\n",
    "g = lambdify([x,y],A2expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\text{True}\\\\\\text{False}\\\\\\text{False}\\\\\\text{True}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ True],\n",
       "[False],\n",
       "[False],\n",
       "[ True]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix(np.array([g(-1,-1),g(-1,1),g(1,-1),g(1,1)]) > 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above expression is symmetric, but not positive definite because some outputs are less than 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\text{True}\\\\\\text{True}\\\\\\text{True}\\\\\\text{True}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[True],\n",
       "[True],\n",
       "[True],\n",
       "[True]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix(np.array([g(-1,-1),g(-1,1),g(1,-1),g(1,1)]) >= 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that some values are equal to 0 though, so we say only $\\geq$ holds, and thus is symmetric, positive *semidefinite*. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.3 Lengths and Distances \n",
    "Norms can be used to find the length of a vector, to find *some* norms, e.g. Manhattan Norm does not require an inner product. A norm can be found by taking the square root of the inner product. \n",
    "### Example 3.5 (Lengths of Vectors Using Inner Products)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1],\n",
       "[1]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = np.vstack([1,1])\n",
    "Matrix(x0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.41421356]])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sqrt(np.dot(x0.T,x0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Using the inner product noted above, we get a \"shorter\" length.\n",
    "x0[0]*x0[0] - (x0[0]*x0[1]+x0[1]*x0[0])+2*x0[1]*x0[1] "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.4 Angles and Orthogonality\n",
    "#### Figure 3.4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(np.linspace(0,np.pi,100),np.cos(np.linspace(0,np.pi,100)))\n",
    "plt.grid(alpha=.1)\n",
    "plt.title(r\"When restricted to $[0,\\pi]$, then $f(w) = cos(w)$ returns a unique number in the interval $[-1,1]$.\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3.6 (Angle between Vectors)\n",
    "\n",
    "$$\n",
    "cos\\omega = \\frac{\\langle x,y\\rangle}{\\sqrt{\\langle x,x\\rangle\\langle y,y\\rangle}} = \\frac{x^Ty}{\\sqrt{x^Txy^Ty}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1],\n",
       "[1]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1],\n",
       "[2]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.vstack([1,1])\n",
    "y = np.vstack([1,2])\n",
    "Matrix(x)\n",
    "Matrix(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "18.434948822922017"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cosAngle = lambda x, y: (np.dot(x.T,y)/np.sqrt(np.dot(np.dot(x.T,x),np.dot(y.T,y))))[0][0]\n",
    "angle = lambda cos: np.arccos(cos)*(180/np.pi)\n",
    "angle(cosAngle(x,y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Figure 3.5\n",
    "*The angle $\\omega$ between two vectors $x,y$ is computed using the inner product.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Figure setup.\n",
    "plt.grid(alpha=.1)\n",
    "plt.legend([\"x\",\"y\"])\n",
    "plt.axis([-scale/5,scale/5,-scale/5,scale/5])\n",
    "plt.title(r\"Angle of %1.2f$^\\circ$ for vectors $x$ and $y$.\" %round(angle(cosAngle(x,y)),2))\n",
    "# Figure setup.\n",
    "\n",
    "plt.quiver(*origin2D, *x,scale=scale,color=\"lightblue\", width=.005)\n",
    "plt.quiver(*origin2D, *y,scale=scale,color=\"blue\", width=.005)\n",
    "plt.annotate(str(round(angle(cosAngle(x,y)),2))+r\"$^\\circ$\",xy=(scale/50,scale/30));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1],\n",
       "[1]])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-1\\\\1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-1],\n",
       "[ 1]])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.vstack([1,1])\n",
    "y = np.vstack([-1,1])\n",
    "Matrix(x)\n",
    "Matrix(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "90.0"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "angle(cosAngle(x,y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Figure setup.\n",
    "plt.grid(alpha=.1)\n",
    "plt.legend([\"x\",\"y\"])\n",
    "plt.axis([-scale/5,scale/5,-scale/5,scale/5])\n",
    "plt.title(r\"Angle of %1.2f$^\\circ$ for vectors $x$ and $y$.\" %round(angle(cosAngle(x,y)),2))\n",
    "# Figure setup.\n",
    "\n",
    "plt.quiver(*origin2D, *x,scale=scale,color=\"lightblue\", width=.005)\n",
    "plt.quiver(*origin2D, *y,scale=scale,color=\"blue\", width=.005)\n",
    "plt.annotate(str(round(angle(cosAngle(x,y)),2))+r\"$^\\circ$\",xy=origin2D);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.vstack([1,1])\n",
    "y = np.vstack([-1,1])\n",
    "inner = np.vstack([[2,0],[0,1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.33333333]])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([[109.47122063]])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "omega = (x.T@inner@y)/npl.norm(x.T@inner@x)\n",
    "omega\n",
    "angle(omega)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By changing how inner is induced, such as with the matrix $\\begin{bmatrix}2&0\\\\0&1\\end{bmatrix}$, we find the vectors are no longer orthogonal, despite their being orthogonal with respect to another inner product."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.5 Orthonormal Basis\n",
    "We can use Gaussian elimination to find a basis for a vector space. If we are given a set of non-orthogonal, unnormalized basis vectors, we can concatenate them to construct an audmented matrix, and apply Gaussian elimination to it, to find an orthonormal basis. This process is called the *Gram-Schmidt* process. \n",
    "\n",
    "### Example 3.8 (Orthonormal Basis)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Data and figure setup.\n",
    "plt.grid(alpha=.1)\n",
    "b1 = np.vstack([1,1])*(1/np.sqrt(2))\n",
    "b2 = np.vstack([1,-1])*(1/np.sqrt(2))\n",
    "plt.title(r\"Orthonormal basis since: $angle(dot(b_1^T,b_2))$ = %1.2f,\" %angle(np.dot(b1.T,b2)) + \" and both norms are %i \" %round(npl.norm(b1)) + \"and %i.\" %round(npl.norm(b2)))\n",
    "# Data and figure setup.\n",
    "\n",
    "plt.quiver(*origin2D,*b1,scale=scale, color = \"lightblue\")\n",
    "plt.quiver(*origin2D,*b2,scale=scale, color = \"gold\")\n",
    "plt.legend([r\"$b_1$\",r\"$b_2$\"]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.6 Orthogonal Complement\n",
    "#### Figure 3.7\n",
    "Any plane in $\\mathbb{R}^3$ can be described by some vector orthogonal to it, this vector is said to span it's orthogonal complement, and is known as the normal vector of a plane."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Figure setup.\n",
    "fig = plt.figure()\n",
    "ax = fig.gca(projection = '3d')\n",
    "scale3D = 10\n",
    "ax.set_xlim3d(-scale3D,scale3D)\n",
    "ax.set_ylim3d(scale3D,-scale3D)\n",
    "ax.set_zlim3d(-scale3D,scale3D)\n",
    "point  = origin3D.T[0]\n",
    "ax.grid(b=None)\n",
    "# Figure setup.\n",
    "\n",
    "colors = [\"r\",\"g\",\"b\"]\n",
    "normalVecs = np.vstack([[-1,1,1],[1,-1,1],[1,1,-1]])\n",
    "for c,i in enumerate(normalVecs): # Using index of vectors to iterate colors.\n",
    "    d = np.dot(-point,i)\n",
    "    xx, yy = np.meshgrid(np.linspace(-scale3D/5,scale3D/5,10), np.linspace(-scale3D/5,scale3D/5,10))\n",
    "    z = (d-i[0] * xx - i[1] * yy) /i[2] #ax+by+cz = d ∴ z = (d-ax-by)/c\n",
    "    ax.plot_surface(xx, yy, z, alpha=0.2,color = colors[c])\n",
    "    ax.quiver(*point,*i, length=scale3D/2,color=colors[c])\n",
    "    \n",
    "ax.set_title(\"Normal Vectors with their Planes\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.7 Inner Product of Functions\n",
    "An inner product of two functions u and v mapping from $\\mathbb{R}$ to $\\mathbb{R}$, can be defined as the following.\n",
    "$$\n",
    "\\langle u,v \\rangle := \\int_a^b u(x)v(x)dx\n",
    "$$\n",
    "### Example 3.9 (Inner Product of Functions)\n",
    "$$\n",
    "u = sin(x)\\\\\n",
    "v = cos(x) \\\\\n",
    "f(x) = u(x)v(x)\n",
    "$$\n",
    "\n",
    "#### Figure 3.8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.linspace(-np.pi,np.pi,100)\n",
    "y = lambda x: np.sin(x)*np.cos(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x,y(x))\n",
    "plt.grid(alpha=.1)\n",
    "plt.title(\"sin(x)cos(x)\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function is odd, f(-x) = -f(x), so the integral from -_ to _ will evaluate to 0, thus sine and cosine are orthogonal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.37840124765396416, 0.37840124765396416)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y(-2), -y(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An integration of the function sin(x)cos(x) with respect to x, over -pi and pi, as stated above, evaluates to 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "symbX = Symbol(\"x\")\n",
    "integrate(sin(symbX)*cos(symbX),(symbX,-pi,pi))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "round(sum(np.sin(x)*np.cos(x)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It also holds that the collection of functions $\\{1,cos(x)...cos(n\\times x)\\}$ is orthogonal if we integrate from $-\\pi$ to $\\pi$, i.e. any pair are orthogonal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(cos(symbX)*cos(3*symbX),(symbX,-pi,pi))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.8 Orthogonal Projections\n",
    "*This section begins with an implementation, and procedes with an explanation and reimplementation.*\n",
    "#### Figure 3.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/vb/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:5: FutureWarning: `rcond` parameter will change to the default of machine precision times ``max(M, N)`` where M and N are the input matrix dimensions.\n",
      "To use the future default and silence this warning we advise to pass `rcond=None`, to keep using the old, explicitly pass `rcond=-1`.\n",
      "  \"\"\"\n"
     ]
    }
   ],
   "source": [
    "# Initializing data and using least squares regression as fit line.\n",
    "xs = np.vstack([-3.2,-2.2,-2,-1.7,-1.7,-1.5,-1.3,-.4,-.3,0,.5,.6,.8,1,1.1,1.2,1.3,1.7,2.3,3.2])\n",
    "ys = np.vstack(np.random.normal(0,.8,len(xs)))\n",
    "points = np.hstack([xs,ys])\n",
    "lsq = npl.lstsq(points,np.random.randn(20))[0] \n",
    "# Initializing data and using least squares regression as fit line.\n",
    "\n",
    "posPts = [p for p in points if p[0]>=0] # Splitting into positive and negative to keep a 0,0 origin.\n",
    "negPts = [p for p in points if p[0]<0]\n",
    "maxNeg = np.hstack([xs[0],[xs*(lsq[0]/lsq[1])][0][0]]) # -Line for data to project onto.\n",
    "maxPos = np.hstack([xs[::-1][0],[xs*(lsq[0]/lsq[1])][0][::-1][0]]) # +Line for data to project onto."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Figure setup.\n",
    "plt.axis([-4,4,-4,4])\n",
    "plt.plot(xs,xs*(lsq[0]/lsq[1]),c=\"k\")\n",
    "plt.scatter(points[:,0],points[:,1])\n",
    "plt.grid(alpha=.1)\n",
    "# Figure setup.\n",
    "\n",
    "# Iteratively finding each projection and plotting it.\n",
    "for i in negPts:\n",
    "    newQ = (np.dot(i,maxNeg)/npl.norm(maxNeg)**2)*maxNeg # Projection step: p = v2 * (dot(v1,v2)/norm(v2)**2).\n",
    "    plt.quiver(*origin2D,*newQ,scale=8,width=.003,color=\"orange\")  # Projection vector.\n",
    "    currV = newQ-i \n",
    "    plt.quiver(*i,*currV,scale=8,width=.003,color=\"orange\")\n",
    "    plt.annotate(\"   \" + str(round(npl.norm(currV),1)),i)    \n",
    "for i in posPts:\n",
    "    newQ = (np.dot(i,maxPos)/npl.norm(maxPos)**2)*maxPos # Projection step: p = v2 * (dot(v1,v2)/norm(v2)**2).\n",
    "    plt.quiver(*origin2D,*newQ,scale=8,width=.003,color=\"orange\")  # Projection vector.\n",
    "    currV = newQ-i \n",
    "    plt.quiver(*i,*currV,scale=8,width=.003,color=\"orange\")\n",
    "    plt.annotate(\"   \" + str(round(npl.norm(currV),1)),i)\n",
    "# Iteratively finding each projection and plotting it.\n",
    "    \n",
    "plt.title(\"Orthogonal projection of a two-dimensional dataset onto a one-dimensional subspace.\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Figure 3.10\n",
    "*Examples of projections onto one-dimensional subspaces.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1],\n",
       "[2]])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2\\\\1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[2],\n",
       "[1]])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1.6\\\\0.8\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1.6],\n",
       "[0.8]])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.6\\\\-1.2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 0.6],\n",
