{
"cells": [
{
"cell_type": "markdown",
"id": "86a364b0-b071-49ac-91cd-910ed1ac0836",
"metadata": {},
"source": [
"# 2025-09-19 Projections, Rotations, and Reflections\n",
"\n",
"* Orthogonality\n",
"\n",
"* Projections\n",
"\n",
"* Rotations\n",
"\n",
"* Reflections"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "b6c325c7-6e5d-4b47-bc6b-35b2eaf235a1",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"vander (generic function with 2 methods)"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using LinearAlgebra\n",
"using Plots\n",
"using Polynomials\n",
"default(lw=4, ms=5, legendfontsize=12, xtickfontsize=12, ytickfontsize=12)\n",
"\n",
"# Here's our Vandermonde matrix again\n",
"function vander(x, k=nothing)\n",
" if isnothing(k)\n",
" k = length(x)\n",
" end\n",
" m = length(x)\n",
" V = ones(m, k)\n",
" for j in 2:k\n",
" V[:, j] = V[:, j-1] .* x\n",
" end\n",
" V\n",
"end"
]
},
{
"cell_type": "markdown",
"id": "25c26217-d4aa-4656-9278-c1c743520ac5",
"metadata": {},
"source": [
"## Matrices as linear transformations\n",
"\n",
"Linear algebra is the study of linear transformations on vectors, which represent points in a finite dimensional space.\n",
"The matrix-vector product $y = A x$ is a linear combination of the columns of $A$.\n",
"The familiar definition,\n",
"\n",
"$$ y_i = \\sum_j A_{i, j} x_j $$\n",
"\n",
"can also be viewed as\n",
"\n",
"$$ y = \\left[ A_{:, 1} \\vert A_{:, 2} \\vert \\cdots \\right] \\begin{bmatrix} x_1\\\\ x_2\\\\ \\vdots\\\\ \\end{bmatrix} = \\left[ A_{:, 1} \\right] x_1 + \\left[ A_{:, 2} \\right] x_2 + \\cdots $$"
]
},
{
"cell_type": "markdown",
"id": "f6cee22a-89f5-452f-834d-2ed62f322c13",
"metadata": {},
"source": [
"## Inner products and orthogonality\n",
"\n",
"The **inner product**\n",
"\n",
"$$ x^T y = \\sum_i x_i y_i $$\n",
"\n",
"of vectors (or columns of a matrix) tells us about their magnitude and about the angle.\n",
"The **norm** is induced by the inner product,\n",
"\n",
"$$ \\left\\lvert \\left\\lvert x \\right\\rvert \\right\\rvert = \\sqrt{x^T x} $$\n",
"\n",
"and the angle $\\theta$ is defined by\n",
"\n",
"$$ cos \\left( \\theta \\right) = \\frac{x^T y}{\\left\\lvert \\left\\lvert x \\right\\rvert \\right\\rvert \\left\\lvert \\left\\lvert y \\right\\rvert \\right\\rvert} $$\n",
"\n",
"Inner products are **bilinear**, which means that they satisfy some convenient algebraic properties\n",
"\n",
"$$ \\left( x + y \\right)^T z = x^T z + y^T z $$\n",
"$$ x^T \\left( y + z \\right) = x^T y + x^T z $$\n",
"$$ \\left( \\alpha x \\right)^T \\left( \\beta y \\right) = \\alpha \\beta x^T y $$"
]
},
{
"cell_type": "markdown",
"id": "c815d66f-a7f9-44e5-bb4b-b16213914454",
"metadata": {},
"source": [
"## Examples with inner products"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "7e83d1b1-6749-49c0-a423-216265cde8ae",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x' * y = 1\n",
"y' * x = 1\n"
]
}
],
"source": [
"x = [0, 1]\n",
"y = [1, 1]\n",
"\n",
"# Let's take some inner products\n",
"@show x' * y\n",
"@show y' * x;"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "b0b532fb-d34a-4a46-8e20-1334e43e5c0e",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ϕ = π / 6 = 0.5235987755982988\n",
"y = [cos(ϕ), sin(ϕ)] = [0.8660254037844387, 0.49999999999999994]\n",
"\n",
"cos_θ = (x' * y) / (norm(x) * norm(y)) = 0.49999999999999994\n",
"θ = acos(cos_θ) = 1.0471975511965979\n",
"rad2deg(θ) = 60.00000000000001\n",