       "[-1.2]])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v1 = np.vstack([1,2])\n",
    "v2 = np.vstack([2,1])\n",
    "projec = lambda v1,v2: (np.dot(v2.T,v1.T)/npl.norm(v1.T)**2) * v2 # v2 * (dot(v1,v2)/norm(v2)**2)\n",
    "v3 = projec(v1.ravel(),v2.ravel()) # Projection step.\n",
    "v4 = v3.ravel() - v1.ravel()\n",
    "Matrix(v1)\n",
    "Matrix(v2)\n",
    "Matrix(v3)\n",
    "Matrix(v4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.rcParams[ \"figure.figsize\" ] = (7,7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 504x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Figure setup.\n",
    "plt.axis([-scale/2,scale/2,-scale/2,scale/2])\n",
    "plt.suptitle(r\"(a) Projection of $x \\in \\mathbb{R}^2$ onto a subspace $U$ with basis vector $b$.\",size=15)\n",
    "plt.title(\"Displacement of %1.1f.\" %npl.norm(v3))\n",
    "plt.grid(alpha=.1)\n",
    "# Figure setup.\n",
    "\n",
    "# Plotting projection.\n",
    "plt.quiver(*origin2D, *v1, scale=scale, color = \"k\")\n",
    "plt.quiver(*origin2D, *v2, scale=scale, color = \"orange\")\n",
    "plt.scatter(*v3,color=\"k\")\n",
    "plt.quiver(*origin2D,*v3,scale=scale, width = .005,color=\"orange\")\n",
    "plt.quiver(*v1,*v4,scale=scale,width=.002,color=\"red\")\n",
    "# Plotting projection.\n",
    "\n",
    "plt.annotate(\"w = \" + str(round(angle(cosAngle(v1,v2)),2))+r\"$^\\circ$\",xy=origin2D)\n",
    "plt.legend([\"x\",\"U\",r\"$\\pi_U(x)$\",\"b\"]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. The projection $\\pi_U(x)$ is closest to $x$, where \"closest\" implies that the distance $||x-\\pi_U(x)||$ is minimal. It follows that the (red) segment is orthogonal to $U$, and therefore the basis vector $b$ of $U$. The orthogonality condition yields $\\langle \\pi_U(x) - x, b\\rangle = 0$, since angles between vectors are defined via the inner product."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```python\n",
    "'''\n",
    "v3 = pi_u \n",
    "v1 = x \n",
    "v2 = b\n",
    "'''\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.0"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "round(np.inner(v3 - v1.T,v2.T)[0][0],5) # Checking orthogonality condition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. The projection $\\pi_U(x)$ of $x$ onto $U$ must be an lement of $U$ and, therefore, a multiple of the basis vector $b$ that spansU. Hence, $\\pi_U(x) = \\lambda b$, for some $\\lambda \\in \\mathbb{R}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1],\n",
       "[0]])"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.693877551020408\\\\0.720093010790912\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0.693877551020408],\n",
       "[0.720093010790912]])"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.693877551020408\\\\0.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0.693877551020408],\n",
       "[              0.0]])"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.0\\\\-0.720093010790912\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[               0.0],\n",
       "[-0.720093010790912]])"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v1 = np.vstack(np.array([1,0]))\n",
    "v2 = np.vstack(l2normData[34:35][0])\n",
    "v3 = projec(v2.ravel(),v1.ravel()) # Projection step.\n",
    "v4 = v3.ravel()-v2.ravel()\n",
    "Matrix(v1)\n",
    "Matrix(v2)\n",
    "Matrix(v3)\n",
    "Matrix(v4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Figure setup.\n",
    "plt.axis([-scale/2,scale/2,-scale/2,scale/2])\n",
    "plt.grid(alpha=.1)\n",
    "plt.scatter(l2normData[:,0],l2normData[:,1],s=.5,color=\"k\") # Circle data from prior figure on the L2 norm.\n",
    "plt.title(\"(b) Projection of a two-dimensional vector (black) with ||x|| = 1 \\n onto a one-dimensional subspace spanned by b (orange).\" + \" \\n Projection magnitude %1.1f.\" %npl.norm(v3));\n",
    "# Figure setup.\n",
    "\n",
    "# Projection plotting.\n",
    "plt.quiver(*origin2D, *v1,scale=scale, color=\"orange\", width=.005)\n",
    "plt.quiver(*origin2D, *v2,scale=scale, color=\"k\", width=.005)\n",
    "plt.annotate(\"w = \" + str(round(angle(cosAngle(v1,v2)),2))+r\"$^\\circ$\",xy=origin2D-.2, size = 8)\n",
    "plt.quiver(*origin2D,*v3,scale=scale,width=.002,color=\"red\")\n",
    "plt.quiver(*v2,*v4,scale=scale,width=.002,color=\"blue\");\n",
    "# Projection plotting."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Three steps to determine a projection between any vector x and some basis vector, b, for a subspace, U.\n",
    "\n",
    "1. Find the scalar, $\\lambda$ for b. Given that some projection, p, exists, the inner product between x-p and b, will equal 0. <br>\n",
    "    a. We know that the projection, p, can also be written as a scalar operation of our basis vector, b, so we say $\\langle x-\\lambda b,b \\rangle= 0$. <br>\n",
    "    b. We can then isolate the scalar, $\\lambda$, by seeing that the above is equal to the inner product of x and b minus $\\lambda \\langle b,b \\rangle = 0$ <br>\n",
    "    c. $\\lambda = \\frac{\\langle x,b \\rangle}{\\langle b,b \\rangle}$ which can also be written as the inner product between b and x divided by the squared norm of b, $\\frac{\\langle b,x \\rangle}{||b||^2}$. \n",
    "2. Find the point on U that the projection, p, will create. Seeing as $p = \\lambda b$, we take (c) from above to find $p = \\frac{\\langle x,b \\rangle}{\\langle b,b \\rangle} b$.\n",
    "3. Find a projection matrix $P_p$. We know a projection is simply a linear mapping, so we should be able to find a matrix, $P$, that takes any vector, $x$, and maps it, or creates a new projected vector, $p$, to our subspace or line, such that $p = P_p x$, and $P_p = \\frac{p}{x}$. <br>\n",
    "    a. Seeing as $p = \\frac{\\langle x,b \\rangle}{\\langle b,b \\rangle} b$, from 2. above, $P_p  = \\frac{1}{x}\\frac{\\langle x,b \\rangle}{\\langle b,b \\rangle} b = \\frac{bb^T}{\\langle b,b \\rangle}$\n",
    "    \n",
    "**Applying these steps to Figure 3.9.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3.2\\\\3.11596518977999\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[             3.2],\n",
       "[3.11596518977999]])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3.2\\\\-3.11596518977999\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[             -3.2],\n",
       "[-3.11596518977999]])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3.2\\\\0.222895607313388\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[             -3.2],\n",
       "[0.222895607313388]])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = np.vstack([negPts,posPts])\n",
    "pB = np.vstack(maxPos)#basis vector\n",
    "nB = np.vstack(maxNeg)\n",
    "Matrix(pB)\n",
    "Matrix(nB)\n",
    "sampleX = np.vstack(data[0])\n",
    "Matrix(sampleX)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Step 1.** Find lambda. Recall lambda is the scalar applied to B that represents the coordinates of the projection."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.47848768]])"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sampleLam = np.dot(nB.T,sampleX)/npl.norm(nB)**2 # b^T x / ||b||^2; recall inner(x,b) = dot(b^T,x)\n",
    "sampleLam"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Step 2.** Find projected point using lambda. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-1.53116056464251\\\\-1.49095094355936\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-1.53116056464251],\n",
       "[-1.49095094355936]])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sampleP = nB*sampleLam\n",
    "Matrix(sampleP)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Step 3.** Find projection matrix, P. For any data point, x, the dot product between P and x will yield the new data point projected on the line. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.513302786496735 & 0.499823004544031\\\\0.499823004544031 & 0.486697213503265\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0.513302786496735, 0.499823004544031],\n",
       "[0.499823004544031, 0.486697213503265]])"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pMatrix = np.dot(nB,nB.T)/npl.norm(nB)**2 #bb^T/||b||^2\n",
    "Matrix(pMatrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-1.53116056464251\\\\-1.49095094355936\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-1.53116056464251],\n",
       "[-1.49095094355936]])"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-1.53116056464251\\\\-1.49095094355935\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-1.53116056464251],\n",
       "[-1.49095094355935]])"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix(sampleP)\n",
    "Matrix(np.dot(pMatrix,sampleX))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.axis([-scale/2,scale/2,-scale/2,scale/2])\n",
    "plt.grid(alpha=.1)\n",
    "plt.scatter(data[:,0],data[:,1],color=\"orange\")\n",
    "plt.plot(xs,xs*(lsq[0]/lsq[1]),c=\"k\")\n",
    "plt.quiver(*origin2D,*maxNeg,scale=scale)\n",
    "plt.quiver(*origin2D,*maxPos,scale=scale)\n",
    "plt.quiver(*origin2D,*sampleP,scale=scale,color=\"r\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3.10 (Projection onto a Line)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\2\\\\2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1],\n",
       "[2],\n",
       "[2]])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = np.vstack([1,2,2])\n",
    "Matrix(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 2 & 2\\\\2 & 4 & 4\\\\2 & 4 & 4\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1, 2, 2],\n",
       "[2, 4, 4],\n",
       "[2, 4, 4]])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "9.0"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.111111111111111 & 0.222222222222222 & 0.222222222222222\\\\0.222222222222222 & 0.444444444444444 & 0.444444444444444\\\\0.222222222222222 & 0.444444444444444 & 0.444444444444444\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0.111111111111111, 0.222222222222222, 0.222222222222222],\n",
       "[0.222222222222222, 0.444444444444444, 0.444444444444444],\n",
       "[0.222222222222222, 0.444444444444444, 0.444444444444444]])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Invoke step 3 from above.\n",
    "Matrix(np.dot(b,b.T))\n",
    "npl.norm(b)**2\n",
    "P = np.dot(b,b.T)/npl.norm(b)**2 #bb^T/||b||^2\n",
    "Matrix(P)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.555555555555556\\\\1.11111111111111\\\\1.11111111111111\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0.555555555555556],\n",
       "[ 1.11111111111111],\n",
       "[ 1.11111111111111]])"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.vstack([1,1,1])\n",
    "Matrix(np.dot(P,x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.rcParams[ \"figure.figsize\" ] = (10,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = fig.gca(projection = '3d')\n",
    "ax.grid(b=None)\n",
    "scale3D = 10\n",
    "ax.set_xlim3d(scale3D,-scale3D) # Inverted graph orientation for figure clarity.\n",
    "ax.set_ylim3d(-scale3D,scale3D)\n",
    "ax.set_zlim3d(scale3D,-scale3D)\n",
    "point  = origin3D.T[0]\n",
    "ax.quiver(*origin3D, *b,length=scale3D, color = \"lightblue\",lw=3)  \n",
    "ax.quiver(*origin3D, *x,length=scale3D, color = \"orange\",lw=3)  \n",
    "ax.quiver(*origin3D, *np.dot(P,x), length=scale3D, color=\"lightgreen\", lw=2)\n",
    "ax.set_title(\"Example 3.10\", size=15);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Steps 1-3 revisited for projection onto a general subspace.**\n",
    "\n",
    "1. Find lambdas of the projection with respect to the basis of the subspace. <br>\n",
    "    a. $\\lambda = (B^TB)^{-1}B^Tx$. Where $\\lambda$ is the set of all lambdas, and $B$ is the set of all bases. \n",
    "2. Find projection by multiplying lambda by B.\n",
    "3. Find the general projection matrix by dividing the projection in 2 by x. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3.11 (Projection onto a Two-dimensional Subspace)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\1 & 1\\\\1 & 2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1, 0],\n",
       "[1, 1],\n",
       "[1, 2]])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}6\\\\0\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[6],\n",
       "[0],\n",
       "[0]])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}5\\\\2\\\\-1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 5],\n",
       "[ 2],\n",
       "[-1]])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\-2\\\\1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 1],\n",
       "[-2],\n",
       "[ 1]])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "basis1 = np.vstack([1,1,1])\n",
    "basis2 = np.vstack([0,1,2])\n",
    "B = np.hstack([basis1,basis2])\n",
    "x = np.vstack([6,0,0])\n",
    "proj = np.vstack([5,2,-1])\n",
    "projDisplacement = x - proj\n",
    "Matrix(B)\n",
    "Matrix(x)\n",
    "Matrix(proj)\n",
    "Matrix(projDisplacement)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The projection was found using the steps noted above. \n",
    "1. $\\lambda = (B^TB)^{-1}B^Tx$.\n",
    "2. $\\pi = B\\lambda$\n",
    "3. $\\mathbb{P} = \\frac{B}{x}\\lambda$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}5.0\\\\-3.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 5.0],\n",
       "[-3.0]])"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}5.0\\\\2.0\\\\-1.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 5.0],\n",
       "[ 2.0],\n",
       "[-1.0]])"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.833333333333333 & 0.333333333333333 & -0.166666666666667\\\\0.333333333333333 & 0.333333333333333 & 0.333333333333333\\\\-0.166666666666667 & 0.333333333333333 & 0.833333333333333\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 0.833333333333333, 0.333333333333333, -0.166666666666667],\n",
       "[ 0.333333333333333, 0.333333333333333,  0.333333333333333],\n",
       "[-0.166666666666667, 0.333333333333333,  0.833333333333333]])"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}5.0\\\\2.0\\\\-1.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 5.0],\n",
       "[ 2.0],\n",
       "[-1.0]])"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 1. \n",
    "lam = npl.inv(B.T@B)@B.T@x\n",
    "# 2. \n",
    "pi = B@lam\n",
    "# 3. \n",
    "P = B@npl.inv(B.T@B)@B.T \n",
    "# Note that B/x is handled by removing the x from the equation used for lam.\n",
    "Matrix(lam)\n",
    "Matrix(pi)\n",
    "Matrix(P)\n",
    "Matrix(P@x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "xTest1 = np.vstack([1,-.5,3])\n",
    "xTest2 = np.vstack([-2,1,0])\n",
    "pTest1 = P@xTest1\n",
    "pTest2 = P@xTest2\n",
    "displ1 = xTest1 - pTest1\n",
    "displ2 = xTest2 - pTest2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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kk3j22WfR0dGBr33ta/jlL3+537q7KIp46qmnsGHDBsybNw8dHR244IILMDU1NeP37733HubMmYOBgQE8+uijAGKbSl133XVoaWnZb2txu89fzNeSzuxx5vp4mV7nbNmOZzJVVXHFFVego6MDPT09GBsbw/XXXz/jNmvWrMEJJ5yA+fPnY/78+bj66quxdOlSfOtb38JRRx2F7u5u/OMf/8i622nUPwlfeyeanTKm1H3fXAdbvPjXr19S0PuUaowAMo4z02vPdAztHLNMrrvuOrhcLtxwww148MEH4XK5ErkR27dvT3sc87lfqvtc/8MfAgCsApe17rrrLkQiEXR1deFf/uVfcPfdd8/4Rp7LmLK9jq1bt+JnP/sZNmzYgJ6eHni9Xni9Xjz00EM5jTnT8Z2tWOePNWvW4IUXXpgxuwNk/oxlOucBM88Hd9xxR9HOc8m++93v4vDDD8eiRYuwZMkSHHrooTM2Jkx3LDN9LjId/2yfp3x88YtfxDPPPINIJPt2DPnI9B4qioLf/va3uO+++9Da2opHH30Up5xyyoz7P/nkk1i5cmViRvrTn/50TueSqpUpo7ncqdXVYnBwkL/00kuVHgbJoJgbnFmWxd8c93POOf9wb4j7o3pRHreaORwO3tTUlKhkzERVVb548WKuaVrW2xZ6v/GQykcCEf7JT36Se71eftxxx2W8fS6vI98x5fv6Oee2Xkchj1/v6uH9TefKK6/kt912W873K4ePfvSj/B//+Eelh5GvtDEN9dKaZXx8HHPmzMHmzZsxZ86cSg+HpDF37lz8/Oc/xyc/+cmiPN5bEwEs7fDBsCy8uyeEJR2lTXIlqXHO8eZEAB/pbKr0UAghtYl6adnx+uuv44ADDsAll1xCwU6DkgQBFo8lL5PyY4xBEhh0K/tGhIQQkgua4SEEwNsTARzY7oXAGPZGNYQ0EwNNxd/2nWTnV3XsjeoYarbXQZ0QQpLQDA8hmbhlEWHdBAC0OGRMqoWVR5P8NTlkBLTa6P9ECKkdFPAQAsCnSImLLGMMbllEiC66FeMQBUQNs9LDIITUEQp4CAHgVSQEkwKcQZ8L22108Cal0e9zYTiQW/d6QgjJhAIeQgAoogCntG+TPlmMtToodE8Ykh+3LCJCMzyEkCKigIeQaYOzkpR7PU7sCha28y/Jn0eeOetGCCGFoICHkDTaXDJ2R7RKD6MgW7ZsAWMs0SJh5cqV+PnPf17hUc2Ubkz9PictaxFCioYCHkLSiCcvx6u3CsE5x/z587F06dIijKx2XXvttfjCF75g67aKKEC3rII7qJfC3Llz8cILL1R6GISQHFDAQ0gGA00ubPcXnrz8pz/9CWNjY/jggw9SNkslqbU6ZeyN2tsigHM+ozN3tUrXaZwQUloU8BCSgaNIswz3338/Pve5z+HEE0/E/fffP+N39913H+bPnw+fz4d58+YlGk8ODQ3hjTfeAAA8+OCDYIzhrbfeAgD8/Oc/x8knnwwAsCwLN9xwAxYsWID29nasXr0ae/bsyXmMr732Go466ii0tLSgt7cXF198MTQttqTHOcc3vvENdHV1obm5GQcddFDaLvQjIyM46aST0NbWhoULF+Lee+8FADz33HO4/vrr8eijj8Lr9eLggw9O3Gfr1q045phj4PP5cMIJJ2BiYgIA0ONx4q+b38fRRx+NlpYWHHzwwVi/fn3ifitXrsRVV12FY445Bm63O9GpO+6GG27AaaedNuNnX//613HppZcCAKampvClL30Jvb296O/vx9VXXw3T3Dejd++992LJkiXw+XxYunQp/va3v+Hss8/Gtm3b8NnPfhZerxc33XQTgFjDxWXLlqGlpQUrV67E22+/nXicuXPn4sYbb8RBBx0Ej8cDwzBw4403or+/Hz6fDwceeCBefPHFnN4vQkiOMjXaKmu7L0Kq1FgoyncGInnfPxQKcZ/Px59++mn+2GOP8fb2dq6qKuec82AwyH0+H9+0aRPnnPORkRG+ceNGzjnnZ599Nr/llls455yvXbuWz58/n991112J3/34xz/mnHN+22238SOOOIJv376dR6NR/uUvf5mfeeaZnHPOP/zwQw6A63qsIeqKFSv4vffem3Kcf/3rX/mf//xnrus6//DDD/nixYsTzQ2fe+45vnz5cr53715uWRZ/6623+MjISMrH+fjHP84vvPBCHolE+N///nfe0dHBX3jhBc4559/73vf4WWedNeP2K1as4PPnz+ebN2/m4XCYr1ixgl9++eWcc8537NjBz7/yWv77p5/mpmnyP/zhD7ytrY2PjY0l7js4OMg3btzIdV3frwHkli1buMvl4lNTU5xzzg3D4D09PfzPf/4z55zzz33uc/zLX/4yDwaDfHR0lB9++OH8nnvu4Zxz/utf/5r39fXx1157jVuWxd99912+ZcsWzvn+zWs3b97M3W43/8Mf/sA1TeM33ngjX7BgQeJ9Hhoa4gcffDDftm0bD4fDfNOmTXxgYIAPDw8n3qf33nsv5fEkhOQkbUxDMzyEZNHhUjBRQPLyb3/7WzgcDpxwwglYtWoVDMPA008/nfi9IAjYuHEjIpEIent7sWzZMgDAihUr8NJLLwEAXn75ZVx55ZWJf7/00ktYsWIFAOBnP/sZfvjDH2JgYAAOhwPXXnstHnvssZyXTg477DAceeSRkCQJc+fOxVe+8pXE88myjEAggE2bNoFzjiVLlqC3t3e/x9i+fTteeeUV3HjjjXA6nTjkkENwwQUX4IEHHsj43Oeddx4WLVoEl8uF1atXY8OGDQBiM1tNIsc/rfh/IQgCjj/+ePzTP/0TnnnmmcR9zz33XCxbtgySJEGW5RmPOzQ0hOXLl+N3v/sdAGDdunVwu9048sgjMTo6imeffRa33347PB4Purq68I1vfAOPPPIIgNgs2r/+67/i8MMPB2MMCxcuxNDQUMrxP/roo/jMZz6D448/HrIs49vf/jYikQj++7//O3GbSy+9FIODg3C5XBBFEaqq4q233oKu65g7dy4WLFiQ7S0ihBSAAh5CsmCMwSEKee8Lc//992P16tWQJAkOhwOnnHJKYlnL4/Hg0UcfxT333IPe3l585jOfwaZNmwDEAp6XX34Zu3btgmmaOOOMM/Dqq69iy5YtmJqawiGHHAIgthz0+c9/Hi0tLWhpacGSJUsgiiJGR0dzGuc777yDVatWoaenB01NTfjOd76TWFo67rjjcPHFF+Oiiy5Cd3c3vvzlL8Pv9+/3GCMjI2hra4PPt6/b/NDQEIaHhzM+d09PT+L/3W43gsFg4rXdc+tNuPK6HyVe3yuvvIKdO3cmbj84OJjxsdesWYOHH34YAPCrX/0Ka9asSTy2ruvo7e1NPPZXvvIVjI2NAYgFb3aDkJGRkRnBkCAIGBwcnPG6k8e5cOFC3H777bj22mvR1dWFM888EyMjI7aeixCSHwp4CLFhMM/k5R07dmDdunV48MEH0dPTg56eHjz22GN45plnEsHEpz71KTz//PPYuXMnFi9ejLVr1wKIXRTdbjfuuOMOfPzjH4fP50NPTw/uueceHHXUUdB1HaZpYnBwEM8++ywmJycT/0WjUfT39+c01gsvvBCLFy/Gu+++C7/fj+uvv35G7tKll16KN954A2+++Sbeeecd3Hzzzfs9Rl9fH/bs2YNAIJD42bZt2xJjYSxtX7+UBgcH8YWzzsL/961vYWLPXkxOTiIUCuGKK65I3CbbY55++ulYv349duzYgccffzwR8AwODsLhcGBiYiJx3Px+P958883E799///2Ujzn7Ofv6+rB169bEvznn2L59+4z3YPZ91qxZg1deeQVbt24FYwyXX365jSNCCMkXBTyE2OCURGhm7snLDzzwABYtWoTNmzdjw4YN2LBhA9555x0MDAzg4YcfxujoKJ588kmEQiE4HA54vV6I4r4dn1esWIE777wTK1asAOccH/vYx3DXXXfh6KOPRiQSQSAQwDnnnIMrrrgC77zzDgzDwNjYGJ544omcX2MgEEBTUxO8Xi82bdqEu+++O/G7119/Hf/zP/8DXdfh8XjgdDpnjDNucHAQRx99NK688kpEo1H83//9H/7zP/8TZ511FgCgu7sbW7ZssV1N9YUvfAFPPfUUPvzH3zDsDyMajSaCF7s6OzuxcuVKnHfeeZg3bx6WLFkCAOjt7cUJJ5yAb33rW/D7/bAsC++//35iGe+CCy7ALbfcgjfeeAOcc7z33nuJoKa7u3tGgvTq1avx9NNP48UXX4Su67j11lvhcDhw9NFHpxzT5s2bsW7dOqiqCqfTmVjmIoSUDgU8hNjU6XZgPJxbLs/999+Pr33ta4nZnfh/X/3qV3H//ffDsizceuut6OvrQ1tbG1566SXcddddifuvWLECgUAAxx57LFRVxTHHHINAIICPfexjEEURoijia1/7Gv75n/8Zn/nMZ9Da2oojjjgCL7/8MqLRaE55PLfccgt+9atfwefzYe3atTjjjDMSv/P7/Vi7di1aW1sxNDSE9vZ2fPvb3075OA8//DC2bNmCvr4+fP7zn8f3v/99HH/88QBisy0A0N7ejuXLl2cd0+DgIJ544gnceO01uO7mH2NwcBA333xzzuXna9aswQsvvJCY3Yn75S9/CU3TsHTpUrS2tuK0005LLJedfvrpuOqqq7BmzRr4fD6cfPLJieq3K6+8Etdddx1aWlpwyy234MADD8SDDz6ISy65BB0dHXjqqafw1FNPQVGUlONRVRVXXHEFOjo60NPTg7GxMVx//fUAgIceeiiRx0UIKR6W5Rtr9e34RUiFcM7x5kQAH+lsKutzGoYBwzDAGEssixiGkfaiH69IsCwrcXtRFCHLMiRJgiiKOS8tVYN39gQx1OSGQ6LvaYSQtNKe3KRyjoKQWsYYgyIIUA0TDqn0yw+WZUHX9UTgYjdIid9WEGKBQTz4iUQiNR0A9XudGA5GML/FU+mhEEJqEAU8hORgoMmF7YEoFraW7qLLOYdpmtD12A7DuQQ7qaQLgKLRfX2q4gGQKIqQJKkqAyCPIiE8Fa70MAghNYoCHkJy4JZFRHQTnPOSBAWc80T1VaGBTjqzHzdbACSKYiJYqjS3JCKkG/DIdOoihOSGcngIydFIMApFENDhTp2Qmq/4rE48mMoU7GTK4SlU8s6kcYIgzFgCq1QApBoWtvrDWNTmrcjzE0KqHuXwEFIsPR4H3poIFC3gmZ2YXOnZlFQzQJxzqKoKVVUBxAKg+M7G5QyAHJKQ2B6gGpfdCCHViwIeQnIkMAZZiF14FbGwC32+icnllC4A0jQt0Vx0dgBUytfS7JAxpRpoccrZb0wIIdNoSYuQPIR0A7uCKhbkmbxcaGJyKZe0chU/hySPhzG23xJYsQIgw7Lw7p4QlnT4st+YENJoaEmLkGLyyBLCejivpZVyJCaXU3KpO7AvANI0bcYSWLECIEkQYE7PMtX6sSOElA/N8BCSp+FABC5JRJvLfi5PLonJmVTTDI8d8Uqw5NctSVLiv1wDoF2hKETG0Ol2lHDUhJAalPZEQgEPIXmyOMfbEwEss7Hzcrodk/NVawHPbPEAKC7XACiXY08IaSi0pEVIsQmMQWAMumVBzlClVAuJyeXGGJvRLDO+zJec05S8G/TsAEiYPo6mxSEKdDwJIdnRDA8hBQhoBibCGua1uPf73ezE5GKWbpumCdM0i/Z41SbVXkDJVWCiKGJvVEfEMNHvc1VwpISQKkMzPISUgk+RsDVFu4N42TbN6uQnVSm8aZozur/LoohtQR3dLrlm+oERQiqHAh5CCtTskDEZ1RP7wsRzdQpNTCb7pGuHAdPEXn8AssBqsiEqIaR8aEmLkAKZFsfmPUEsafcWNTE543PW+ZKWXSHDwqRqos8tJYKgWu4ITwgpGC1pEVIqosDAOUc4qkIAzeqUk0cSMBLW66YjPCGkdCjgIaQA8dySTkXAzpCKQZ+z0kNqOA6BIWpacCa1+cjUET6+1BgPfCrdEJUQUh4U8BCSp+TE5CaHhJGwXukhNaRul4RdEQND3vQbQKYLgGY3RK2GjvCEkNKggIeQHKXrg+WVRQQ0Az6F/qzKySEK0Kzc0g3z6QhPS5WE1DY6M7qmBvcAACAASURBVBOSg0w7Jvd6FHzgj1LAUwEeSUBQt+CV85uVqbaO8ISQ4qMzMyE2WZYFTdPSlptL08nLtPtv+XU5JWwL6fDK9vuaZZIqAAIwIwAqZUd4QkjxUcBDSBazZ3Uy5Xb0uBXsCmvo91JTy3KSBFbSDurl7ghPCCk+CngIySDXHZObHRJGQhr6yzQ+sk+zImJKs9DiELPfuECzAyAA+y2BFdoRnhBSXBTwEJJCusRkOzyygJBuwiOX/sJL9ulwiPggoJUl4EnFTkNUCoAIqRwKeAiZJX6hMk0zr8TUXo8DW/xRHNBCTS3LSZh+nyzOE/9fSakCIMMwbHeEJ4QUFwU8hCTJlphshzydT1ItF95G0u6QsFs10emsvlNbpoao8c9b8gwQtcMgpLiolxYhyFxuno+9qoGoYaHXU5yqodmol1ZqnHO8F9BwQFPtJY3HS+GpHxghBaFeWiQ/pap6qSaWZUHXdduJyXa0KCI2hbSSBTz1/p7kizEGkTEYFodUY1sDpOsHFolEKAAipAho73SS0fMfjmMqqlV6GCURn9VRVbWowQ4Qu3i5JAFhnWZhyq3LKWIsalR6GAWLBz/JZe7xACgYDGJqagqBQADRaDSxLEYISY8CHpJRk0PCpt0hhPTav4AkS66giV9Yiv1tud+jYCRUn8FiNfPKIkKGVelhFF26ACgaje4XAMVnLAkh+9CSFsmo1Skjaph4d08Ii9u9cEq1X2odLzcvJDHZDlkUYFDyckUoAoNqWnCI9fudLlNH+Pi/Zy+BUUNU0sgoaZlktDMYxXhYxdxmNz6YDGNZhw9yjV5Eip2YbMfuqA7D4uh2FzeXx7IsGEZ9zboVU9S0MBY1MKdEOVS1IJ4EnXyOn90PjAIgUofSntjp004yalIkeGQJI8Eolnb48OZEAGaOnamrQbzcvJzBDgC0OSTsqYN8klrjFAWoZu19TospvgQmimIiuInvBh0KheD3++H3+xEOhxNLYJQHROoZLWmRjNyyCNWM5QIIjOHANi82jvvx/3Q11cQyTSE7JhcDYwxOSUDUsOCU6PtFOblEASHDgoeOOwDqCE8InQlIRowxcACDPhd2+CNwySIWtHrw1kSg6r8Nxk/mpUxMtqPPo2A4pJb9eRtdt0vCWIRm19JJNQMExBqiBoPBGTNAmqbBNM2q/5snJBOa4SG2+BwStkyFwTmHV5Ew4HNh0+4gFrd7q/IbYLkSk+1wiAJ0s3SdvElqssBglLCDer3J1hE+/nc0OweIji2pFRTwkKwUgUEzLXS4FUxENHS6HWhxyrA4h8UBsYrOd7MTk6slKbPDJWMiqqPT1bhJtJXQJIvw6xaaldqvLiy3dB3hqSEqqVXVcTUgVa3JIcOvGujxODCatDTT5lIgVtFutpVKTLaj3SlhgpZXyq7DKWKCksaLJt7wNP4fYwy6riMSiSAQCMDv9yMUCkFVVVoCI1WHZnhIVk2KhJ2hKDrcClySiLBuwC1Xz0dndmJytczqJGOMQRHrf2+YaiNO56DRXkilka0jPIAZS2DUDoNUEp15SVZOSUBkeufaOU0ubPNHKjyifVIlJlerfo8Dw0HaebncWh0i9qrU4qMcUiVCm6Y5ox1GMBikdhikIqrnazqpWsnfyGRRgMVRFc0Z47k61ZCYbIdTEqCaFiXRllmbIuK9gIZ2J53uyi1VKXx8NpYaopJyozMAsS1+oR7wObHDH8HcFnfFxlGNicl2tDtjGxG2u+RKD6VhMMYgMAbT4lWVc9aIqCM8qaTauVKQiortXBtb1mpyyAholZmOtiwLqqpWZWKyHZ0uGeNRPfsNSVF1OkSMq5S8XG2oIzwpJ5rhIbY0OSQENSPRPDS5RL0canlWJxljDBKLlfkrBSQv11qgV2k+WcBo1ECPq9IjIZmkmwGKN0QF9s0Azd4wkZBsqHkoyQvnHG9OBPCRzqayPJemabAsqyZndWaLGCZ2hXXMa3Lm/Rjx/VCIfVsCGvo8MhRa1qpZ6RqiUkd4kiTtHzjN8JC8MMZKXqJe6T5YpeKSREQNlZKXy6xrutXEgIfyp2pVun5gqqpCVWN7hFFHeJIOzfCQvOmmhfcnQ1jc7iv6Y89ewqq3wGBXSIMiMrQ587v40gxPft71qzigqTzLsKT80s0AUUPUhkIzPKT4SlWiHt8xuVbKzfPR5ZbxzmQk74CH5McpxvaUclEH9bqUagYIAHWEJwAo4CEFKmaJer0kJtshMAaRMeiWBbmOX2e16XZKGInomOulnmaNIFtDVCB1DhAFQPWJAh5SkCaHjK1TkYLzUSzLgq7rdZOYbEefR8FIUMNQAcnLJDeKyKBb1EG9UVFH+MZGAQ8pWCEl6vWamGyHRxaxLZBf8jLtR5I/ryQgaFjwydRBvdFRR/jGQgEPKViPx4E3JwI5BzzxE4tpmg0V6CRrdoiY0ky0OOhPsVy6nBK2hHQKeEhKqRqiUgBUH+gsSwq2r0TdhNvmRSQ+q1PPicl2dLsVvDcZyTngadTjVQyiwGBxWtYi9lBH+PpBAQ8pijlNLlsl6o2UmGyHOB3sVUMz1kbSqojYq5loo5k1kqN0DVENY1/rkuQZIAqAqgf9tZOiiJeoZ2rQ2IiJyXb0uRXsDGkY9NH+MOXS5hDxQUCjgIcUjDrC1w76aydFM+BzYnsggrnNM0vUGzkx2Q6vImJ7UK30MBqKMP35szhP/D8hxUAd4asXBTykaFKVqFNisj0+RYRfM9Ck0J9kuXQ4JUxETXS56JiT0smlISoFQKVFrSVIUe0MRiEJDJ1uByUm58C0ON6biuDAVnsbOFJricJxzvFeQKNWE6SiUrXDoI7wBaHWEqQ8ejwObBwPoEUWKDE5B/G8p0w5UKS4GGOQWGwjQpmOOamQVDlAs2eAqCN8cdAMDykqy7Lwzu4AupwS3DJNzebCrxnwayYGvNlnHGiGpzhChoVJzUS/m3qakepkpyEqBUAz0AwPKa3kxOQ+t4RtQR0HtNDHKxdNioThoFbpYTQUjyRgJEyBI6leqWaAOOfUEDUPdEUiBYv/8cXLzR2SBItrtDyTB68sIqiZ8Cq0C3C5OASGqGnBKdK3ZFL97HSEZ4wllsAkSaIAaBotaZGCpEtMDmgmJlWD9pbJkWFxfOiP4oAWV9bbxk9upDCqaWFXxMAQdVAndSB+TbcsK/H/DdYRPu0Lo680JC/xHJLkKdXkPyCfIiKom9TkMkfSdNsDk45b2ThEAZpFx5vUh/gXT1EUZ8zwaJqGUCgEv9+PU089tdLDrAgKeEjOLMuCpmmJKqx03xTanBL2qEbK35H0ut0KRsOZZ28okCwujyQgqFuVHgYhJZEcAKmqil27dlV6SBVBAQ+xLd4HS1VVcM6zTot2uWSMUUJozpoVEVOqWelhNJQup4SxKAXnpP6FQiG43fb2+6o3FPAQW+KJyfH+MHbWfxljcEoCIgZdvHPBGINbEhDS6biViyQwmLNKfwmpR+FwmAIeQtKJz+rk0/RzwKNgB5Va56zPq2AkRMetnJplEVO0rEXqXDgchsfjqfQwKoICHpJWPDE5vsFdPpn9sS7qHCYlheZEFgSYnMOiGYey6XCKmKBlLVLnIpEIXK7sVaD1iAIekpJlWVBVNWtish19HgfNVuSBcqDKK7mDOiH1KhQK0QwPIcC+WR1VVQHkN6szG5Wo56fVIWEvVbmVVbtDwm5KGCd1jJa0CMG+xORizOrMRiXquaOk7/JrUQRManS8Sf2igIc0tHi5eTQazSsx2Q5anslPv0dJ2V+LMUYzZiXAGIPIGAzKOSN1iqq0SMNKTkxmjJVsy3GarciPIgowrNTl0nW8NXxFdTpFjFPyMqlTFPCQhhRPTDZNsyzN5ahEPT+dLhnjEZodKxefLCJoUHk6qU/hcBher7fSw6gICngaUCkSk+2gEvX8tDkl7KEZh7KSBQbNpKCH1B8qSycNw24frFKhEvXcMcagiAKiNOtQNt1OCaMUZJI6FAqFaIaH1LfkPlilSky2g0rU89PvVTAcUis9jIbhkgRETfqMkvoTiUQoh4fUr3IlJttFJeq5c4gCNJN6PZWTSxQQplk1UmeoLJ3ULdM0y5qYbAeVqOenwyVhNy2zlE23S8JohI43qS8U8JC6E5/V0bRYvkylZ3WSUYl6fjqcMiaSqrWq5f2sV7LAYFAHdVJnKOAhdaXSicl2UIl67hhjkEUGlaqHyqZJFuGnDuqkjlDAQ+pCtSQm20El6vnppyq3sqIO6qTeRCIRCnhIbYv3waqWxGQ7qEQ9d04pVp5OyyzlITIGC6DjTeqGrutQFKXSw6gICnjqQDwxudpndWajEvX8tDmpi3o5tTlE7KEO6oTUPAp4alg1JybbRSXquet0yRijVhNl06aI2EMd1EkdaPQvlxTw1KhaSEy2g0rUcycwBokx6JT/VBaMMQgAzAa/WJD6UavXi0JRwFNjZicm1+KsTjIqUc9Pn0fBTgoUy6bTKVEHdVLzOOc1fb0oFAU8NSRVYnI9oBL13LllEVGTkpfLxScLCFB5OqlxqqrC4XBUehgVUx9XzAZQK+Xm+aAS9fw0K7RHTLkwWkYkdSAcDjdsp3SAAp6ql9wHC6jNxGQ7qEQ9d7TMUl7UaoLUulAo1LB78AAU8FQ1y7KgqmrNJybbQSXquRMYg8AYDJp1KAu3JCBCu1yTGtbIndIBCniqUnxWR1VVAPU7qzMblajnrtslYRfNOpSNUxQQoQ7qpEbRDA+pKvHE5EaY1ZmNStRzwxiDRxIQpgtw2XQ7RYzSMiKpUcXI4TFNE4ceeihWrVpVpFGVj1TpAZAYzjlM00zk6jRasAPMLFF3SWKlh1MzfLIAv26iSaZjVmqKKFDiMqk5jz/+OHbv3g1RFAte0vrJT36CJUuWwO/3F2l05UMzPFUgvrdOLfXBKhUqUc9dl0vCGC1rlY1XEhDQad8oUjsOPvhgcM7x4IMP4oEHHsCZZ56Je+65B5s2bcopb3LHjh14+umnccEFF5RwtKVDMzwVFt8xOb4hVKMGOnHJJeqi0NjHwi6RMTCAjlmZdDolbAvp8NGMGqkR8+fPx/z58+HxeDA5OYlVq1Zh/fr1+O53v4v3338ff/nLX2w1FL3ssstw0003IRAIlGHUxUcBT4XEZ3XiuTr1solgMcRL1Ad9jbtBVq66XTJGowb63HKlh1L3JIHB5Lzhd60ltSccDsPr9WLZsmVYtmwZLrroItuf49///vfo6urCYYcdhvXr15d+sCVAV9kKqJc+WKVCJeq588oCgrQJYdm0KCL2anS8SW0Jh8P7VWnZvf68+uqrePLJJzF37lyceeaZWLduHb7whS+UYpglQwFPGc3ug0XBTnptTgl7qUQ9J7Ggh3JLyqHdIdIWCqTmRCKRvMvSf/SjH2HHjh3YsmULHnnkERx33HF48MEHizzC0qKAp0ySd0xu9MRkO7pcMkapRD0n3U7aCbhchOm/XYtmIUkNCYfDtPEgKS3TNKGqKkzTpFkdm6iLeu5EgYEDMOkiXBYdTgkTUfp8ktqRakkrHytXrsTvf//7IoyovCjgKaH4rI6mxcqsaVYnN1SinrtuKlEvm2ZZwBQtIZIaQjM8pCQoMblw1EU9dz5ZRJB2Xi4L6qBOak2xZnhqFQU8RUaJycVFXdRz5xIFhCjoKYsup4QxajVBakS8LL1RUcBTRJSYXHxUop67HhclL5eLR6ZeZqR2ULd0UhSUmFw6VKKeG0lgsDinCqIycQgMUZOCHlL9QqEQzfCQ/FFiculRiXpqmT5nnU4J41RBVBbUy4zUCk3TbLWQqFcU8BSAEpPLg0rUc9ckC5jS6HiVg1MUoFLiMqkRjdzGqHFfeQFmJybTrE7pUYl6bhhjcEkCIpRfUhZuSUCIWnuQKkZ5kBTw5IxzDk3TZiQmk9KjEvXc9bgk7KKllrKgai1SCxp9JYKu1jmIJyZTuXll9HkUKlHPgSwwGJS8XBbxY03fokk1a/TPJwU8NlBicnXwKRKVqOeowyFht0q5POXQLIuYomUtUqV0XYcsy5UeRkVRwJMFJSZXFypRz02LImAvBTxl0eEUMUHLWqRKhUKhht6DB6CAJy1KTK5OVKKeG8YYnCLtE1MO1EGdVLNG76MFUMCTEiUmVy8qUc9dj0vGrjDNPJRDGy0hkirV6H20AAp4UtI0jRKTqxiVqOdGERl0Sqgti1ZFwCTtf0SqEM3wUMCTEuecgp0qRiXquWtziDTzUAaMMYiMwaDPJqkyFPBQwENqFJWo56ZNEbGXZh7KotMpYpySl0mVoaRlCnhSopmd6kcl6rlhjEEWGFRKXi45nywiSDtckyoTiUQoh6fSAyAkX1Sinpte2nm5bGSBQaPgklSRUChEAU+lB1CNaIanNlCJem4cogDVpOTlcuh2ShilbvWkikQiEVrSqvQAqhFjjC4KNYBK1HPX6qBcnnJwSQLtfUSqSigUgtfrrfQwKooCHlLTGrlEPZ+ZyA6q1ioblyggTLk8pErQDA8FPKTGUYl6bhhjkBiDRser5LpdEkYpZ4pUCdp4kAIeUgeoRD03vW4Juyj3qeSogzqpJjTDQwFPSpS0XFuoRD03TlFAlJKXy8InCwhQB3VSBahKiwKelCjgqT2tVKKekxZFxJRGF+JS63RKtAkhqQq00zIFPKROdFOJek46nCLGKUAsOZExWADNppGKi0QiVKVV6QEQUgxUop4bYbrnk07JyyXXpojYQ1sBkAqjpGUKeFKiJa3aNOBRMNygJer56KGdl8uizSFiL20FQCqMAh4KeEgdkUUBJpWo2+aWBEQMi5ZbSowxBgbApONMKigajcLlclV6GBVFAQ+pK1SinpsmRYCfqohKrsMpYYJaTZAK4pxDEBr7kt/Yr57UHSpRz00XVRGVRZMswK9TwEMqg86HMRTwpEA5PLWNStTtExgDYwwGLQOWVHyHa0oSJ5XU6Nc2CnhI3aES9dz0OKkFQjlQqwlCKosCHlJ3GGNwilSibpdHFhCiJpcl55YERKiDOqkA0zQhimKlh1FxFPBkQOuetWvASyXquYi1QKAAsdScIkOEgktSZrTLcgwFPCmw6bwGUruoRD03XS4JY7TcUnLdTgmjlCROyiwUClHAAwp4SB2r9xL1Ygbl4vRj0V4xpaWIAiUuk7KjTukxFPCQukUl6rnpoqTasvBKtHxIyotmeGIo4CF1jUrU7fPJIoK0CWHJxTqoU8BDyodyeGIo4EmDcnjqA5Wo58YjCxT0lJgkMJic08wjKRvqoxVDAQ+pa/tK1Okibgcl1ZZHiyJiUqPPJCkPWtKKoYAnDZrhqR+xEnW10sOoCZLAwDmHRbMPJdXuELGbllpJmUQiEZrhAQU8aTHGaMq5TtRriXqpPp9dTgljNMtTUsL0FyoKLEk5FLKktX37dnziE5/AkiVLsGzZMvzkJz8p8ujKR6r0AAgph3iJ+qDPUemhVL0mRcSo30CPq9IjqW/xDupdLjoNk9L44Q9/CF3XMTU1hUMOOSSvx5AkCbfeeiuWL1+OQCCAww47DMcffzyWLl1a5NGWHs3wkIZAJeq5cYkCwpT3VFLNsoApKk8nJXTxxRfjiCOOwMaNG3HbbbfhE5/4BP7t3/4Nr776KjTN3h5lvb29WL58OQDA5/NhyZIlGB4eLuWwS4ZluQA07NVB13UYhgFBoJiwXuwKaVBEhjanXOmhFAXnHLpemgo03eLYHtIx36eU5PGrnq5DmgzA6Gwr6dN8ENAwxyNDEihnkJTOddddh6OOOgpHH3001q9fj3Xr1mFychKPPPJITo+zZcsWfPzjH8fGjRvR1NRUotEWLO0fE82lpkFJy/Wn2y1j095I3QQ8pSRPl05bnCfyTRqGbsDz5odgFgffPo7Q8gNL9lTxfKk+N30mSenEc3i6urqwevVqrF69OufHCAaDOPXUU3H77bdXc7CTEU1fpEEBT/2hEvXcdE7nmDQUzuF+dxsEi8e+Jsql/U7opU71pAwKLUvXdR2nnnoqzjrrLJxyyilFHFl5UcBDGko/lajb1iwLmNQaKOCxLLjeH4YY3bdMaHqcJX9aRWBQTQp6SOkUUpbOOceXvvQlLFmyBN/85jeLPLLyooCHNBSlTkvUS4ExBqfEGmJGjGk6HDvGofa2wxL3nRa5Uvqlpm7qYUZKrJCy9FdffRUPPPAA1q1bh0MOOQSHHHIInnnmmSKPsDwohycNWtKqX1Sibl+vS8aOkI55dZy8LPpDEINhqAOdYLoBZlrgAKJze2G5Sv8ZcYoCVArASQkVEvAce+yxdVPdSjM8pOH4FAkBKlG3RRYYjHrt+8Q5lF27wQwTWl8nIAhQxvaCATDam2C0NZUl4AEAt0S5PKR0wuEwvF5vpYdRcRTwkIbU5qAu6na1O0RMqPWVy8MMA44dYzCavTDapitODBPy7ikAgNZV2nL02bqcEsZoWYuUSCQSgctFO4lSwEMaUrebuqjb1aqIdZW8LIQiUHbtgdrXMWMGR5mYBLM4DJ+7bDM7cXU9k0YqzrIsSBJlsFDAkwbl8NQ3KlG3jzFWH5VEnEMen4QYjkLt7wREcd/vrNjvAEDrLu/sTlyzLGJKr/FjTEgVo4CHNKxaL1EvZ1De65Kws5aXXEwLjuFxWC4FemcrMOvYSXv9EHQDplOB6ct/v5JCdDhFTFDTVlJkNGu4D81xkYaVXKIu0tb+GSmiAN2KLbnU2uynEFEhT0xC62kHT7WRIOdQxvYCmM7dqdDrS+6g3nC7W5OSq7W/21KgGZ4sKDqub/ES9VpU7s9mmyJiT43l8kh7/JCmglAHulIHOwDEYARiRIUliTDafGUe4UxtDhF76ixBnFSWZVkU7EyjgCcNxhh9SBoAlajbV1MXY8uCMjIBLgrQetozztooo3sAAHpnC1DhZsGtioi9NRZUkuoWiUQKaitRTyjgIQ2PStTtYYxBZgyaWd3BIVNjuybr7c0wmzPvPcKiGiR/CJwx6B0tZRphhvEwBoEx2gmcFE04HKaS9GkU8JCGRyXq9vW4JeyMVO+xEqeCkHdPQh3oBHdkbwuhjE3P7rQ3pV3yKrcup4gxSl4mRRIKhfLeZbneUMBDGh6VqNvnFAWoZhXuFxPfNdm0ErsmZ8MMA/JuPwBA72ot9Qht80oCgvRZJEVCS1r7UMCTAeXwNI5aL1Evp1aHiEmtei7ITE+xa7IN8vgUGOcwmjywnNXTV40xBlmo/qVDUhvC4TAFPNMo4CEE1EU9F7FWE9Wx5CIEI1BG90Dt68xtd2TLgjw+XYreXT2zO3HdTgmjtKxFiiAUClHAM40Cngxohqex9HoU7AzXZol6OQmMQWIMeiWDQ84hj++FGInvmpzbqUzaG4BgmDBdDpje6rsYuCQB0Vrf2ZpUhUI6pdcbCniyqLpcBVIyTYoEv1Y7JeqVDMh7XBJ2VSp5eXrXZNPlTLlrclacJ0rRta487l8mLlFAmHJ5SIEo4NmHAp4MaIan8VCJuj0uSUDEKH/yshBR4RgZh9bdBsubX6mtGAhDjGqwZBFGq/2cn3LrclEHdVI4SlrehwIeQpJQibp9zUp5m13a2TXZjkQpemcrUMUtRRSBQacO6qRAVJa+DwU8hCRhjMFBJeq2dJar2aVlQRkZt7VrcjZCRIXkD4MLDFoVbDSYjU8WEKAO6qQAkUiEAp5pFPBkQEtajWmAStRtEaZ3BTZKmLy8b9fklqy7JtshTzcJ1dubAUks+PFKrdMhYbxW2nmQqkRl6ftQwJMBBTyNiUrU7YslL5dmlie2a/KU7V2Ts2G6AXmPHxyA1ll9peipiAKDRctapAC0pLUPBTyEpEAl6va4JQGRYpdPz9g1uaNoDT3l8cnYRoPNXnCnUpTHLIda7FJPqgctae1DAQ8hKdRaiXoleSUB/iJdkPPdNTkry4I8PgkA0Ktwo8FMWh0i9tKyFskTLWntQwEPIWnUQol6NSy7drmkojS7FIPh/HZNtkHe44dgmjDdTpie2uocLTAGBsCk4JvkgWZ49qGAJwPGWFVcUEhldLlljFGJelZi/IKcb87T9K7JQkTLa9dkW48/Ot1Gooo3GsykwylhIkqzPCR3lMOzDwU8WdCSRuMSGINCJeq2dLtk7Mpnlsc0Y7smu53QO1tKEoyI/hBEVYMlSzBafUV//HJokgX4dQp4SO5ohmcfCngIyYBK1O3xygJCOe4XE9s1eSK2a3IJl5mUsdqe3QFis80V719GahK1ltiHAp4MaDmLUIm6fV5ZQNDmLIS0ewqiP1TwrsnZCOEopEBso0G9o7lkz1MO3S4Jo9RqguTIMAwoSu1UJZYSBTyEZEEl6vZ0O21ckC0LyvA4uCRB724r+YxLfHZH72gBxOrfaDCTkmwBQEgDoYCHkCyoRN0eUWDgSF9NxFQttmtyZwvM5tJPsTNNh1RjGw1m4xAZohT0EJvonDUTBTw20IeG1EKJejXoTtPhO7Zrsj+2a7JS+K7Jdsjjk2AAjBZfUXZqrga2ZtEImYXSM2Io4MmAPiQkjkrU7fHJ4sxmlyXaNTkr04IyEdtoUKuxjQYzcYgCNMonIzbRl/WZKODJgoIeAlCJei7ckoCQYQGWBceHI8XfNdkGec8UmGnB9DhLWgFWCR7JfnI4aWyqqsLpdFZ6GFWDAh5CbKISdXt64tVEggAxFAXK/Z2B86RS9LYyP3npdTkljNEmhMSGUChEbSWSUMBDiE1Uom6PNN3h2+Ic3CGDl2sZK/78U0EIqg5LkWG0eMv63KUSNS28H9CgmRySwGBSB3ViA/XRmokCnixoSYskoxJ1ezqdEsajJvT2ZkhTobI+d623kUjFKQoY9MjYEdbxYUCDVxIwqdHyKsmMNh2ciQIeQnJQbSXq1RqQN8kCpjQTRnszmFa+ZG8hFIEUioCLAvT22t5ocDZFYJjvU9DnljClGfggQMurJDOa4ZmJAp4sqvWCRPWfEQAAIABJREFUQiqHStSzY4zBJcWSvLksgWnlOV6J3J2OluI3Ia0CEcPC9pAOCwwCiy0dEpIO5fDMVH9nhBKolm/zpDp0uWWMUol6Vj0uCbsiBowWL6TJQMmfj2k6pL0BcCDWiLSOhA0L7/lVjEUNDHlkiIxhrlfGhErJyyQ9muGZiQKeLGiGh8wmMBbb8ZZK1DOSBQaDc5iyBKaXfoZHGdsb22iw1Ve2zQ1LLaRbeNevYkI1MM+rYMirYFK30OEQ0aKImNIo4CHpUaf0mSjgISQPA14HdlCJelYdDgkTqgmuSKXN5TFNyBNTAACtu/ZL0QO6iXemVOzVTMz3KZjjUWKtOzjHHtVEm0MEYwwiYzCoapCkQUtaM1HAQ0geGqFEnYWjEMcnC3qMFmU6ebnFB2kyWKSR7U+emAKzLBheFyx37W605tdigY5ft7CwScHA9PJV3GjUQLdLSsw8x/bkoXwykhrN8MxEAU8WtKRF0qnnEvVY76spOHbthjI8nvfjMMbgEBgiglC6ZS3OoYzX9kaDk5qJzVMqQkYs0Ol3yxBmnXs45/BrFlqUfV3fvfL0rtaEpBAKheD11sdeVMVAAU8WFPCQdJoUCYEqKlEvinjvK8uCNtiNyAGDEKIqWDT/wK7HJWNX2ABXZDC1+Mta0mQQgmbAcshl6cJeLLHlKQObp1RETQuLmhT0pgh04hhjWNik7PdzRWBQqYM6SSESiRS0pPXcc8/hwAMPxMKFC3HDDTcUcWSVIVV6AITUstbpEvU2Z+0nyTLdgLJrN/TOFlhOBwCAOxVE5/fDMTwOrbMF3LH/BTcbRWTQOYfe7IG8NwC9mDk2nEMZ3QOgdjYa5Jxjt2pit2qiVRGxqEmx/cUqVTDUPd3KY4439/eG1KevfvWrCAaDCAQCmDdvHjjnOX95N00TF110EZ5//nkMDAzg8MMPx0knnYSlS5eWaNSlRzM8hBSgXrqoi8EwlNE9UPs6E8FOAmNQ+zuhjE3mnXjc5hAxYTEwo7hVRUIoCjEcrYmNBjnnGI8aeMcfmy1b1KSgKykfJ19OUYBax7lkJHd33303rrnmGqiqivvuuw+HHnoo1q5di0cffRTj4/aWqF977TUsXLgQ8+fPh6IoOPPMM/HEE0+UeOSlRQEPIQWId1Gv2RJ1ziGP7YUQ1aD2d6bfrI8xqP0dUEb35JWL06aI2KuZ4A65oOWx2ZSx6dmdzhagzD277OKcYzQSC3REFgt0OpyFBzrJ3CLl8pB9GGNYtGgRFixYgNtuuw1//etfcf7552Pz5s342c9+ZusxhoeHMTg4mPj3wMAAhoeHSzXksqAlrSwoh4dkM+BVsC2gYmGLq9JDyY1pwrFzN/RWHyyPjbELAtT+ztjyVm87uGT/9MEYgywwBJu88O71Q3cWvqzFVA3SZBCcAXpna8GPV2zWdKDj1y10OSUc2OzIfqc8dbkk7AjpmOejZS2yT7wsXZIkHHXUUTjqqKNs3zdVbmKtXw+r8ysRITVEEYXYBns1tKwghKNwjExA7Wm3F+wk7ihA7euEMrIbyHF5qtclYZdmFW1Za99Gg03gcvV8d7M4x0hYx3t+DW5JwIHNDrQ6xOx3LEB8k8e6SqAnBYtEInlXaQ0MDGD79u2Jf+/YsQN9fX3FGlpFUMBDSBH01VCJurR7CmIgDHWgC5DyuBCLAtS+DjhGJoAcqoMcogDN4jAdCoRogZs2Gibk3dMbDXZVx+yOxTl2hGKBjk8WsKjZgWaltIFOsiZZhF+nZS2yTyHd0g8//HC8++67+PDDD6FpGh555BGcdNJJRR5heVHAk0WtT+GR8qhUiXpOn0/LgjI8Di5LsUqpQj7bkgitrx2O4XHAsn+RbVVEjLtdBW9CqExMglkchs9d8Y0GTc6xPaTj/YCGFkXEomYHfHL5Ap24DqeICdqEkCQpJOCRJAl33nknPvWpT2HJkiVYvXo1li1bVuQRllf1zAMTUuNaqrhEnakalLG90LrbitZniksStN5Y0KP2d9pKGm53iHgvYKKnkH1jOIc8vQN0JTcaNCyO4bAO3eLodcvwSJV930XGwBGbaUq3lw9pLJqmQVHyz+s68cQTceKJJxZxRJVFMzyEFEl3lZaoS5MByHv8UAe6it5Uk8sStO622EyPjdktxhgkxhB1KBAi+S1rSXsDEHQDplOB2VT+PkG6xbElqGFLUEOXU8LCJgc8UnWcStscIvZQB3WSRKjS6sVKoCNhEyUDkmyqrkSdcyg7dwMc0Ho7SrYpH1dkaF2ttoOeXreEHQ4npKk8lrWSNhrUy7zRoG5xfBjQsC2ooccVC3RcVRLoxLVOl/8TQtes/dGSVhaUw0NyUS0l6vFdk7XOVnBn6UuVuUOB3tG8b3krw9+NUxQQtQBuWrEAKYe/MTEYgRhRYUki9LamYgw9K820sCMcy43pd0twpNurqAowxiAwBtPiEAU6dxG6hiWjgMcG+sAQu5JL1Ct1wREDYYj+ENS+DBsJloDldEBva4IyMgGtL/OMUosiYo/DgeaImlPScXyjQb0MGw1GTQvDYQMCgAG3DEWsjfNAl1PEWNRAr7v6cskIqaTq/apCSI2qWIn69K7JTNVjAUcFZiIstxNGqw/Krt0Zl7c6nCJGZQXSVMj2Y7OoBmkqBM4Y9I6WYgw3pYhh4T2/il0RA3M8Mub5lJoJdgDAKwkIVsuyKqkYTdMgyxT0JqMZHkKKrEmRsCOo5dWwL2/G9K7J7U0VL9O23E4YVizXRutpT3kbgTEIggDdsr+spYztBQDobaXZaDBsWBgJ65AFhrleBVKNLgnFd7XWTF5TgRoprnA4XFCn9HpEMzw20JIWyVW8i3o5COEoHDsnoPa2VzzYibO8LpgeF+TpBONUelwSdihOCOFo9gdM2mhQL/JGg0Hdwrt+FROqgXleBUM1HOzEdTsljNKePA0tFArlvQdPvaKAh5ASKEuJOucQxycL2zW5hEyfG5bTAXl8b8rfuyUBIVmGaGNZSxmfBOMcRpMHlqs4PakCuol3plRMaibm+xTM8Sh1k+jrkgREC9nriNQ8muHZHwU8NtAMD8lVrESdla5E3bQgbh8Fd8iF75pcQmazB1yWIE9Mpvx9k0PEFEfmcnbLSgRNxWgjMaXFAh2/bmFhk4IBjwyxSo9fIZyigAjl8jSsQnZZrlcU8NhEexqQXA14HdgRLLBnVAosokLathNmTzuspuo/oRktPnBRgDS9JJWsyylhl+yAEEq/rCXtDUAwTJguB0xf/t9YJzUTm6dUhI1YoNPvlut6R2Ja1mps4XAYLldlt8eoNhTw2EAzPCQfiRL1IgbLwp4pCLsnYQz1AUXeNbmUjNbYnjnS3sCMnwuMxfYJ8qdZ1kraaFDLY6NBzjn2qAY2T6lQTY5FTQp66zzQiVNEBt2iDuqNimZ49kcBDyEl1OdRsDNUhBJ1i0PcMQoAMAe6gRrMNTHam8FME+KsxqE9bhkjXEi5rCUGwhCjGixZTARNdnDOMRE18I5fg8GBRU0Kul1Sw3158ckCArSs1ZAoh2d/FPAQUkJNigR/oV3UNR3S1hFY7S2w2pqLN7gK0DtaIGg6xKQZHY8kIKgoEEOR/W6fKEXvbLUV5HHOMTYd6ACxQKfL2XiBTlynQ8J4lFpNNCKa4dkfBTyElFirQ8JkniXqbCoIcdduGHN6wVNUJ9XihVzvaoUQUSEGwomfedwOhPwzAx4hokLyh8AFBi3LRoOcc4xGYoGOxGKBTkcDBzpxosD+f/buO07K6t4f+Oc8bcr23sAg1S5RiQQBC2CDXa4KYkGx96iYePOLpN2bQIzRxJfJKyHJK8k1xiQ3yi4lCgFsiUajKF6KQURQ2N53Z6c97fz+GGbY3uaZeaZ836+Xr8TdZeYwzj7PZ875fs+ByWlZKx15vV6a4emHAs8o0GmzJBolbhmNY21R5xxiQwtYUIMxscSWXZNjSSvOg9jjh9ATCjnFbgkN/Za15MhGgzlDttybnKPBp+FgtwqHwDAjx4F8BwWd3uhA0fTk8/mQmZlp9zASSmpdRQlJQAJjcIylRV3XIX7eADM7E2acTwSPG8agluZD6vZC8AUgMgauyIAntNTFNB1yezc4Bm9FNzlHnU/DoW4VbknAjBwHch2JtQ9Rosh3iGgPUuBJN36/n5a0+qHAQ0gcjKpFXTfAur0Qa5thTCgBz0jxllLGoJYVhNrO/UEU5bjQ3BOqvZHDGw3mZPY57d3kHLXeUNDJlgVMz3EgR6GgMxyBMTDA0m5BkvioaHkgOkuLkDjo3aI+2CZ3Qn0LxLpmMMZgZmeABVRAYOACA5gQKtgVWGi2J/xPKmAMankhHHUtyC7MRTMYig0jslFh+BgJw+So92sIGBxlLhkTMpKnJT8RFDoltAYMlLjokp8uaIZnIHr3jwLVAxArhFvUJ2QOLD6WjjWBHf8ELrZ3Q2zvHvJxOBAKPAIDBAGchf6dC0Lof9mJcMQj/ytE/syw3+/1tdDPCkP+GcuCF2MIVhTBUdeCDMUJtb4NWboBw+1A0O1EXY8KzeQod8twSzQpPR7ZsoDmgEqBJ43QDM9A9O4nJE6GO0XdzHBC6PGDAeCMhTqyOAczOcBNwOShf7gJFj6KweCAYcLOOM4HDUnhIDZISOr//V5fMxUJE9s7cJhJyP78CA7OmQPVq6HCLcNFQScqjDFILLQRoZyEeziRsaO29IEo8BASR+EW9Txn3yUZ/YypgGmCef0A5+DZw3RX8HD44TB1HYamg4VDUSQkhX6GcR56XN7/a7xPoBrwZzgHM81h/kzoayd+1rrXSPT58MlH+zHpqbWQZ54NY+FC6BddBOQO35pOhlfsktDk12k5ME1Q4BmIAg8hcVTilnGwwz8g8AAIzXpkjeICxRggHv+UfrzOx5Zy1CFDlBkJRIOHMHPAnzENE17NgMA5yvfuQoamQvAHgN//Hvj978FFEcasWaHws3AhzJkzAdouYkwyJAH1Y90egSQtquEZiI2wIRWV9SO0qVkgEKD9eIglDnf5UZ7hgNOCZRrTNKHryXtAZMAwUefVIDCGigwZyvHlloPtPpx59AjYhx+A/eVPEN95B6zX39MsKICxYAH0hQthLFgAXlRk118hqRztUVHskuBMsX2dyECLFi3CK6+8ko6hZ8g1Wwo8o0CBh1hJNUwc9QQxNTf6tvNkDTx+3USdT4MkMFS45QF1JbVeDfkOEdmdHjBwaIoA6e9/h7hzJ6SdOyEcPdrn540vfhH6ggUwFi2CMWsWINHk9WCCholGv44vZCoj/zBJavPmzcP7778PUUy7bRso8EQjHHgYY9SxRSxxoMOHabmuQVvUxyLZAo9PN1Hv06AIDOVuGdIQBbS6yXHUq2FylgKxywvRH4Bakh9azuMcwiefQNyxA9LOnRDffBMseGKPI56TA/2ii0LLXwsWgE+YEK+/XlL4pDuIadkDOwVJapk3bx52796djvcsCjzR8vv9FHiIZbpVHd2qMWiL+lgkS+Dp0Uw0+DU4RYZylwxxFJ1Ch7qDmJylQGAMgi8Aub0bwfLCgbU7Ph/Et94KhZ+dOyF+8kmfbxunnhqp/TG+/GXA6bTyr5Z06n0asmUBmXLaffJPG5xzzJ8/nwJP/29Q4BmdQCAAgPbkIdb5qN2HU/NcUb2nEj3weDQDDT4dbklAuVuCMIa/a6dqQDU4io/vHcNUDUpjeyj0DHG2FgCwzz6D9MoroeWvN94A6+mJfI+73TDmzYN+PADxKVPG/5dLUr1nz0hqCgeeDz/80O6h2IECT7Qo8BCrNXhVOEU2eMfWKCVq4OlSDTT5dWTKAkpdYws6YZxzfNKtYnpOr1kww4CjvhVqcR64YxQ3bFWF+K9/RWp/xL17+3zbPPnkSPgx5s0D0uSwxU+6g5iapdD1LEVR4BniGxR4RocCD7GayTkOdvhxSv74d0NNtMDTGTTQFNCRLQsoGWfQ6e2oV0WRQ+q78aDJoTS0Qs/NhDnG88ZYQwPEV1+FtGMHpFdfBevsjHyPKwqMOXNC4WfhQpinnpo6R3j00xLQITGGPDpwNSVpmoZFixbh/ffft3sodqDAE61gMDjoDrmERCPaFvVECDycc3SoBloCBnIVEcVO0bLfE80MHRZ6cv/lF86hNLXDcDpg5I5zVsYwILz/PqRw59f770eO9wAAs7w80vauX3QRkDfw1PZkZXKOwx4VU6l4OSV1dXXh+uuvxz/+8Q+7h2IHCjzRosBDYiHaFnXOOTTNns3kOOdoCxpoCxrId4godFgXdHobbvlFausCMzm0Igt2YW5rg/Taa5HiZ6G5OfItLggwZ82KLH+ZX/xi0m982LsonKSW+vp6PPjgg/jb3/5m91DsQIEnWhR4SKxE06JuR+DhnKMlYKBDNVDgEFEQo6AT1h40YHCOIufge+uI3V6IXj/U0gLrlqBME8K+fSc6vwbb+PCSS6AvWpS0Gx92BA3ow7yuJHl9+umn+O///m9UV1fbPRQ7UOCJlqqqME2TAg+xXDQt6vEMPJxzNAcMdKoGipwi8pTYBp3ez3vIow67d4zgD0Ju6xq8bd0K3d3Db3w4c2ak9seYNQuQE/+8qtG8riQ57dmzBz//+c/x/PPP2z0UOwx5UaJoP0qMMZrhITEx3CnqicDkHE1+HR7NRLFLwoyc+N4gGWNQBIaAYQ55JILpckAtzoOjtgVqeQG41TstZ2dDX7IE+pIlCHIOduhQqPD5+MaH4ocfQvzwQ+DJJ8Gzs09sfLhwYcJufMgYg8gYdJMPuQEkSU50jtbgaIZnlDRNg67rdLwEiYnxtqjHcobH5BwNfh1ezUSJS0KOYl9Hj2qYqPfrmDTSkQjhtvWiPHBnnPaZ8fshvvnm0BsfnnLKiY0P58xJqI0PezQD3ZqJcnfiz0iR0Xv11Vfx2muv4emnn7Z7KHagGR5CEtmwp6jHmcE5Gnw6/IaJUpeEigS4GSqiAM3kI8+CiSKCE4qh1LdCz8mEmRn9eWUjcrlCZ3gtWgSg18aHr7wC6fXXIR44APHAASg/+xm4yzVw40MbZ/UyZREN/sTZ1oBYw+fz0QzPIGiGZ5RohofE2nha1K1sSzdMjjq/hqDBUeaSkSkn1nu9LRD6exaMpsiWc8jNHeCKDD0vK8YjG4aqQnz33RMbH+7Z0+fb5qRJofCzaJFtGx9+3qOi1CXBQSeop4y//OUvaGxsxJo1a+weih2oaDlauq5D0zQKPCRmxtOibkXg0U2OOp8GzeQod8twj3NPoFgbT5Gt1N4NZhjQCnMTYhNB1tgYmvnZuTO08WFHR+R7XJb7bnx42mlxGXPAMNHs13ESnaCeMp599lnouo6HH37Y7qHYgQJPtAzDgKqqFHhITI21RT2awKMdDzq6yVHhlvvuZpygPutRUe6SoYijDwKixwfR44NaZmHbuhUMA8IHH5zY+HDXrr4bH5aVRcJPrDc+pBPUU8vPf/5z5Ofn484777R7KHagwBMtCjwkHsbaoj6eomXVCAUdE0CFWxqy8ykRBQwTTX4dXxjjbITgD0Ju7UKwIkZt61Zoa4P0+usnNj5saop8K9YbH9Z6NeQ5RGQkQeglI3vyyScxY8YM3HDDDXYPxQ4UeKJFgYfEy1hOUR9L4AkaJmp9OhiACW4JShIFnd7Ge/Al03QoDW1QywrA5QTv1+C878aHb789+MaHx4++4MXFUT3dkEd4kKT0ve99DxdccAGWLl1q91DsQF1a0UrE/VFIaspzSOgMGsizaAfcgGGizqtBYAwTM2QoSb7nSq4iokM1kT/Ggy+5LCFYUQRHfUt829bHgzGYZ54J9cwzgdWrQxsf/uMfoeLnHTsgHD0K4YUXIL/wAoDoNz6UBQadj6ILjiQ0r9eLjIwM+Hw+uN3jP5Q47NFHH8WWLVugKAqmTJmC3/3ud8jNteAYF5vQDM8omaaJYDBIMzwk5sZyivpwMzx+3USdT4MkMFS4ZchJHnTCTM7xaTQ7BPPQaetGdgaMzOhvCnE3yMaHLBA48e3eGx8uWAA+ceKoHrbJr8MpMlv3WyLReeCBB7B3716oqorbb78d99xzD1yu8W/NsH37dlxyySWQJAlf//rXAQA//OEPrRpurNCSVrQo8JB4Gm2L+mCBx6ubqPdpcAgM5W45JXfRPeJRMSEjihDHOeSWTnBZhJ6Xbe3g4s3vh/jWWyeWvw4e7PPt0W58aHCOzzwqplDxclLz+Xy49tprMXHiRBw4cAAFBQW49NJLUVVVhWnTpo37cWtqavDiiy8mw3EVFHiiRYGHxNNoW9R7B54ezUSDX4NTFFDulsZ1GGmy8OsmWoM6JmZEtywldXjAND102nqKvF7s889DGx/u3Anp9dfBenoi3xtp40M6QT01rFy5Ek899RSmTp2K2tpabN++HVlZWVi+fPm4H7OyshIrVqzAypUrLRxpTFDgiRbnHIFAgAIPiZv+LeqfdPoxrV8AMgwD7b4gmgIG3FIo6KTLzWq8xcv9iT0+iN1eqGWFKRN6Isa48WG77ITJgUI6QT2pXXXVVfj973+P8vLyEX924cKFaGxsHPD1tWvXRoqe165di127dqG6ujoZarwo8ESLAg+Jt66gDo8WalHvCuro0QxU9GpX7whoqPX44WIcJU4RoiBAOP5POmjy63CIDLkW1JywgAqlpQPB8iIgSbvXRmOkjQ/1Cy7ARw/9J6aW5sVt40Nivcsuuwzbtm1Ddnb0y7XPPvss1q9fj1deecWSQug4oMATLQo8xA7hFvWPO/2YnuuCwBjaAxoavCqyJIZSlwTGGAzDgGmaCP8+c84hiiIYYyn7njU5x2GPiqkW1ZwkVdu6FYbY+PDf9z+Cab/5BYS83LhtfEisNX/+fLz33nuQpOjex9u2bcMjjzyCN954A0VFRRaNLuYo8EQrHHgYY8kwpUdSRINXhYDQqdY5DglNPhW5iogiR2gmp/97MRx6wgEICL13hRSd/TnsUXFShoWF2aYJR10rtMIcmK40K949vvGhd99H8NU34aQ//T7yrVhvfEisNW/ePOzevTvqe9XUqVMRDAZRUFAAAJg9ezbWr19vxRBjiQKPFQLHWz8p8JB4MTnHW/VdyFFEFLsVFDrEyF4po3kfmqY5YPYHQCQsJXsA8ukm2oMGJmRYeKI751Aa2mBkuWFkJcUUvqU45zjUrWLG0YNx2/iQWMuqwJOkKPBYgQIPscOeZg84Y5ie44DEMO5ZxqFmf5J96etgVxDTc6yfjZFbOsBFEXp+kretj8OAM8s8Hkh//3ufjQ97i3bjQ2Idzjnmz59PgWewb1DgGT0KPMQOQd3A4S4/TA7kOyUUu6y5mYQPHuXHd9gNS7bZn0a/BrcoIDsGG+ZJnR6woAatOC+tCnhDbf8GJg42cxajjQ+JNcKB58MPP7R7KHahwGMFCjwk3sJLUp90BTEt1wmPZsZkJ1zTNPv8E5YM4cfq4uX+hB4/pK4eqOUp2LY+jFGfoG7RxofEGoZh4OKLL8bu3bvtHopdKPBYIRgM0lkzJC4455HwwRiDRzPh0QxURLnR3miEA89gsz+iGApbiRaAPvWomJQhQ4zRrtIn2tYLATE9jl445tVQ6BDhGuMJ6tFsfEii5/F4sHz5crz11lt2D8UuFHisQIGHxEP/sBN+vx3o9GNGjjPu77/+sz/h34FEmv3p0Ux0aQYq3LGrHWG6DqW+DWppPriS+jUqqsFR79cwKTOKkD3GjQ+RmRnlqElTUxPuuece7Ny50+6h2IUCjxVUVY3chAiJhaHCDgA0+jQ4RYZch317xIQDT+/wEw5AoijaGn5iVbzch2nCUd8KLT8bpjv1l2as2s06bKSND405cyLFz7Tx4fgcPnwY3/72t7Fp0ya7h2IXCjxW0DQNuq4nxCdaknrCHVRDtZ2bnOOTriBm5CbOjTYcfMLjDrNj9qfepyFbFpApx3jJiXMoje0wMpwwsjNi+1w2a/BpyJQFZMXiNe2/8eH774P1qh8zy8po48Nx2L9/P5555hn88Y9/tHsodqHAYwUKPCRWwqEBGL4+5kh3EGUZMpwJePxB79mf8N8lLB6zP4bJ8VlP/E77lls7wRmDXpATl+ezg2FyfObVMCUr9rVj4Y0Pw8XPQlNT5Fu08eHovffee3j++efxm9/8xu6h2IUCjxUo8JBYCO+LM5r9dVTDxDGviinZiTPLMxQ7Nj081B3EyVlK3E6KFzt7IAaCUEvyU3b5xeplrVHhHMK+fbTx4Ti88cYb2L59O5555hm7h2IXCjxWoMBDrDRcvc5wDnYGMDXHkVSnosdr9sejGejRTJTFsHi5P8EbgNzRHTp4NEZdYnZqDegQGJBvY+1Yn40Pd+6E8Pnnfb5NGx+e8PLLL2P37t14/PHH7R6KXSjwWEHXdWiaRoGHRG28YQcAulUDPZqB8ji0qMdK77ofq/f9GfX+MRZiQRVKcweCZYWAlFpt67He52jMrNr4sLMTyM2N06DjZ8OGDTh27Bi+9a1v2T0Uuwx5MU2DI4GtQ91ZxAojFSePJFsRUe9TUZbEWyT0Psh0sCMvwrNA45n9cYkCvLqJjDHuHxMN7lAQLCuEo7415drWBcbAABicx22pcFiMgU+bBm3aNGj33Tfoxofy5s2QN28GMPjGh8L//R/cl10G7c47EVy9GsjPt/kvZR2fzwe3O/3OgBsNmuEZA8MwoKoqzfCQcRttcfJIEqFFPVb6z/6Er1GjPfNLNzk+j1ehbX8p2rbepRoIGBwlrsR/v7GjR/tufOjxRL4X3viQiyLkrVtDX8vNhfrII1DvvhtwuewatmXWr1+PrKws3HPPPXYPxS60pGUFCjwkGmMpTh5JIraox8Jgsz8AIsFnqN/FQ91BTM5S7Klz4hxKUzsMlwNGTmpspMc5xyGPGvelwqiNsPEhdzojy2FmWRnUNWug3XADICV+sBvKT37yE5x88sm46abBd2Q6AAAgAElEQVSb7B6KXYb8pac79xgk6/IBsVfvG7YVYQcILTMoAkPAMEf+4SQmCAJEUYSiKHA6nVAUJTLTYxgGNE2DqqoDAlGRU0JLwBjmkWOIMailBWCaAam1y54xWIwxBokxaGaSfQZWFBhz50L97nfhe/NN9Bw8CP8vfgHtyivBMzL61P4IDQ1wPvAAMqZOhfzkk8DwkwEJi5a0hkaBh5AYiqY4eSQVGTLqvaplj5cMBEGALMtQFAWKokCWZYiiGFkGU1UVuq4jUwwtw9hJL8wBl0UojW1Je/Psrdglodmvj/yDCYyXlkK/8UboS5aAeb2D/ozQ3g7HS1sBPTn/rj6fD5l0RMegknfejpAEF21x8kgUUYBuhpa3kqlF3Srh5azwgaa99/0xTRMK4+gOqMiQR1f7EwtGTia4LMFR1xI6eDSJl8MzJAH1Ps3uYViCl5VBv/xyIBgEgkGwYBDw+cBOngq27AZwX0/StrbTDM/QKPAQEgNWFSePpNQto9GnJXWLulX6d35ViCaO9gThEkNhMLyH1nC1P7Fgup1QJTEUepK8bd1xfBk1EXf6HgtjwQL4Fyzo8zWhqwdKUzu4IiP4hVKbRhY9v9+PjIzUPvJkvJL7XRtnVMNDRsMwDBiGEZdZhWxFRLdm79JNIhIEAU5ZApgIxeGAoiiQjheihmt/NE0bUPsTK1yRQ23rDa1gweRdhixxSWhK8mWtwQi+AORuL8xMdyjsJPG1nmZ4hkYzPIRYJJb1OsPJVSR0BvWUbFGPVpFLQktAR4lLHnHfn5gfeCqJCFYUQ2lohZ6bCTMj+VqgHaIANdkKl0cgtXdDUDUEJxQnddAJ83q9VMMzBJrhIcQCvY9NiGfYAULFpKn4qdsKuYqIzmDf16Z351f4H0EQIiEoprM/AoNaXgjJ44PY2WPtY8dJhiSgR0uB7kDOoTS0ggsC1NKClAg7QGhJi2Z4BkeBZxxG2LuIpJneswXhWYJ4SpcW9fFgjMEhCgjog7824XoeRVHg6LX0JQgCTNOMHCdjafgJt60bBuSWTmseM46KnRKaA8kdsJmmw1HbDC0vG0Zuas2GUJfW0CjwjEG8P7mTxMc5h67r4JzbEnbC0rFFfbTK3fKou4sEQYAkSX1mf0RRHDD7E166jIZekAPTIUNpaE2qtnVJYDA4T9oPfoLXD6WpHcHyInBn6hX7G4YBOUk7zGKNFv0JGafen/rt3n073VvUh6OIAjSTR7YHGK3B2t7Dy5bhDrxoa3+M7AzwcAdXErWt5ygiulQTuY7k6jiT2rrADBPBiqKUWcIio5ccv12EJJD+OyfbHXbCwi3qZKBCp4TWKJdhwrM/4aWv8KaHg83+jIXpdkItzoOjtgUsSTa7K3SIaA0mx1gBAKYJpa4FXJagFeelbNhJ1lm3eEmMKzUhScLO4uSRUIv60PIdItotvEGHC59lWY4EoP5HXoyl9ocrMoIVhVAa2sACib80GZ5FNJPgBstUDY7aFmiFuTCy02N/mkS6LiUSCjxjRG+k9GV3cfJohFvUSV+MMSiCgGCMCrvDR170n/0Jh+NRzf6IIoITiiG3dUHo8cdknFYqcEhoCyZ2wBY9PsgtnQhOKAJ3pH5dC83wDI8CDyGjkCjFySOhFvWhlWfIqPfGfsmv9+zPUAeeDjn7w0Jt66LXD6nDE/OxRiNXEdBp83llQ+IccksHhKAKNYlqo6IVCATgdDrtHkbCSo93gYUS9UZHYscwDOjHaysSpV5nKNSiPrTQpnlm3D8FDzb7E257H3T2hzFoJfmRm3aidnAxxiAyBj3RNiI0TTjqWmC6HNAKc1O2XmcwXq+X9uAZRmJfvRMUTRumh0QtTh4JtagPLd8hod3GZZjemx72nv0RBCESflRVDZ38npsJ0+mA0pC4p60XO8WE2pOHBVU4alugFufDyEy/Gz8dKzG85LiCJxCa4UkPiVycPJLeLeqkLyu6tawUnv0J7/nTv/Yn4JQRyHZDOdYMxOHMr7HKlEV4h9jUMd7Ebi/ktm4EJxSDK+m544rP56ODQ4dBgYeQfpKhOHkk1KI+OMYYZIFBTcAlv6Fmf7jLAV9hNuSjjdB8/rgdeDpaisBiVgw+KpxDbmoH0/Tj9TrJ9/tqFTpWYngUeAjpJVmKk0dCLepDG8vOy3bqPfsjZ7ihTSxFRmsX4A9Elr90Xbc9/JS4JDTZNWtmGHDUtcDIckMvyLFnDAmkp6eHAs8wKPCMUbLeAMnIkqk4eTSoRX1wTinUnp5MtXiCIEBUZBiTypHhCcAZ1CM7QPev/Yl3AHKKAoJG/F9LwR+Eo74Vamk+TDd1JgGhGR5a0hpa8l/V44wCT+pJ1uLkkVCL+tByHVLitlQPhzFoFUWQAkE4Pf4h9/1RVTWusz8uUYhrLY/U6YHU2ROq15HSs15nMFS0PLzUuLITMk7JXJw8EmpRH1qRU0JLsoZBxqCXFAAApMY2CIwNqP2RjoeA8ez6PB4lLgnN8Xg9OYfS2AZwQC0rSKuW89GgouXhUeAhaSsVipNHQi3qgxMYgygAWqLtITMGRn42zAwn5LqWPm3rI+37E4ulL1lg0GN9grpuwFHbDD0nE3peVuyeJ4nRDM/wKPCQtBQOO+ETtFMx7ADUoj6cMreCBl9yh0EzKwN6QQ7ko03AIDN5Q3Z+9Tvw1IoAlC2L6NZidHSHLwBHQxuCZYUwXY6YPEcq8Pl8yMzMtHsYCYsCzxil8s0xXYSLk5O9E2u0qEV9cG5JgF9PruLlwXCXA1p5IeTaJkAd/r+zIAhQFCUy+yNJUmT2R9f1qMJPoVOMyR5HUns3RI8PwQlFgCRa/viphNrSh0eBZxyS/QKZrlK1OHkk1KI+tBxFQncyFi/3J0vQJpRAbmwD8wdG9UcEQYAkSZFNDx0Ox6CzP6ZpjioAiYyBw8LZRM6h1LeCi0LoqI0U/2BiBa/XSzU8w0iPKz5Je6lcnDwa1KI+uJTqZBMFaBNLILV3Q+j2jumPCoIwoPYnPPsTnhEdzexPnkNEhwVHdzBNh6O2GVp+NowcWqIZyZEjR6DruuVt6U8++SQYY2htbbXsMe1E/XxjlG43ylTQu14nHZawBlPskvBJVxC5DvqV701gDAIDdJNDSoUdehmDVlEMqTm087Axzs34wgEIAMzjB66GZ3rCy1/seHdY75nSfEXEIY+KAuf432dCjx9ypwfB8iJApM/ko/Hcc89h69at6OnpQWlpKaZPn46ysrKoHvPYsWPYsWMHTjrpJItGaT96N5GUli7FySOhFvWhlbplNKRYjZNeHFoCkhqjP3g0XPjcv/OLMTag8JlzDoExGOPpfuMccmsnRF8AwQoKO2Px7W9/G//85z8xdepUeL1e3HTTTTj//PPxjW98A3v37h3XY65evRpPPPFESl0z6eMeSVm9p9/TpV5nOBUZMmq9KiZn0660vWXKIupSsHXfyM+G0OODXNsMraLYsjOmwrM/kiT1mf0xjNBSVp7I0egNoixDGf3vnWlCaWiDkZ0BI4uKbseDMQZZlvGVr3wFX/jCF+DxePDqq6+ivb19zI+1efNmVFRU4Oyzz47BSO1DgWecwjMGJPH0nn5P51md/nq3qAv0mvSRJYvoVg1kK6nVBWRmusElCfKxRmgTigHR2r9fONCEZ4BM00SOoKPFE9rrJxyCwkvJgwUgpmpQGtuhluaDK7Kl40s3vTcezMrKwtKlS4f82YULF6KxsXHA19euXYt169Zh+/btMRunXSjwjBHdPBMbhZ3hhVvUyzMUu4eSUErcMg53B1Mu8AAAdyrQyosg1zZDLyuMaagIt707ZQ6IEmQBfWZ/wv8brv0RPT6I3d5QyznNwkZtLEXLO3fuHPTre/fuxZEjRyKzO7W1tTjnnHPw7rvvorS01LKx2oECzzjQTTQxUb3OyLIVEfU+FeV2DyTBiMffK4bJIaZC8XJ/sgRtYkko9BTmgsf4sM1St4zmgIaTMh19Zn/Cv5+GrkNq64bJGNSSPDBQQakVVFWFokT3YebMM89Ec3Nz5N8nTZqEXbt2obCwMNrh2Y7eYyQlUNgZPWpRH1ypS0ajP7WKl/sQQm3rYqcHQldPTJ8qtKlj38LlcNu7IkrIbOkCsjOgF+XCHGTfHzJ+VK84NJrhIUmPipPHhlrUB5d1fPYrpTEGvbwIYnMHxNZOGIW5MXsql8jg1024pBO/kyygQmpqg1ZeBCZLCM9F9Jn9Ob7vDxBa+kqnTUKjEasNcT/77LOYPK4d6F00DjR7kBjSdefkaIVb1IPUoj5AhiSiJw12pTaK8wBJhNTQGnXb+lD6H2kidHogtXVCm1gKyH3DdmT25/iuz+Ezv8JBSNM06LpOsz+jQPenodEdgiQlKk6OTkWGjDpvCi/fjFM6nTtm5GbBzMqAXNsMxODUeEUUoJomuGlCamgFM8xRtccPdeBp731/YnHie7KjI49GRnPa48AYozeXjaheJ3qhFnVOLer9SMKJ86DS4XUxM13gshhqW68otvxwziyRwV/bAqEwZ9yF0oPt+tw77Oi6HvmZdJ7ltaJgOdWl77sjShR47EFhxzrpNJsxFsUuGU2pXLzcD3co0CqKIdc1gwWt+3szfwAV7R2oz8qyrCtssNkfSQp9bu+/63O6zf74fD64XC67h5HQKPCMA91k7REuZkznM7GsRKeoDy5HEVPjBPWxkMTQwaNNbWBef9QPJ7Z3Q+zoAT+pFAYTYvYBsf+Bp7IsQxCESO2PqqppU/tDJ6WPjAIPSXhUnBw71KI+OJckwJtuYTDctt7VA6HTE/raWIMC55DqWwAG6OWFAGPId4hot+AE9ZGk++yP3++H203HcgyH7hwkoVFxcmwVuyQ0+Snw9FfuVlLuQNFROd62zjQdYksHxC4vxI7u0f1ZTYd8tBFGXjaMvOzIlwucEtpsCNX9Z38kSeoz+5Nq4YdmeEZGRcvjQDfd+KB6ndjr3aLuoNOpIySBweTpU7zcn1GUB6HTA8EXADjAnEFwl2PIn2deP6S2rkHP6xIYA4O9r2X/wmcAkeXx3kdfDHfmV6Lz+Xw0wzOC5PuvmgDoxht7FHbih1rUB1fsktCcxrNfZm4WjJxMMF2H1NQGGIMvS4mtnRA9PmgTS4Y8nLTIKaElQV7LcPhRFGXQ2R9d15Ny9ocCz8go8JCEQ8XJ8dW7RZ2ckKOI6FIT4yZtC84htXfDzMoAOId8rLnvJoUmh1zXDC5L0EsLgGF+T3MUEZ0JWgguCAIkSRqw6WH4Q1fvIy8SOQD1PimdDI4CD0kYVJxsnxJqUR+AMQanKMCvJ+5NLqYYCy1RgQNMAAJByHWhQyWZqkE+1gi9IBdmTuYoHopBEgAtBhscWilc+Ny/9ie86WEi1/7QDM/IqIaHJAQqTrZXjiKigU5RH6DMLeOYV8WU7NieLp6wGIORnwMjLxtiSyfkxlawoAauyNAmlABjqPsqcclo8mmYkJk8m+OFl78kSYpsehi+TvWu/RFF0fYPaFS0PDIKPONAN2NrUb1OYgi3qNOhoieElvvSt3g5gjEYxXkwCrIhtHXBLMobdglrMJmymNS1YuFAIx6vUwoHnvD/2l347Pf7UVpaGtfnTDa0ZkBsRWEncVCL+uCKnBJaA/S6AABEEWZx/pjDTphDZAikyBLhYJse9j/wNFz7Ew9UwzMyCjzENlScnFjoFPXB5TlEdNDmjJYodctoTMFjO3rX/vQ+8FQQhLhteuj3+ynwjIDmrseBbszRoXqdxFWRIaPWq2Fy9tB7rqQbxhgcooCAYcJJexVFxSkKCBqJXbhshZEOPI1F7Q8VLY+MfntJXFHYSWzUoj64creMhiSuP0kkGbKAnjQ6tmOwIy/6z/6oqhr17A8taY2MAg+Jm/5t5xR2EhO1qA+kiAJUk8fsEMx0Uppmp9H3F679Ce/5M1jtz3gOPKUZnpFR4IkCXfxGj4qTk0cOnaI+qAKnaMuZUKlGEhh0k66fwNCzP+F9f1RVHfXsj8/nQ2bmyHsipTMKPONAN+uxoeLk5JOjiHSKej8FDglt1K1liVxH4u68bKf+sz/hADSa2R86LX1kFHjGiW7aI6Odk5NXiUumFvV+GGNQBIG62CxArf4jG2r2B8CgnV9er5dmeEZAdyASE1ScnNyoRX1wZRkyGqi+KWoCY+AAFcePwWD7/vQ+8LS+vj4SiMjgKPAQy1FxcmqgU9QHCrVVm1R/YoFCB83yjFfv2R9FUfDYY4/hvPPOg8NB20kMhwLPONFNfHBUnJw6qEV9cHkOCR1Bqj+JVmhDR3odo2GaJlavXg2n04kNGzbQ9XYEFHiIZag4OfVQi/pARU4JLTQzETXGGEQG6Al+gnqiMgwDDz/8MDIyMvCTn/yEaiRHgV6hcaKb+QlUnJy6qEV9IMYYZAFQqb4paiUuGc1pvCfPeBmGgdWrVyMrKws//vGP6Zo7SvQqRYHW8ak4OR1Qi/pAZW6FipctkKWI8GgUHMciPLOTnZ2Np556isLOGNArNU50Y6fi5HRBLeoDuaTQ2Vr0oSd61A04eoZh4KGHHkJubi6efPJJCjtjRK8WGRcqTk4f1KI+uBxFos3zLFBKdWKjYhgGHnzwQRQUFOBHP/oRhZ1xoFeMjFl43wcqTk4f1KI+ULFLQgvNfEUtNFtGM2XDMQwDX/nKV1BUVIQf/vCHFHbGiV41MmrhWR3DMKg4Oc1Qi/pAAnUZWcYtCfBScfygDMPAAw88gJKSEjz++ON03Y0CvXLjlG5vOipOJtSiPlBZhox6n2r3MJJeiUuiOrFBGIaB+++/H2VlZfjBD36Qdvcdq9GrR0ZExckEoBb1wbglEX6dapuipYgCNJNTEXgvhmHgvvvuQ0VFBdatW0dhxwL0CpJhUXEy6Y1a1AfKVkR0UfFy1ChQn6DrOu69916cdNJJWLt2LYUdi9CrSIZExcmkP2pRHyj0mtBSX7SKnBKa6b0VCTuTJk3C97//fQo7FqJXcpxS+eZPxclkKNSiPpDAGBioeDlaosDAeXpv6KrrOu6++26cfPLJ+N73vpfS9xk70J2M9EHFyWQk1KI+UBkVdFsi3ymhLc2WTLdv347a2tpI2Jk6dSqFnRihwEMiKOyQ0aAW9YEyZRFenepPolXgENEeSK/Xsba2NrKEdfToUSxatAiGkV6vQbywEaYP6Yo2BNM0EQwGU2a5h4qTyVh0qQa8moHyDMXuoSSMOq+KbFlEliLaPZSk9klXAJOzHBCF9LgG6bqOO++8E5MnT8b555+PrVu34p133sFpp52G+++/H3PnzrV7iMlmyDcOBZ5x4pwjEAikROAxTTPyiSIV/j4kPg50+nFKrsvuYSQMw+Q47AliWo7T7qEktS7VgE83UOZO/TCtaRruvPNOnH766fj2t78d+aDJOcfevXshiiJOP/10m0eZdIYMPFI8R0ESCy1hkWiEW9RzHXQZARCZkTBMnjazE7GQLQto9Gkoc9s9ktjSNA133HEHzjzzTHzrW9/qc/1ljOGss86ycXSpiT7ORylZOwoo7JBoUYv6QNSiHj3GGGSBQU3hTkBN03D77bfjrLPOGhB2SOxQ4BmnZH6DUtghVqAW9YGyFREe2jwvaqXu1D1qQtM03HbbbZg5cya++c1v0vU3jijwRCEZ36h0TASxErWoD+SWRPRQ6ImKWxLhS8EjO8Jh55xzzsGaNWvo+htnFHjSCO2cTKxGLeoD0Z481nCJLKXOKVNVFbfeeivOPfdcPPbYY3T9tQEFnjRAOyeTWKJT1PuSBAYToBAYpVR6X4XDzpe+9CV84xvfoLBjE7rzRSEZ3rRUr0NijQ59HKiEzoWKmkMUoKbAcR2qquKWW27B7Nmz8fWvf52uwTaiwJPCKOyQeMlRRHSl2ZEAwwmdoE6vR7SyZAGeJD6JXlVVrFq1CnPmzMF//ud/0jXYZhR4opDIb14qTibxVOKS0UgzGhGMMbgkISULb+OpOInb/IPBIG6++WbMnTsXjz76KF2DEwAFnhRExckk3qhFfaAyt4IGn2r3MJKaJDAYSXiCejAYxKpVqzB//nx87Wtfo2twgqDAEwXGWEL9IlJxMrETtaj3JQsMhknFy9HKd4joCCbPslZ4ZufCCy/EV7/6VQo7CYTuiCmC6nWI3ahFfaAil4SWAC31RaPAKaE1SerDgsEgbrrpJlx88cV45JFH4nIdfvLJJ8EYQ2tra8yfK9lR4EkBFHZIokilVmIr5B4/b4yMn8AYGBJ/piwQCGDlypVYsGABVq9eHZfr8LFjx7Bjxw6cdNJJMX+uVECBJ8lRcTJJJNSi3hdjDA5RSKkN9OxQ6JTQksBF8eGws2jRIjz88MNxuw6vXr0aTzzxBF33R4kCTxTsfpNRcTJJRNSi3le5W0YDzXpFJVcR0Zmg7emBQAA33ngjLrvsMjz00ENxuw5v3rwZFRUVOPvss+PyfKlAsnsAycyugEFLWCSRlbhkfNIVRI6DLi9AqLZJMzk45/S7Ok6MMUgCoJsckpA4r2E47FxxxRX4yle+Yvl/34ULF6KxsXHA19euXYt169Zh+/btlj5fqmMjdBkl9qKpzQzDgKqqce2GorBDksGR7iDKM2Q4RJpEBoC2gA6TcxS5ZLuHkrR6NAOdqoEJGYrdQwEA+P1+3HjjjViyZAnuv//+uF6L9+7diwULFsDtdgMAamtrUV5ejnfffRelpaVxG0eCGvI/BAWeKMQ78FDYIckiaJio82qYnO2weygJgXOOg10BzMh12T2UpPZxZwAzcp12DwN+vx833HADqqqqcN9999l+LZ40aRJ27dqFwsJCW8eRIIb8j0Efv6IQzzc5FSeTZOKgFvU+GGNQRIE2ZoySQ2QI2Pwa+v1+XH/99Vi6dGlChB0yejTDEwXTNBEMBmM+w2OaJgwjVLBHmwmSZNGlGvBqBsoTZAnCbkHDRL1Xw8k06zVuAcNEo0/DpCx7XsNw2PmP//gP3HvvvRR2EhPN8CQj2jmZJDNqUe8rdPq3mVC7sycbpyggaNjz+vl8Plx33XW4+uqrKewkKbqDJiiq1yGpgFrU+8p3SGhPomMSElGGJKAnzkE6HHaWLVuGu+++m67HSYoCTxRi9aansENSBZ2i3lehU0IrHTURlRJ3fE9Q93q9WLFiBa699lrcdddddD1OYrRRRoIJL2OF9+ygXy6SzHqfok4t6qEPSbLAoBomFHo9xkUWGHQTMd3X6KmnnoJpmrjwwgvxX//1X7jhhhtwxx130PU4ydFvXAKhnZNJKiqnU9T7KHfLqKedl6OSq4joiuHOy7fccguKi4uxYsUKHDp0CB988AFefvll+P3+mD0niT0KPBaItgiRipNJKqMW9b6cUqg9nYqXxy/Wp9A7nU688MILWLduHQ4fPozrrrsOr7/+OubOnYuvfe1rMXteElvUlh6lQCAAYPz1PFSvQ9IBtaj31ezXIAsMeXT8xrh90hXAlGwHBIuvmT09PbjuuuuwcuVK3HbbbQO+393djezsbEufk1iK2tITEYUdki6oRb2vRD/9OxkUOCS0WTzL09PTgxUrVuCmm24aNOwAoLCTxCjw2IR2TibphlrUTxAYgygAmkmT6OOV5xDRYWGLf09PD6699lqsWrUKt956q2WPSxIHBR4bUHEySUclLhlNNKsRUeZW0OBV7R5G0mKMQWChE9Sj5fF4sHz5ctx666245ZZboh8cSUgUeKI0lrBCxckknQnHW7LpPKkQtyTAR8XLUSlxyWiOck8ej8eDa6+9FrfffjtWrVpl0chIIqI7bpxQvQ4h1KLeX64ixbS9OtVlKSI82vgDdHhm5/bbb8fNN99s4chIIqLAE6XRBBcKO4SEUIt6X8UuCc2083JUlOMbOY5Vd3c3li9fjrvuuovCTpqgwBMlxtiwU9JUnExIX/E+GiCRCYxBgDV1KOmq1C2jcYzvp3DYufvuu7Fy5coYjYwkGgo8MUTFyYQMlBPjXXKTTZlbRgPtvDxuLkmAXx99YOzq6sLy5ctx77334sYbb4zhyEiiocATI1ScTMjQQqGHlnIAIEMW4dMpAEbDLQnw6SMva4XDzn333YcbbrghDiMjiYTuxBajJSxCRlbiktHko8ATliWL6KZZr3ErcUloHGGWrKurC8uWLcP999+P66+/Pk4jI4mEAk+UegcaKk4mZHSoRb0vqmuKjiIK0Ew+ZD1lZ2cnli1bhgcffJDCThqjwBOlcKihmR1CxqY8Q0Y9tagDAMTj1wuDipfHLVsR0D1Ii3pnZyeWL1+Ohx56CCtWrIjpGB599FGccsopOOuss3DVVVehs7Mzps9HxoYCjwU451ScTMgYOY5/KqcW9ZDxdBuRE4qdMlr6vX4dHR1YtmwZHn74YVx77bUxH8OiRYuwb98+7NmzB9OnT8cPfvCDmD8nGT0KPFHat28fbr/9dmzevBl+v9/u4RCSVGgp54QsWUQPHbA6bqLAYHJElrXCYWf16tVYvnx5XMZw6aWXQpIkAMDs2bNRW1sbl+clo0OBJ0pnnXUWHnzwQXz44YdYtGgRrr/+ejz//PPo6Oiwe2iEJDxqUe8rQ6LQE418p4j2oBEJO1/96lfjFnb6++1vf4srrrjClucmg2MjnONCc81jwDnH/v37sWHDBrz00kvIycnBkiVLUFlZiZKSElrqImQQDT4VbklAjiLZPRTb6SbHEU8Q03Kcdg8lKXHOsbepEw/fcA0effRRXHPNNZY/x8KFC9HY2Djg62vXrsXSpUsj/3/Xrl2orq6m6378DfmCU+CJEc45Dh8+jJqaGmzatAmccyxevBiVlZU4+eST6ZeAkONMznGoK4jpuXSTB4BPugKYnO2IFDKT0Wtvb8dP/vACLpgxCVdfdZUtY3j22Wexfv16vPLKK3C73baMIc1R4LET5xyNjY3YuHEjampq0NHRgcsvvxyVlZU47bTTaGNCkvaOdAdRniHDIdLvQg0jkX0AAB3NSURBVJdqwKsZKM9Q7B5KUmlvb8eyZcvwn9/5b5w/dx4qslxxH8O2bdvwyCOP4I033kBRUVHcn58AoMCTWDo6OrBlyxbU1NTgyJEjuPjii1FVVYXzzjsPoijaPTxC4i5omKjzapic7bB7KAnh404/ZuTG/4adrNra2rBs2TJ84xvfwNKlS7G/1YMzirLjPo6pU6ciGAyioKAAQKhwef369XEfR5qjwJOovF4vtm3bho0bN2L37t2YM2cOli5dirlz50KWZbuHR0jcfNwZwLQcBwRaysGxHhX5DhEZMn0AGklbWxuuueYarFmzJlJD83FbD07OdUOhGcN0RIEnGaiqildffRUbN27Em2++iZkzZ6KqqgoLFiyAy0Wf9khq61IN+HQDZW5aytFNjs89QUyh4uVhtba2YtmyZfjmN7+JqqqqyNd7VB0tPhUn51INTRqiwJNsDMPA22+/jerqauzcuRNTp05FZWUlLr/8cuTk5ER+7tixYygqKoLTSRdGkvwOdPpxCi3lAAAOdgYwlWa8htTa2oprrrkG3/rWt/qEnbB9Ld22LGsR21HgSWamaWLPnj3YsGEDtm7dioKCAlRWVqKiogLf+c538Nxzz2HatGl2D5OQqFGL+gmdQR0Bg6PUTUvb/bW0tGDZsmX4zne+gyVLlgz6M592eFGW6YSblgXTDQWeVME5xyeffILvfve72LJlC0477TRcffXVWLp0KSZOnEjt7iSpUYv6CZxzHOwKUPEyQo0eXq8XEyZMiISd7373u1i8ePGQfyagGzjW7ce0/Mw4jpQkgCFvgvQxKgn95S9/QXd3N+rq6tDd3Y2amhrcf//96OnpweWXX46qqiqccsopFH5I0ul9inq6t6gzxuAUBfh0E24pvV+Luro6PPbYY2htbUVzczMeffTREXcxdkoigsbAw0RJ+qIZniTzzW9+Ex6PB0899VTkzJawtrY2bN68GTU1NTh27BgWLlyIqqoqfPGLX6S9fkjSCBom6r0aTqYWdaiGiWNeFVOyacarubkZV111FS6//HLU1dXhww8/xOzZs1FVVYVFixbB4Rj4fjna5UOOU0aOg5YF0wgtaaWKrq6uPkXLQ/F4PNi6dSuqq6uxf/9+zJs3D1VVVZgzZ86AoERIoqEW9ROoeBloamrCsmXL8P3vfz8ys6PrOt5++21s2bIFX/3qV1FSUjLgz+mmiUMdXpxSkBXvIRP7UOBJZ8FgEDt37kRNTQ3eeecdnHfeeaisrMTFF19M3V0kIVGL+gntAR065yh2pecsRTjsrF27FpdffvmY//y+lm6cXphFS/zpgwIPCdF1HW+++Saqq6vx+uuvY8aMGaiqqsKll16KrCz6FEQSB7Woh6Rz8XI47Kxbtw6XXXbZuB6joScAWRBQSOE5XVDgIQOZpokPPvgA1dXV2LZtG8rKylBZWYkrr7wShYWFdg+PpDlqUT/hM08QpS4ZzjQqXm5sbMSyZcvw+OOP49JLLx3345ic499tPTi9kD7QpQkKPGR4nHMcOHAA1dXVeOmll+B0OrF48WJUVVWhvLycpoNJ3FGL+gmqYaI2jc4asyrshH3U6sEpBZlpXQeVRijwkNHjnOPo0aOoqanBpk2bEAgEcOWVV6KyshLTpk2j8EPihk5RP+HjTj+m5zhT/vcvHHZ++MMfYtGiRZY8ZptfRdAwUZ5J4TkNUOAh48M5R0tLCzZt2oSamho0NTVh0aJFqKqqwllnnUXt7iSmqEX9hNaADoCj0Jm6xcsNDQ1Yvnw5nnjiCSxcuNCyx+Wc23aCOok7CjzEGl1dXXjppZdQU1ODjz/+GBdddBEqKysxe/ZsiCJt4U6sR23ZIaHi5SBmpOgSXzjs/OhHP8KCBQssf/wDbT2YkuuGTLOFqY4CD7Ge3+/H9u3bUVNTg127dmH27NmorKzE/PnzB90EjJDx6FJ1+HSTWtSRukt84bDz5JNP4pJLLonJc3QHNXQENHwhh05QT3EUeEhsaZqGf/zjH9iwYQP+/ve/4/TTT4/sgJqRkWH38EiSoxb1kIBhotGnYVJW6nygqK+vx/Lly/HjH/8YF198cUyfi05QTwsUeEj8mKaJ9957Dxs2bMCOHTswceLESLt7Xl6e3cMjSSjUoi4iR6Fl01QqXq6rq8O1114bl7ADAIfavZiQ7YRTCr2PmrxBlGSkTngkACjwELtwzrF//35s2LABL7/8MrKyslBZWYnKykqUlJT0uWgHAgHa+ZkMilrUT2j2axAZQ4EzufcnCoedn/zkJ7jooovi8px+zUB9TwBT8kKzzh+1enAa7c+TaoYMPKm1EEwSDmMMZ5xxBr7zne/gnXfewS9/+UtomoZVq1bh0ksvxdNPP41PP/0UBw8exIIFC9DS0mL3kEkCEhiDdPwU9XRX5JSOd2wlr9raWixfvhxPP/10zMLOtm3bMGPGDEydOhWPP/44AMAli/DrRkyejyQ+CjxJ5sknnwRjDK2trXYPZcwYY5gyZQq+9rWv4e9//ztefPFF5OXl4Y477sCcOXMwa9YsNDU1wTTppkYGqsiQUe/V7B6G7RhjkIXQZoTJqLa2FitWrMAzzzyDCy+8MCbPYRgG7r//fmzduhUfffQR/vSnP+Gjjz4CAGQqEjxqKDAKjMEwaSEjXVDgSSLHjh3Djh07cNJJJ9k9lKgxxlBWVobi4mIAwLvvvou5c+fi8ccfx9y5c7FmzRr861//gmHQpzES4hAFaCaHOfwyfFoocyto8CVf+Dt27BiuvfZaPPPMM5g/f37Mnufdd9/F1KlTMXnyZCiKguuuuw6bNm0CAFRkOlHvCQAAnJKAIF1j0gYFniSyevVqPPHEEylRrAgAP/3pT/GrX/0K27dvxxlnnIGbb74Z1dXV+Oc//4l58+bht7/9Lb785S/j4YcfxmuvvQZNS74LPLFWiVtCk5/eBy5JQMAwMUINZkI5evQoVqxYgZ/+9KeYN29eTJ+rrq4OEydOjPz7hAkTUFdXBwCQRQGaGXrtnKKAgJ6cM2Vk7CjwJInNmzejoqICZ599tt1DsczMmTOxZcsW5OTk9Pm62+3G1Vdfjeeeew4ffPABrrnmGmzZsgUXXHAB7rrrLmzZsgU+n8+mURM75SgSulT6RA4AuYqEziR5LcJh52c/+1nMww6AQYNg7w+KeU4FHQENTkmkwJNGkrvMP8UsXLgQjY2NA76+du1arFu3Dtu3b7dhVLEzmgufoii47LLLcNlll8EwDLz99tuorq7GunXrMGXKFCxZsgRXXHHFgNBEUleOIqJLNdK+Rb3IJeFQVxB5jsS+jH/++ee47rrr8POf/xwXXHBBXJ5zwoQJOHbsWOTfa2trUV5eHvn30gwHPm7vwcm5brQH1LiMidiP2tKTwN69e7FgwQK43aEdQsO/vO+++y5KS0ttHp09TNPEnj17sGHDBmzbtg35+fmorKzE4sWLUVxcnDLLfmQgalE/4dOuAE7KckAWEvP9Hg47v/jFLzBnzpy4Pa+u65g+fTpeeeUVVFRUYNasWXj+j39E2eRpKHCFduze3+LBKQUZONjuxanUmp5KaB+eVDJp0iTs2rULhYWFdg8lIXDOcejQIVRXV2Pz5s0QRRGLFy/G0qVLMXHiRAo/KehwdxAVKXjEwlj5dAOtAR0nZSbe5nmfffYZrr/+eqxfvx5f/vKX4/78L7/8Mh5++GEYhoHbbrsNa9aswcH2HuQ5ZRS5HWjxBWFwjna/RnvxpBYKPKmEAs/QOOeor69HTU0NNm7ciO7ublxxxRWorKzEqaeeSuEnRdAp6id83OnHjAQ7dsPusDMUzjkOtntR6FKQ75Kxv9UDgTEKPKmFAg9JT21tbdi8eTNqampw7NgxLFy4EJWVlTjnnHMgCOk9O5Ds6BT1kEQ7duPIkSO44YYb8Mtf/hKzZ8+2ezgDcM5xoK0HJRkONHlDszx0vlZKocBDiMfjwdatW1FdXY39+/dj3rx5qKqqwpw5cyBJiV34SQbqCurwGXSKusk5Pu0OYlqO/TVNhw8fxg033IBf//rXOP/88+0ezpA45/h3Ww+yZAl1PX7Mrsi3e0jEOhR4COktGAxi586d2LhxI95++22ce+65qKqqwsUXX0zneSUROkU95FBXAJOyHJHjN1r8OiZkxjcIJkvYCeOcY3+LB03eIC6ZVEjL3amDAg8hQ9F1HW+++SZqamrw2muvYfr06Vi6dCkuvfRSZGXR2n4iq/eqyJATZznHLj2agc6ggQmZSqiIP84zPp9++ilWrlyJX//61/jSl74Ut+eNFuccb9W240tluVCk9H4PpRAKPISMhmma2L17NzZs2IC//e1vKCkpQVVVFa688koqEk9A1KJ+Qu/i5U+6AjENPJzzyIzIoUOHsHLlSvzmN7/BrFmzYvachIwSBR5CxopzjgMHDqC6uhp//etf4XQ6sWTJElRVVaG8vJymwBMEtaiH1HlVZMsishQx5oHntttug2EYOPfcc/Hcc8/h2WefxXnnnRez5yNkDCjwEBINzjmOHj2KmpoabNq0CYFAAFdeeSUqKysxbdo0Cj82ohb1EMPkOOwJLWXFY4bnr3/9Kx588EGUlJSgtLQUVVVVqKqqSonDjUlSo8BDiFU452hpacGmTZuwceNGNDY2YtGiRaiqqsJZZ50FQRAQCATwm9/8Bvfddx+FoTj4uDOAadSijk+6Apic5cARTxCTs2P3ehw8eBA333wzfve73+Hcc8/FsWPHsHnzZmzevBlf+MIX8Ktf/Somz0vIKFDgISRWurq68PLLL6O6uhoff/wxZs+ejV27dmHVqlW488477R5eWqAW9RCPasCjGTA4UOiU4JKsX+b7+OOPsWrVqkjY6c8wDIgiFQAT21DgIYnj0UcfxZYtW6AoCqZMmYLf/e53yM3NtXtYlvjss89w+eWXo6ysDC0tLTj//PNRVVWF+fPnw+FI7yWXWEvEHYfjQTc5OoI6ilwygNDrUOCUIDCGfIsPFg2Hnf/5n//BOeecY+ljE2KRIQNPelf5EVssWrQI+/btw549ezB9+nT84Ac/sHtIljh06BCuvvpqrF+/Hq+99hp2796NG2+8Edu3b8f8+fNx2223YePGjfB6vXYPNSVlyaFT1NONJDAEDI4GX+jU7wxJhMmBgG5a+jwHDhzAqlWr8Oyzz1LYIUmJZniIrWpqavDiiy/i+eeft3soUeGcY/HixVi3bh1mzpw54PumaeK9997Dhg0bsGPHDkycOBFLlizBlVdeifx82uXVCuneol7vVWFyoNQt43B3AKLAMCXbmtfi3//+N2655RY899xzg76/CUkgtKRFElNlZSVWrFiBlStX2j2UqPXem2Skn9u/fz+qq6vx0ksvISsrK9LuXlJSQkXOUejdos45R7dmIEdJn2NDmnwaAoaJoMkBzjHdgiU+CjskyVDgIfG1cOFCNDY2Dvj62rVrsXTp0sj/37VrF6qrq9P2Js85x5EjRyLt7oZhYPHixaisrMTkyZPT9nUZr/4t6rFuz05ErQENzX4NAd3EOUWZUT3WRx99hFtvvRV/+MMfcPbZZ1s0QkJiigIPSSzPPvss1q9fj1deeQVut9vu4SQEzjmampqwceNG1NTUoL29HZdeeimWLl2K0047jU53H6XeLerpGHgAoD2gYV9HAPPLxn80ykcffYTbbrsNf/jDH3DWWWdZODpCYooCD0kc27ZtwyOPPII33ngDRUVFdg8nYXV0dOCvf/0rampq8Omnn+KSSy5BZWUlZs2aRW2/w+jdop6ugQcI1Y2NNySHw84f//hHnHHGGRaPjJCYosBDEsfUqVMRDAZRUFAAAJg9ezbWr19v86gSm8/nw7Zt21BTU4Pdu3djzpw5qKqqwty5c6Eo6b33zGDCp6h/2h3AyVm0IeFY7N+/H7fffjuFHZKsKPAQkipUVcXrr7+O6upqvPnmm5g5cyYqKyuxYMECWh48LnyKulczkK2IyJRpRmw09u3bh9tvvx1//vOfcfrpp9s9HELGgwIPIanIMAy8/fbbqKmpwc6dOzF58mRUVlbiiiuuQE5Ojt3Ds024Rb3YJUE1OYqPb8pHhkZhh6QICjyEpDrTNLFnzx5UV1dj69atyM/PR2VlJRYvXozi4uK06Piq86ooccmQBIbD3aHA0xrQMSmLdrkezt69e3HnnXfiz3/+M0477TS7h0NINCjwEJJOOOc4dOgQqqursXnzZoiiiMWLF2Pp0qWYOHFiyoYf1TBxqDt0cCYDUNejwgDStnB5NPbs2YO77roL//u//4tTTz3V8sc/duwYbr75ZjQ2NkIQBNx111146KGHLH8eQo6jwENIuuKco76+HjU1Ndi4cSO6u7txxRVXoLKyEqeeemrKhR/D5DjYFcDETAX1Xg2ANRvwpaJw2PnLX/6CU045JSbP0dDQgIaGBpxzzjnweDw499xzsXHjRppJIrFCgYcQEtLW1obNmzdj48aNOHr0KBYsWICqqiqcc845KbPXD+cch7qDcAoMrUEdXyzMsHtICef//u//cPfdd8c07Axm6dKleOCBB7Bo0aK4PSdJKxR4CCEDeTwebN26FTU1Ndi3bx/mzp2LpUuXYs6cOZCk5D6SgXOOz3tUHO1RMa80M+VmsqLx4Ycf4p577sELL7yAGTNmxO15P/vsM8yfPx/79u1DdnZ23J6XpBUKPISQ4QWDQbzyyiuoqanBP//5T5x33nmoqqrCxRdfDKfTia1bt+Kzzz7Dvffea/dQx+RwdwAnZciQaLNGAMDu3btx7733xj3s9PT04MILL8SaNWtw9dVXx+15SdqhwEOIlbZt24aHHnoIhmHgjjvuwP/7f//P7iFZStd1vPXWW6iursarr74Kt9uNrq4ubNq0CRMnTrR7eGScwmHnxRdfxPTp0+P2vJqmYcmSJbjsssvwyCOPxO15SVqiwEOIVQzDwPTp07Fjxw5MmDABs2bNwp/+9KeULcL8xS9+gWeffRZz587Fa6+9hpKSkki7e0FBAS0VJYkPPvgA999/P1588UVMmzYtbs/LOceqVauQn5+Pp59+Om7PS9LWkBek5F6kJ8QG7777LqZOnYrJkycDAK677jps2rQpJQPPE088gbfffhuvv/46nE4nOOc4cOAAqqurcd1118HhcGDx4sWoqqpCRUUFhZ8E9f777+OBBx6Ie9gBgLfeegvPPfcczjzzTMycORMAsG7dOlx55ZVxHQchFHgIGaO6uro+yzoTJkzAv/71LxtHFBt1dXU4cuQIXnjhhUgBM2MMp556KtasWYPHHnsMR48excaNG3H33XcjEAjgiiuuQFVVFaZNm0bhJ0Hs2vX/27u3kKjWPo7jv2lLOzoRUnaYzmwMi+hgEUUmdLJs1hQRVCBFIV0V2oHKrOxEQUSHi+iqm4roprV0ZsywHK20k0LRAQqKCg0LskClLB3Xe1W7eLe7HeksXfP93MlcrP+N49dnrWc9VVq/fr1M09Rff/0V9evPnDlTP7mTAESFO/agAlH0T1/ebvzj7vV6derUqTZ3a3k8Ho0YMUJZWVkKh8MKBAIaMmSIcnNzlZKSor179+r+/ftqbW2N8uT4qqqqShs2bHAsdoDOhOABftHQoUNVXV397eeamhoNGTLEwYmc5/F4NGDAAGVmZqqwsFClpaWaOHGijh07phkzZmjbtm2qqKhQJBJxetSYUVlZSewA3+GhZeAXtbS0KDExUSUlJfJ6vZo6darOnz/PgYttaGpqUnFxsUzTVFVVlaZNmybDMJSamqo//+SMq45w9+5dZWVlybKsb8+aATGCXVpAe7p06ZKys7MViUS0du1a5ebmOj1Sl9Dc3KwbN27INE1du3ZN48aNk2EYmjdvnnr37u30eK5w9+5dZWdny7IsjRo1yulxgGgjeAB0Lq2traqsrJRpmiouLtawYcPk8/mUnp6u+Ph4p8frku7cuaONGzcqPz9fI0eOdHocwAkED4DOy7ZtPX78WKZp6tKlS+rVq5cMw5BhGBo0aJArHwpvD7W1tdq8ebN8Pp8GDBigvLw8YgexjuAB0DXYtq0XL17IsiwVFBQoEolo0aJFMgxDo0ePJn6+Y9u2nj59qpMnT+rChQtKTk7W8uXLtWTJEnm9XqfHA5xA8ADoemzb1tu3b5Wfny/LslRXV6e0tDT5/X6NGzfONae7/46bN29qy5YtKigoULdu3VRQUKD8/Hx9+vRJeXl5mj9/vtMjAtFE8ADo+j58+KBQKCTLsvT8+XPNnj1bhmFo6tSp+iMGDwf9GjuBQEDDhw//4bO6ujp9/vw55l+ZgJhD8ABwl48fP+ry5cuyLEv37t3T9OnTtXjxYs2cOVPdu3d3erwOV1FRoa1btyoQCHCgK/A3ggeAe3358kVlZWUyTVPl5eWaMGGC/H6/5syZo549ezo9XrsrLy/Xtm3biB3g/xE8AGJDJBLR7du3ZZqmrl69qtGjR8swDC1YsED9+vVzerzfduPGDW3fvp3YAf4ZwQMg9rS2turhw4e6ePGiioqKFB8fL8MwtGjRIiUkJEiS9u3bp8zMzC6xq+n69evKyclRMBjU0KFDnR4H6IwIHgBtq66u1qpVq/TmzRt169ZN69atU1ZWltNjtSvbtvXs2TOZpqlgMCiPx6Pm5maNGTNGJ0+e7PQ7vq5fv64dO3YoEAgQO0DbCB4AbautrVVtba0mT56shoYGJScnKz8/X2PHjnV6tA7R0tKijIwMvX//XpFIRA0NDVq4cKEMw1BSUlKne9fP19gJBoNdYiUKcFCbv7xx0ZwCQOc0ePBgDR48WJLUp08fJSUl6fXr164MnpaWFq1Zs0ZjxozRnj175PF4VFdXp2AwqP379+vVq1eaO3eu/H6/Jk+e7PjKz7Vr17Rz506FQiG2mAO/gRUeAD94+fKlZs2apUePHqlv375Oj9PuTp8+rTdv3rR54GtjY6OKiopkWZYePHiglJQULV68WDNmzFBcXHT/RywrK9OuXbsUDAaJHeC/4ZYWgJ9rbGxUamqqcnNztXTpUqfH6RC2bf/nW1afP39WSUmJLMvSrVu3lJycLMMwNHv2bPXo0aND5ywtLdXu3bsVCoW+rb4B+CmCB8C/a25uls/nU1pamjZt2uT0OJ1OS0uLKioqZJqmSktLlZiYKMMwlJaW1u4rYeFwWHl5ecQO8OsIHgBts21bq1evVnx8vI4fP+70OJ1ea2ur7t27J9M0dfnyZSUkJMjv9ys9PV39+/f/rYeew+Gw9uzZo2AwSOwAv47gAdC28vJypaSkaPz48d8e0j148KDS09Mdnqzz+3piuWmaCoVC6t69u3w+n/x+v7xe7y/FT0lJifbu3atQKKRBgwZ12MyRSERTpkyR1+tVKBTqsOsADiB4AKCj2bat6upqWZb17cTy9PR0GYahxMTEf42fq1evav/+/QqFQho4cGCHznn06FFVVVWpvr6e4IHbEDwAEE22bevdu3cqKCiQZVmqra3VvHnz5Pf7NWHChB+2u1+5ckUHDhyISuzU1NRo9erVys3N1dGjRwkeuA3BAwBOqq+vV2FhoSzL0pMnT5SamirDMNTY2KhDhw5FJXYkadmyZcrJyVFDQ4OOHDlC8MBtePEgADipb9++WrlypVauXKmmpiYVFxfrzJkzCofDun///rezvTpSKBRSQkKCkpOTVVZW1uHXAzoTVngAwEG/8l6g35WTk6OzZ88qLi5OTU1Nqq+v19KlS3Xu3LmoXB+IAm5pAQD+VlZWxi0tuFGbwdO5jwcGAABoB6zwAAAAt2CFBwAAxC6CBwAAuB7BA8DVIpGIJk2aJJ/P5/QoABxE8ABwtRMnTigpKcnpMQA4jOAB4Fo1NTUqLCxUZmam06MAcBjBA8C1srOzdfjw4R/OrQIQm/gWAOBK3x+jAAAEDwBXqqioUCAQ0MiRI7VixQqFw2FlZGQ4PRYAh/DiQQCuxzEKQMzgxYMAACB2scIDAADcghUeAAAQuwgeAADgegQPAABwPYIHAAC4HsEDAABcj+ABAACuR/AAAADXI3gAAIDrETwAAMD1CB4AAOB6BA8AAHA9ggcAALgewQMAAFyP4AEAAK5H8AAAANcjeAAAgOsRPAAAwPUIHgAA4HpxP/ncE5UpAAAAOhArPAAAwPUIHgAA4HoEDwAAcD2CBwAAuB7BAwAAXI/gAQAArvc/2BTM4+yhsRAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 3D Figure Formatting.\n",
    "fig = plt.figure()\n",
    "ax = fig.gca(projection='3d')\n",
    "ax.grid(b=None)\n",
    "scale3D = 5\n",
    "ax.set_xlim3d(-scale3D/1,scale3D/1)\n",
    "ax.set_ylim3d(scale3D/1,-scale3D/1)\n",
    "ax.set_zlim3d(-scale3D/1,scale3D/1)\n",
    "ax.set_title(r\"Projection onto the span of bases $[1 , 1 , 1]^T [0 , 1 , 2]^T$, for vector $[6,0,0]^T$ (in red).\" + \"\\n Aswell as other vectors.\")\n",
    "# 3D Figure Formatting.\n",
    "\n",
    "# Span visualization.\n",
    "quiveropts = dict(color='lightblue', linewidths = .5,arrow_length_ratio = .05, pivot='tail')\n",
    "spanScalar = 6\n",
    "ax.quiver(*origin3D,*B[:,0]*spanScalar,**quiveropts)\n",
    "ax.quiver(*origin3D,*B[:,1]*spanScalar,**quiveropts)\n",