"\n",
"cos(ϕ - π / 2) = 0.4999999999999999\n"
]
}
],
"source": [
"# Let's define a vector from an angle\n",
"@show ϕ = π/6\n",
"@show y = [cos(ϕ), sin(ϕ)]\n",
"\n",
"# Let's compute and use the angle between x and y\n",
"println()\n",
"@show cos_θ = x' * y / (norm(x) * norm(y))\n",
"@show θ = acos(cos_θ)\n",
"@show rad2deg(θ)\n",
"\n",
"# And compare to our original angle\n",
"println()\n",
"@show cos(ϕ - π/2);"
]
},
{
"cell_type": "markdown",
"id": "d314fc93-45fa-444a-9cea-a3df08bee93c",
"metadata": {},
"source": [
"## Polynomials can be orthogonal too"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "bf6a0fd5-5953-4668-8329-8299b02992d5",
"metadata": {},
"outputs": [
{
"data": {
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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = LinRange(-1, 1, 50)\n",
"A = vander(x, 4)\n",
"M = A * [.5 0 0 0; # 0.5\n",
" 0 1 0 0; # x\n",
" 0 0 1 0]' # x^2\n",
"\n",
"scatter(x, M, label=[\"\\$0.5\\$\" \"\\$x\\$\" \"\\$x^2\\$\"])\n",
"plot!(x, 0*x, label=:none, color=:black)"
]
},
{
"cell_type": "markdown",
"id": "376bad2b-34c6-4410-9a4c-17ddc32dfecf",
"metadata": {},
"source": [
"Which inner products will be zero?\n",
"\n",
"Which functions are even and odd?"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "f1af0457-cfc9-4c62-8a1e-136d9e7e5cca",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(M[:, 1])' * M[:, 2] = -2.220446049250313e-16\n",
"(M[:, 1])' * M[:, 3] = 8.673469387755103\n",
"(M[:, 2])' * M[:, 3] = -4.440892098500626e-16\n"
]
}
],
"source": [
"# < 0.5, x >\n",
"@show M[:,1]' * M[:,2]\n",
"\n",
"# < 0.5, x^2 >\n",
"@show M[:,1]' * M[:,3]\n",
"\n",
"# \n",
"@show M[:,2]' * M[:,3];"
]
},
{
"cell_type": "markdown",
"id": "f6410a9c-3c01-4134-baaf-ff61ce50fdc6",
"metadata": {},
"source": [
"## Geometry of Linear Algebra\n",
"\n",
"We'd like to develop intuition about transformations on vectors.\n",
"We'll start with a peanut shaped cluster of points and see how they transform."
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "a3428487-be39-4b5d-a5fb-bca7a1df2454",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"default(aspect_ratio=:equal)\n",
"\n",
"# Here's the 'peanut'\n",
"function peanut()\n",
" θ = LinRange(0, 2*π, 50)\n",
" r = 1 .+ .4*sin.(3*θ) + .6*sin.(2*θ)\n",
" x = r .* cos.(θ)\n",
" y = r .* sin.(θ)\n",
" x, y\n",
"end\n",
"\n",
"x, y = peanut()\n",
"scatter(x, y, label=\"peanut\")"
]
},
{
"cell_type": "markdown",
"id": "3b18bf01-0c96-4a58-81b4-bd34e6874ec8",
"metadata": {},
"source": [
"We'll group all these points into a $2 \\times n$ matrix $X$.\n",
"Note that multiplication by any matrix $A$ is applied to each column separately,\n",
"\n",
"$$ A \\underbrace{\\Bigg[ \\mathbf x_1 \\Bigg| \\mathbf x_2 \\Bigg| \\dotsb \\Bigg]}_X = \\underbrace{\\Bigg[ A \\mathbf x_1 \\Bigg| A \\mathbf x_2 \\Bigg| \\dotsb \\Bigg]}_{AX} $$"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "7b54b343-7301-4e48-b9fd-9beb746306f3",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"size(X) = (2, 50)\n"
]
}
],
"source": [
"# Here is our matrix X\n",
"X = [x y]'\n",
"@show size(X);"
]
},
{
"cell_type": "markdown",
"id": "8b868ef5-14d5-47a1-9b96-10ff5bad69df",
"metadata": {},
"source": [
"## Inner products and projections\n",
"\n",
"Consider the operation $f \\left( x \\right) = v \\left( v^T x \\right)$ for some vector $v$.\n",
"We'll apply this same operation to every point in the original peanut $X$."