    "ax.quiver(*origin3D,*-B[:,0]*spanScalar,**quiveropts)\n",
    "ax.quiver(*origin3D,*-B[:,1]*spanScalar,**quiveropts)\n",
    "ax.quiver(*B[:,0]*spanScalar, *((B[:,0]+B[:,1])-B[:,0])*spanScalar,**quiveropts)\n",
    "ax.quiver(*B[:,0]*spanScalar, *((B[:,0]+B[:,1])-B[:,0])*-spanScalar,**quiveropts)\n",
    "ax.quiver(*-B[:,0]*spanScalar, *(-((B[:,0]+B[:,1])-B[:,0]))*spanScalar,**quiveropts)\n",
    "ax.quiver(*-B[:,0]*spanScalar, *(-((B[:,0]+B[:,1])-B[:,0]))*-spanScalar,**quiveropts)\n",
    "# Span visualization.\n",
    "\n",
    "# Projections.\n",
    "ax.quiver(*origin3D, *x, color='red', linewidths = 2,arrow_length_ratio = .05) # <-- Initial vector.\n",
    "ax.quiver(*origin3D, *xTest1, color='pink', linewidths = 2,arrow_length_ratio = .05) \n",
    "ax.quiver(*origin3D, *xTest2, color='pink', linewidths = 2,arrow_length_ratio = .05) \n",
    "\n",
    "ax.quiver(*origin3D, *proj, color='pink', linewidths = .5,arrow_length_ratio = .05); # <-- Projected vector.\n",
    "ax.quiver(*origin3D, *pTest1, color='pink', linewidths = .5,arrow_length_ratio = .05) \n",
    "ax.quiver(*origin3D, *pTest2, color='pink', linewidths = .5,arrow_length_ratio = .05) \n",
    "\n",
    "ax.quiver(*proj, *projDisplacement, color='pink', linewidths = .5,arrow_length_ratio = .05); # <-- Displacement.\n",
    "ax.quiver(*pTest1, *displ1, color='pink', linewidths = .5,arrow_length_ratio = .05)\n",
    "ax.quiver(*pTest2, *displ2, color='pink', linewidths = .5,arrow_length_ratio = .05);\n",
    "# Projections."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Figure 3.11\n",
    "*Projection onto a two-dimensional subspace $U$ with basis $b_1$, $b_2$. The projection $\\pi_U(x)$ of $x \\in \\mathbb{R}^3$ onto $U$ can be expressed as a linear combination of $b_1$, $b_2$ and the displacement vector $x-\\pi_U(x)$ is orthogonal to both $b_1$ and $b_2$.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}6\\\\0\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[6],\n",
       "[0],\n",
       "[0]])"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}5\\\\2\\\\-1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 5],\n",
       "[ 2],\n",
       "[-1]])"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\-2\\\\1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 1],\n",
       "[-2],\n",
       "[ 1]])"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Recall vectors used for above.\n",
    "Matrix(x)\n",
    "Matrix(proj)\n",
    "Matrix(projDisplacement)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "90.0"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "90.0"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "angle(cosAngle(projDisplacement,B[:,0]))\n",
    "angle(cosAngle(projDisplacement,B[:,1]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The norm of the displacement is sometimes called the projection error, and simply reprsents the distance from the original vector to it's projection. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2.449489742783178, 2.449489742783178)"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "npl.norm(projDisplacement),np.sqrt(6) # np.sqrt added for continuity with book."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An important potential trick to note is that if the basis is orthonormal, that is, it is orthogonal and unitary, then $B^TB=I$, meaning that $\\lambda = B^Tx$ and the projection $\\pi_U(x) = BB^Tx$, which in practice on a problem wherein scale is a factor, could save computation time. This process leaves the span unaltered, that is the span of the original basis and the orthonormal basis are the same."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3.12 (Gram-Schmidt Orthogonalization)\n",
    "In the event that we were given a bases and simply wanted to find the orthogonal version, we could do the following.\n",
    "\n",
    "1. Set the first orthogonal vector as the first basis vector. $u_1 = b_1$.\n",
    "2. Project the second basis onto the new vector. $p = \\frac{u_1 u_1^T}{||u_1||^2}b_2$.\n",
    "3. Subtract the projection from the second basis, to find the second orthogonal vector. $u_2 = b_2 - p$.\n",
    "\n",
    "#### Figure 3.12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2 & 1\\\\0 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[2, 1],\n",
       "[0, 1]])"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b1 = np.vstack([2,0])\n",
    "b2 = np.vstack([1,1])\n",
    "Matrix(np.hstack([b1,b2]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2 & 0\\\\0 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[2, 0],\n",
       "[0, 1]])"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u1 = b1\n",
    "p = ((u1@u1.T)/(npl.norm(u1)**2))@b2 # Projection step.\n",
    "u2 = b2-p\n",
    "Matrix(np.hstack([u1,np.vstack([int(i) for i in u2])])) # Output float -> int formatting."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x360 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Figure setup.\n",
    "fig,a = plt.subplots(1,3, figsize=(15,5))\n",
    "plt.setp(a, xlim=(-scale/2,scale/2), ylim=(-scale/2,scale/2))\n",
    "a[0].set_title(r\"(a) non-orthogonal basis $(b_1,b_2)$ of $\\mathbb{R}^2$\")\n",
    "a[1].set_title(r\"(b) first constructed basis vector $u_1$\"+\"\\n\"+r\" and orthogonal projection of $b_2$ onto $span[u_1]$\")\n",
    "a[2].set_title(r\"(c) orthogonal basis $(u_1, u_2)$ of $\\mathbb{R}^2$.\")\n",
    "# Figure setup.\n",
    "\n",
    "\n",
    "a[0].quiver(*origin2D, *b1,scale=scale, width = .005)\n",
    "a[0].quiver(*origin2D, *b2,scale=scale, width = .005)\n",
    "a[0].legend([r\"$b_1$\",r\"$b_2$\"])\n",
    "\n",
    "a[1].quiver(*origin2D, *u1,scale=scale, width = .005, color = \"b\")\n",
    "a[1].quiver(*origin2D, *b2,scale=scale, width = .005)\n",
    "a[1].quiver(*origin2D, *p,scale=scale, width = .007, color = \"orange\")\n",
    "a[1].quiver(*p, *b2-p, scale = scale, width= .002, color = \"orange\")\n",
    "a[1].legend([r\"$u_1$\",r\"$b_2$\",r\"$\\pi b_2$\",\"Displacement\"])\n",
    "\n",
    "a[2].quiver(*origin2D, *u1, scale = scale, width = .005, color = \"b\")\n",
    "a[2].quiver(*origin2D, *b2, scale = scale, width = .005)\n",
    "a[2].quiver(*origin2D, *p, scale = scale, width = .007, color = \"orange\")\n",
    "a[2].quiver(*p, *b2-p, scale = scale, width= .002, color = \"orange\")\n",
    "a[2].quiver(*b2,*-p, scale = scale, width= .005, color = \"orange\")\n",
    "a[2].quiver(*origin2D, *b2-p, scale = scale, width= .007, color = \"b\")\n",
    "a[2].legend([r\"$u_1$\",r\"$b_2$\",r\"$\\pi b_2$\",\"Displacement\",r\"$b_2$ --> -p\",r\"$u_2$\"]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From here we can see that the new bases, $u_1, u_2$, are orthogonal, although not orthonormal as $u_1$ is not unitary."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "90.0"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[2],\n",
       "[0]])"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "angle(cosAngle(u1,u2))\n",
    "Matrix(u1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.8.4 Projection onto Affine Subspaces\n",
    "*Recall affine subspaces from Chapter 2. In $\\mathbb{R}^2$, a line is defined by a support point $x_0$ and a vector $x_1$, with the equation $L = x_0 + \\lambda x_1$. We can intuit if the value of $\\lambda$ is 0 or 1, we have a point either at $x_0$ or $\\lambda x_1$, our support point however, will always \"hold\" our line at a certain point.* "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.rcParams[ \"figure.figsize\" ] = (10,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Figure setup.\n",
    "scale = 10\n",
    "origin = np.array([0,0])\n",
    "plt.axis([-scale/2,scale/2,-scale/2,scale/2])\n",
    "plt.title(\"An affine subspace L. A scaling of the black vector by some value lambda will \\n move the orange vector parallel to the black, 'supported' by the purple.\", size=15)\n",
    "# Figure setup.\n",
    "\n",
    "# Data setup.\n",
    "lam = 1\n",
    "x0 = np.array([-1,1])\n",
    "x1 = np.array([3,.5])\n",
    "val = x0+lam*x1\n",
    "x1s = np.vstack([l*x1 for l in np.linspace(-4,4,25)]) # Grey lines\n",
    "# Data setup.\n",
    "\n",
    "# Plotting vectors.\n",
    "plt.quiver(*origin, x1s[:,0],x1s[:,1], width=.003, scale=scale, color=\"grey\")\n",
    "plt.quiver(*origin, *x1, scale=scale, color=\"k\")\n",
    "plt.quiver(*origin, *x0, scale=scale, color=\"blueviolet\")\n",
    "plt.quiver(*origin, *val, scale=scale, color=\"orange\") #L = x_0 + lambda * x_1 ; Affine subspace\n",
    "# Plotting vectors.\n",
    "\n",
    "# Plotting subspace.\n",
    "# Arbitrary choice of 250 points on the subspace. Subspace is any infinite amount of points along the line.\n",
    "val = np.vstack([x0+l*x1 for l in np.linspace(-4,4,250)]) \n",
    "plt.scatter(val[:,0],val[:,1],color=\"r\", s = .5)\n",
    "# Plotting subspace.\n",
    "\n",
    "plt.legend([\"$x_1*\\lambda$\",r\"$x_1$\",r\"$x_0$\",\"y\",r\"$L = x_0 + \\lambda x_1$\"]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Figure 3.13\n",
    "We replicate the above, this time in $\\mathbb{R}^3$ and with projection onto a plane."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\0 & 1\\\\0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1, 0],\n",
       "[0, 1],\n",
       "[0, 0]])"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1.0 & 0.0 & 0.0\\\\0.0 & 1.0 & 0.0\\\\0.0 & 0.0 & 0.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1.0, 0.0, 0.0],\n",
       "[0.0, 1.0, 0.0],\n",
       "[0.0, 0.0, 0.0]])"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3.0\\\\-2.0\\\\1.5\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 3.0],\n",
       "[-2.0],\n",
       "[ 1.5]])"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3.0\\\\-2.0\\\\0.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 3.0],\n",
       "[-2.0],\n",
       "[ 0.0]])"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.0\\\\0.0\\\\1.5\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0.0],\n",
       "[0.0],\n",
       "[1.5]])"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Data setup.\n",
    "basis1 = np.vstack([1,0,0])\n",
    "basis2 = np.vstack([0,1,0])\n",
    "B = np.hstack([basis1,basis2])\n",
    "support = np.vstack([-2,-1,3])\n",
    "x = np.vstack([1,-3,4.5])\n",
    "P = B@npl.inv(B.T@B)@B.T # Projection matrix onto basis.\n",
    "proj = P@(x-support) # Projection of x onto basis.\n",
    "displ = x - support - proj\n",
    "# Data setup.\n",
    "\n",
    "Matrix(B)\n",
    "Matrix(P)\n",
    "Matrix(x-support)\n",
    "Matrix(proj)\n",
    "Matrix(displ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.rcParams[ \"figure.figsize\" ] = (12,12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fig313(a = True, b = True, c = True):# 3D Figure Formatting.\n",
    "    fig = plt.figure()\n",
    "    ax = fig.gca(projection='3d')\n",
    "    scale3D = 5\n",
    "    ax.set_xlim3d(-scale3D/1,scale3D/1)\n",
    "    ax.set_ylim3d(scale3D/1,-scale3D/1) \n",
    "    ax.set_zlim3d(-scale3D/1,scale3D/1)\n",
    "    ax.grid(b=None)\n",
    "    ax.set_title(r\"Figure 3.13. Affine Projection in $\\mathbb{R}^3$.\", size = 20)\n",
    "    # 3D Figure Formatting.\n",
    "    if (a):\n",
    "        # Basis span visualization.\n",
    "        spanScalar = 6\n",
    "        ax.quiver(*origin3D,*B[:,0]*spanScalar, color='lightblue', linewidths = .5,arrow_length_ratio = .05)\n",
    "        quiveropts = dict(color='lightblue', linewidths = .5,arrow_length_ratio = .05, label='_nolegend_')\n",
    "        ax.quiver(*origin3D,*B[:,1]*spanScalar,**quiveropts)\n",
    "        ax.quiver(*origin3D,*-B[:,0]*spanScalar,**quiveropts)\n",
    "        ax.quiver(*origin3D,*-B[:,1]*spanScalar,**quiveropts)\n",
    "        ax.quiver(*B[:,0]*spanScalar, *((B[:,0]+B[:,1])-B[:,0])*spanScalar,**quiveropts)\n",
    "        ax.quiver(*B[:,0]*spanScalar, *((B[:,0]+B[:,1])-B[:,0])*-spanScalar,**quiveropts)\n",
    "        ax.quiver(*-B[:,0]*spanScalar, *(-((B[:,0]+B[:,1])-B[:,0]))*spanScalar,**quiveropts)\n",
    "        ax.quiver(*-B[:,0]*spanScalar, *(-((B[:,0]+B[:,1])-B[:,0]))*-spanScalar,**quiveropts)\n",
    "        # Basis span visualization.\n",
    "\n",
    "        # L span visualization.\n",
    "        ax.quiver(*origin3D,*support,color='purple', linewidths = 1.5,arrow_length_ratio = .05)\n",
    "        ax.quiver(*support, *B[:,0]*spanScalar, color='grey', linewidths = .75,arrow_length_ratio = .05)\n",
    "        quiveropts = dict(color='grey', linewidths = .75,arrow_length_ratio = .05, label='_nolegend_')\n",
    "        ax.quiver(*support, *B[:,1]*spanScalar,**quiveropts)\n",
    "        ax.quiver(*support, *-B[:,0]*spanScalar,**quiveropts)\n",
    "        ax.quiver(*support, *-B[:,1]*spanScalar,**quiveropts)\n",
    "        ax.quiver(*support+np.vstack(B[:,0]*spanScalar), *-B[:,1]*spanScalar, **quiveropts)\n",