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "7288921e-760d-4697-984d-871de7316a59",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Plot a transformation\n",
"function tplot(X, Y)\n",
" \"Plot a transformation from X to Y\"\n",
" scatter(X[1,:], X[2,:], label=\"\\$X\\$\")\n",
" scatter!(Y[1,:], Y[2,:], label=\"\\$Y = T X\\$\")\n",
"end\n",
"\n",
"v = [1, 0]\n",
"tplot(X, v * (v' * X))"
]
},
{
"cell_type": "markdown",
"id": "85abf8b5-a96d-4c86-98aa-1a206c5f99ee",
"metadata": {},
"source": [
"Question - Are the parenthesis important?\n",
"Why or why not?\n",
"\n",
"$$ v \\left( v^T X \\right) \\overset?= v v^T X $$"
]
},
{
"cell_type": "markdown",
"id": "90f904cf-5b1c-48a5-b871-8b14b5b7aab8",
"metadata": {},
"source": [
"## Exploration"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "e19d8976-980b-4107-9a14-4af0457bd5dc",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v = [1, 1] / sqrt(2)\n",
"\n",
"# What happens if v is not normalized?\n",
"# Discuss with your group and make a prediction before removing the normalization and plotting\n",
"tplot(X, v * v' * X)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "a893e93f-b104-4444-b3a2-fd73989e6a1e",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"norm(v) = 0.9999999999999999\n"
]
},
{
"data": {
"image/png": 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7+/T09Pfv38fGxlKp1NGjR7NYrJaI10g3b96Uk6C5GDZqAn4vM43nkZEFBQVNPYunp6e8vPzZs2erfnz37l16erqPj09Tj1Mfc3NzfX396h/Dw8PV1NRqVj5DQ8O6c9zo6ekNGjTo7Nmznz9/Pnz48KZNmwgEgrAigWYqLCwMuh9W7lr7q0nFwEUhkVHZ2dm4pAKg7RJ+IbSzs7t27dp///3Xt2/fRu5y4MABHo+3devWqsXPjIyM5s+fn5qaeuvWLaHHa7zY2NiOqrJkYqMKQBcNeR6GxcfHN/UsEhISXl5e4eHhKSkpKSkpt27dqnkB13xdu/42sf2TJ08cHBxqVjVra2uBjyH9/PwYDMb8+fM3b95MItV7ZxiI3osXL5hmzkjQV5MyM9dnz56JPhIAbVqrGCwTGhqqrq5ua2tb3TJkyJBly5Y9fvzY29sbr1T5ubnytMYWAEUJCkIoL69xq/j+buLEiceOHdu/fz+RSNy6davAbeLj4//4H5yLi4uenl4DG2RmZiYnJ1c/L2xYjx49JCQk0tLS4JlTa1NaWsqmC56gkkVXLC4pEXEeANo6/Ashm81OSUnp2bNnzcsUQ0NDhBC+r6srKivnpfIauXExk4MQUlJS+osT9erVS0tL6969e1FRUfVde5mbm1eNW2mOsLAw9PsDwvpUVlb6+/svXrx43bp1jx49cnZ2buapgRBpa2tLFT0X+NhAtvibrg5MZQlA0+BfCEtLSzEMU1ZWrtlIpVJlZGSKiorq26