    "        ax.quiver(*support+np.vstack(B[:,0]*spanScalar), *B[:,1]*spanScalar, **quiveropts)\n",
    "        ax.quiver(*support+np.vstack(-B[:,0]*spanScalar),*-B[:,1]*spanScalar, **quiveropts)\n",
    "        ax.quiver(*support+np.vstack(-B[:,0]*spanScalar),*B[:,1]*spanScalar, **quiveropts)\n",
    "        # L span visualization.s\n",
    "\n",
    "    ax.quiver(*origin3D, *x, color='orange', linewidths = 1.5,arrow_length_ratio = .05) # Initial vector.\n",
    "        \n",
    "    if (b):\n",
    "    # Projections.\n",
    "        ax.quiver(*origin3D, *x-support, color='red', linewidths = 1.5,arrow_length_ratio = .05)\n",
    "        ax.quiver(*origin3D, *proj, color='pink', linewidths = .75,arrow_length_ratio = .05)\n",
    "        ax.quiver(*proj, *displ, color='pink', linewidths = .75,arrow_length_ratio = .05, label='_nolegend_')\n",
    "    ## \"Add support point back in\"\n",
    "    if (c):\n",
    "        ax.quiver(*proj, *support, color='purple', linewidths = .75,arrow_length_ratio = .05,label='_nolegend_')\n",
    "        ax.quiver(*origin3D+support, *proj, color='pink', linewidths = .75,arrow_length_ratio = .05, label='_nolegend_')\n",
    "        ax.quiver(*x, *-displ, color='pink', linewidths = .75,arrow_length_ratio = .05, label='_nolegend_')\n",
    "        ax.quiver(*support, *x-support, color='pink', linewidths = .75,arrow_length_ratio = .05, label='_nolegend_')\n",
    "    # Projections.\n",
    "\n",
    "    ax.legend([\"Basis\",\"'Support'\",\"Linear Manifold, L\",\"x\",\"x-Support\",\"projection\"]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig313(b=False,c=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$(a)$ Original setting: we're given a basis, a support, an affine subspace created by the support, and a vector x. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig313(c=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$(b)$ Reduce problem to projection $\\pi_U$ onto basis. Original setting shifted by `-support` so that `x-support` can be projected onto the direction space $U$, our basis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig313()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$(c)$ Add support point back in to get affine projection $\\pi_L$. Projection is translated back to `support`$+\\pi_U$ `(x-support)`, which gives the final orthogonal projection $\\pi_L(x)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.9 Rotations\n",
    "#### Figure 3.14 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.grid(b=True)\n",
    "plt.axis([-2.5,2.5,-2.5,2.5])\n",
    "colors = np.vstack([[i,0,j] for i in np.linspace(0,1,20) for j in np.linspace(1,0,20)])\n",
    "plt.title(\"Figure 3.14 A rotation rotates objects in a plane about the origin. \\n If the rotation angle is positive, we rotate counterclockwise.\", size = 15)\n",
    "\n",
    "R = np.vstack([[-.38,-.92],[.92,-.38]]) # Rotation matrix.\n",
    "original = np.vstack([[i,j] for i in np.linspace(.25,1,20) for j in np.linspace(.25,1,20)]) # Original coordinates.\n",
    "rotated = np.dot(original,R.T) # Rotated coordinates. # Rotation step.\n",
    "\n",
    "plt.scatter(original[:,0],original[:,1], c = colors)\n",
    "plt.scatter(rotated[:,0],rotated[:,1], c = colors);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-0.38 & -0.92\\\\0.92 & -0.38\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-0.38, -0.92],\n",
       "[ 0.92, -0.38]])"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix(R)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Figure 3.15 \n",
    "*Book shows robot arm as a demonstration of an application of rotations.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Figure 3.16 \n",
    "*Rotation of the standard basis in $\\mathbb{R}^2$ by an angle $\\theta$.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1],\n",
       "[0]])"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0],\n",
       "[1]])"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3\\\\1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[3],\n",
       "[1]])"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1 = np.vstack([1,0])\n",
    "e2 = np.vstack([0,1])\n",
    "someVec = np.vstack([3,1])\n",
    "Matrix(e1)\n",
    "Matrix(e2)\n",
    "Matrix(someVec)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.quiver(*origin2D, *e1, scale = scale, width = .002)\n",
    "plt.quiver(*origin2D, *e2, scale = scale, width = .002)\n",
    "plt.quiver(*origin2D, *someVec, scale = scale, width = .002)\n",
    "plt.axis([-scale/2,scale/2,-scale/2,scale/2])\n",
    "plt.title(\"Given a basis, we are looking to perform a basis change, via a rotation matrix.\", size = 15);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Trigonometric identities for sine and cosine work conveniently for rotations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [],
   "source": [
    "E = np.hstack([e1,e2])\n",
    "phiE1 = lambda theta: np.vstack([np.cos(theta),np.sin(theta)])\n",
    "phiE2 = lambda theta: np.vstack([-np.sin(theta),np.cos(theta)])\n",
    "Phi = lambda theta: np.hstack([phiE1(theta), phiE2(theta)]) # Constructed transformation matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.707106781186548 & -0.707106781186547\\\\0.707106781186547 & 0.707106781186548\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0.707106781186548, -0.707106781186547],\n",
       "[0.707106781186547,  0.707106781186548]])"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = Phi(np.pi/4)\n",
    "Matrix(R)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.707106781186548 & -0.707106781186547\\\\0.707106781186547 & 0.707106781186548\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0.707106781186548, -0.707106781186547],\n",
       "[0.707106781186547,  0.707106781186548]])"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix(E@Phi(np.pi/4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3\\\\1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[3],\n",
       "[1]])"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2.82842712474619\\\\-1.41421356237309\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 2.82842712474619],\n",
       "[-1.41421356237309]])"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix(someVec)\n",
    "Matrix(np.dot(someVec.flatten(),R))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.quiver(*origin2D, *e1, scale = scale, width = .002)\n",
    "plt.quiver(*origin2D, *e2, scale = scale, width = .002, label = '_nolegend_')\n",
    "plt.quiver(*origin2D, *E@R[:,0], scale = scale, width = .002, color = \"b\")\n",
    "plt.quiver(*origin2D, *E@R[:,1], scale = scale, width = .002, color = \"b\", label = '_nolegend_')\n",
    "plt.quiver(*origin2D, *someVec, scale = scale, width = .002)\n",
    "plt.quiver(*origin2D, *np.dot(someVec.flatten(),R.T), scale = scale, width = .002, color = \"b\")\n",
    "plt.axis([-scale/2,scale/2,-scale/2,scale/2])\n",
    "plt.legend([r\"$e_{1,2}$\",r\"$\\phi_{1,2}$\"]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rotations in $\\mathbb{R}^3$ can be thought of as rotations of the images of the standard bases of $\\mathbb{R}^3$, so long as the images are orthonormal to each other, and can be implemented with a general rotation matrix by combining the images of the standard basis. \n",
    "\n",
    "The above transformations can be surmised as the following rotation matrix. \n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "\\cos \\theta & \\sin \\theta \\\\\n",
    "-\\sin \\theta & \\cos \\theta\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "In $\\mathbb{R}^3$, rotation about the x,y,z axes is of the following respective formats. \n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "1 & 0 & 0 \\\\ \n",
    "0 & \\cos \\theta & -\\sin \\theta \\\\\n",
    "0 & \\sin \\theta & \\cos \\theta\n",
    "\\end{bmatrix}, \n",
    "\\begin{bmatrix}\n",
    "\\cos \\theta & 0 & \\sin \\theta \\\\\n",
    "0 & 1 & 0 \\\\\n",
    "-\\sin \\theta & 0 & \\cos \\theta\n",
    "\\end{bmatrix}, \n",
    "\\begin{bmatrix}\n",
    "\\cos \\theta & -\\sin \\theta & 0  \\\\\n",
    "\\sin \\theta & \\cos \\theta & 0 \\\\\n",
    "0 & 0 & 1 \n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.707106781186548 & -0.707106781186547 & 0.0\\\\0.707106781186547 & 0.707106781186548 & 0.0\\\\0.0 & 0.0 & 1.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0.707106781186548, -0.707106781186547, 0.0],\n",
       "[0.707106781186547,  0.707106781186548, 0.0],\n",
       "[              0.0,                0.0, 1.0]])"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zRot = lambda theta: np.vstack([[np.cos(theta), -np.sin(theta), 0],[np.sin(theta),np.cos(theta), 0], [0, 0, 1]])\n",
    "R = zRot(np.pi/4)\n",
    "Matrix(R)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1.0 & 0.0 & 0.0\\\\0.0 & 1.0 & 0.0\\\\0.0 & 0.0 & 1.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1.0, 0.0, 0.0],\n",
       "[0.0, 1.0, 0.0],\n",
       "[0.0, 0.0, 1.0]])"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B3 = np.eye(3)\n",
    "Matrix(B3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.707106781186548 & 0.707106781186547 & 0.0\\\\-0.707106781186547 & 0.707106781186548 & 0.0\\\\0.0 & 0.0 & 1.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 0.707106781186548, 0.707106781186547, 0.0],\n",
       "[-0.707106781186547, 0.707106781186548, 0.0],\n",
       "[               0.0,               0.0, 1.0]])"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rotatedB = np.dot(B3,R.T)\n",
    "Matrix(rotatedB)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-2\\\\1\\\\3\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-2],\n",
       "[ 1],\n",
       "[ 3]])"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "someVec = np.vstack([-2,1,3])\n",
    "Matrix(someVec)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-0.707106781186548\\\\2.12132034355964\\\\3.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-0.707106781186548],\n",
       "[  2.12132034355964],\n",
       "[               3.0]])"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rotatedVec = np.dot(someVec.T,R).T\n",
    "Matrix(rotatedVec)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [],
   "source": [
    "P = B@npl.inv(B.T@B)@B.T # Projection matrix onto basis.\n",
    "proj = P@(someVec) # Projection of x onto basis.\n",
    "displ = someVec - proj"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [],
   "source": [
    "newPlane = np.hstack([np.vstack(rotatedB[:,0]),np.vstack(rotatedB[:,1])]) # Span of rotated basis to project onto.\n",
    "P1 = newPlane@npl.inv(newPlane.T@newPlane)@newPlane.T # Projection matrix onto basis.\n",
    "proj1 = P1@(rotatedVec) # Projection of x onto basis.\n",
    "displ1 = rotatedVec - proj1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Figure setup.\n",
    "fig = plt.figure()\n",
    "ax = fig.gca(projection='3d')\n",
    "scale3D = 5\n",
    "ax.set_xlim3d(-scale3D,scale3D)\n",
    "ax.set_ylim3d(scale3D,-scale3D)\n",
    "ax.set_zlim3d(-scale3D,scale3D)\n",
    "ax.grid(b=None)\n",
    "ax.set_title(r\"Figure 3.17 Rotation in $\\mathbb{R}^3$.\", size = 15);\n",
    "# Figure setup.\n",
    "\n",
    "# Bases.\n",
    "quiveropts = dict(color='lightblue', linewidths = .8, arrow_length_ratio = .05, label = '_nolegend_') \n",
    "ax.quiver(*origin3D, *B3[:,0], color='lightblue', linewidths = .8, arrow_length_ratio = .05)\n",
    "ax.quiver(*origin3D, *B3[:,1], **quiveropts)\n",
    "ax.quiver(*origin3D, *B3[:,2], **quiveropts)\n",
    "# Bases.\n",
    "\n",
    "# Rotations.\n",
    "quiveropts = dict(color='gold', linewidths = .8, arrow_length_ratio = .05, label = '_nolegend_')\n",
    "ax.quiver(*origin3D, *rotatedB[:,0], color='gold', linewidths = .8, arrow_length_ratio = .05) # Change of basis.\n",
    "ax.quiver(*origin3D, *rotatedB[:,1], **quiveropts)\n",
    "ax.quiver(*origin3D, *rotatedB[:,2], **quiveropts)\n",
    "ax.quiver(*origin3D, *someVec, color='grey', linewidths = 1, arrow_length_ratio = .05) # Some vector.\n",
    "ax.quiver(*origin3D, *rotatedVec, color = 'b', linewidths = 1, arrow_length_ratio = .05) # Rotated vector.\n",
    "# Rotations.\n",
    "\n",
    "# Projections.\n",
    "quiveropts = dict(color='grey', linewidths = 1, arrow_length_ratio = .05, label = '_nolegend_') \n",
    "ax.quiver(*origin3D, *proj, **quiveropts)\n",
    "ax.quiver(*proj, *displ, **quiveropts)\n",
    "ax.quiver(*origin3D, *proj1, color='blue', linewidths = .5, arrow_length_ratio = .05)\n",
    "ax.quiver(*proj1, *displ1, color='blue', linewidths = .5, arrow_length_ratio = .05)\n",
    "# Projections.\n",
    "\n",
    "plt.legend([r\"Basis $e_{1,2,3}$\",\"Change of Basis\",\"Some Vector\",\"Rotated Vector\"]);"
   ]
  }
 ],
 "metadata": {
  "@webio": {
   "lastCommId": null,
   "lastKernelId": null
  },
  "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": "3.7.4"
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}