uwsDA0NLS0tLTqRzqdPnbs2IafY/F4ja1qVczNzcNC7vL4fFIjHo99zCkhEgl/N36dzWYXFBT4+flV3RluOeHh4aqqqnWHxtTC4XCWLVu2YsUKCQmJnTt3BgQE/LEQYhjW1N/edo/H47XQ70n37t2p32ajimIk+ft1IauMlvCod+817fvPouo3Fh5aCx3v//AOImREIvGPf1vwL4QYhiGE6i68RyaTG/gjKSsr+/btW/XHk5CQGDlyZMOv3HG53CYF8/DwWL9+/ZPU/IEGKn/c+EZidg9bWzU1tSadosrr16+ZTGbDF2rfvn2Liopq+Dh2dnZaWloNbBAeHu7o6Njw3wk+n7927dq5c+dWvb4ybty4o0ePpqSkNPD6IJ/P53A4HA6n4XjipkV/T3avXzlzw8iiCeeQ3P/fMmLkKpydsNl/CZlMbt9/FlW/sVAIhY7L5XK53PY3IIBCofzxQ+FfCKvenyssLKzZyOPxSkpKGrgZqKur6+7uPmbMmMafiEajNSmYra1tP3v7rS+j7XWVaKSGRhWFpeY9Tc27tvNwk45fLTw8XFpautaollq0tbUVFBQaPo60tHQDvRkZGV+/fl24cGED22AYtmrVKl9f3w4dOlS1zJo16/Dhw0ePHt28uc7i2v9HIBBoNBqdTm84nrjhcDgt93vi4z1SUUH+38Xe5SRpTEmfWJQmwS7Zt2m1h7tbC52x9eByuXQ6HQqh0FUVQvH8h4x/IZSQkNDU1MzIyKjZmJmZiWFY9X/HeNl34ECvHj0WPIrd62xV3/DRpIKyuY9iB7m4DB8+/C9OwePx7t2717Fjx4bHpFCp1KpJXv4On8+/fPkyQsjKykrgBhiGff78eeXKlVZWVjW/fxgaGurp6Z04cWLZsmVycnIC9wW4cHYa+PXjwKKiotTUVD09PUVFRbwTAeEIe/Lkyu370XGJ2poaLvY9xo7xEc/iJEr4F0KEUJ8+fQIDA5OSkqpfaLt37x5CqPEvYLSQjh07nrtwYbSPz5hb7zc5mBkp/PYMD+Pzg+Iz1zxLMjQxvXj58l98RT169GhcXJyNjQ1CaOHChdOnTxf6LFnFxcXbt29nMpllZWXTp0+/cOHC3bt3HR0dXVxcam62cePGjIwMdXX1/Pz85ORkIyMjhFBJScnKlSurtvT39+/Zs6evr69w44FmUlBQ+OOtAtBWcLncYb6TXmRjRd3GI6eZqDjrXsijjbvtw24HwtSGLQqHQsjlcqdMmaKvr7927dqqlilTpgQGBm7btu3YsWMEAqGsrOzAgQNycnIjRowQfbxaPD09Q8PCxo31dbrwwl5X2V5bQU2axuLxE/MZD1MLUovKfMeMOXT4cMO3Jeszffp0oQeuRV5efuPGjX/cbOXKlXUb5eTk9u3b1wKhAAACLF657gnfpGLcsp8/y2tW6HdNNXdx8RwT/+55+3t613oI/4V6JpOpqKioqKjYu3dvhNDhw4erfjx69GjVBjwe7+zZs8HBwdW7ODs7T5o06cSJE3369Jk+fbqNjU1cXNz+/ftbyd2ePn36JCZ9ORBwkKhvuePd91kh0QsextzNZg3wHPXq1atz58//XRUEAIBqbDb74vXbFQMX1+7Q7Zyrafvo0SM8QokL4V8RkkgkLy+vuu1Vd9uqNvDz86u1tNCJEyfs7e0DAwM/fPhga2t7+vRpYU22KRRUKnXGjBkzZsxACDEYDBqN1pwndgAAUEtiYiJfxwYRBVz2lXRwDIt809Q5/UHjCb8QUiiUI0eONHRKMnn//v21GgkEwqRJkyZNmiT0PEJXc5kIAAAQChaLxSfXMyiGTKtg4jnxcrsHK9QDAAD+DAwMiFmCJyumZ8d2tjAWcR6xAoUQAADwp6SkZKKpSPjyvHYHq0z2/cVhQzzwCCUuoBACAECrcOHIPq3ghaSPtxCf/7MpO0nxyNDd6/77u3mMQSO1ivcIAQAA6Orqvn/6YO6y1RF7tnCIVBKPraulsfvotl49e+IdrZ2DQggAAK2FiorKxRMHEUJsNhuGposM3BoFAIBWB6qgKEEhBAAAINbg1igAeGKz2ZGRkZ8+xynIynTp0tna2hrvRACIHfEqhKWlpcuWLfvzdqApcnJy8I7QVj18HDrJb1GlUb8SVSsyu1Du+Hp9CuP2xVPq6up4RwNAjIhRIVRVVV23bh2bzcY7SCtStaZuM59GLF26FPcFs9qiDx8+jJm7omDGXSSjghBiI5SHZuUnPunn7vn51dOGl+UCAAiRGBVCMpk8f/58vFO0LhiGVVZWSklJ/XlTIGwzl64u8D5cVQWr8U0df3x7fu7ipckTxuMVDABxA4NlAMABn8//+j0daVnW7Srv6Bl457HoIwEgtqAQAoCDyspKAr2e2dtlVfIL8kUbBwCxBoUQABxISkryK4oF9xWk6WhpiTYOAGINCiEA+LDt0omQEF63Xe7d2YkjYYZlAEQHCiEA+Di4bb1a8GKUHl2zkfbsiBWWMQSWGgBAhMRo1CgArYqenl747Svek2dms0gcrY4kVinh+/tBDr0P3wokEAh4pwNAjEAhBAA3pqamnyJDc3Jy4uLiZGRkLCwsJCUl8Q4FgNiBQggAztTU1NTU1PBOAYD4gmeEAAAAxBoUQgAAAGINCiEAAACxBoUQAACAWINCCAAAQKxBIQQAACDW4PUJAABoFUpLS68GXYt4F81ks/t26+jlOVxVVRXvUGIBrghBG4NhWGZmJo/HwzsIAMIU9iTcxNZ+1qP8s7JDA9V850Wyrfq6XrwShHcusQBXhKDN+PDhw7I5swpyslQkafkVLHkVtY279nbv0QPvXAA0V0ZGhs8/8/P+uYdkfl4Ccg1s83qOm7NhsJlxhy5duuAbr92DK0LQNjx98mSq19BlBsSbbqbHHPRvuJmuNKL4+Xo/fHAf72j1Ki4ufvnyZXR0NJvNxjsLaNXW79iX7+RfXQV/okkVDN+1dP12nEKJEbgiBG0AhmHz/pl2wtFQVYoW/j3/zKe03AoWQkiSTJowyispPVNGpp5FbnHy7ds370n/ppeyeJpWJHY5P+3jMNeB+7aup9PpeEcDrdGTZy/4U/0FdOh2jgtKEnkcsQOFELQBb968sVakqUrRdr/+mpDPWOdgriMrgRAqrGTvfPXVylD/a2YOmdxa/jJnZGT0cRuRNfIg0u/2s4nPP/c0IHG4z9OQm7hGA60Um81GZMFfkjAES5G0OLg1CtqAlJQUAwliVFbxx+ySQ242VVUQIaQoQd3oaN5fU3rWjGn4JqxpzvI12W4bf1VBhBCBwHTwi8bUg4Pv4JcLtF7a2too76uADiZDWgLuIrQ4KISgDZCUlCzH0JW4H3N7dCDWWatvcS/jh7db0ZXWi9dv+RZOdduLbSeeuNKKcoLWw2/SaLmI/XXbJZ8dnjDKU/R5xA0UQtAG9OzZ83kmI7mwzEpFtm6vLI2McbmiTyUQk8nk06QF9ynrpWVkiDYOaBtGeY20peVLhaxHXNbPJoxHjzhsnPF48dxZuEYTC1AIQRugpqZm3bM3g83lYPxaXRifn1pcweLy7t+/z20F5ZBGo/FZ5YL7GPmKCoqijQPaBgKBcP/6pSW2slr7+qkGOKkdGqSx226aRs6r0Hs0Gg3vdO1faxlfAEDD9h89Yayp+jazyEFPuaqFx+cfepdyOynbUEHKTlt+6USfAhZv56Fjo3x8cMxJIBA66OnkZcYhTYtaXRLRN0a6DcAlFWj9SCTSqmWLVi1bxGAwuFyugoIC3onECBRC0DbQ6fSA0+cXTPQNGaMgQSYhhFaExSlKUO+N7kUm/nxqGJ/PmPzPFCVl5YEDB+IYNWDLGpcJ/xZMvY6klaobCckvNJPuTjofgWMw0Ca0tneBxAEUQtBmDBky5OXUGYNOHPmvtzGNRCyo5GwZYFlzA3NlmcODOs6ePD4+LROvkAihrl27ntmxesocZ6a5S4lqRxK7TCH9pSYr4+7da3CbC4BWCAohaEs2b9sxeuz49f7/RYaGbelvXncDGzU5ZhmDxWLhW3LcXV2in3f+8OHD++hYBTmZbl3mdOvW7c+7AQDwAIUQtDEdO3a8evuOoZqSjqykwA3UpGh5eXna2toiDlaLhISEi4uLi4sLvjEAAH8Eo0ZBmyQhKVHMFDyBZzGTI+IwAIA2DQohaJMc3YbcSsqu255TzmKwOD5OfbuaG1++eEH0wQAAbQ4UQtAm7di1+3Fa8dO0gpqNFRze9DsfaCQCsaKUU5i7au7MFUuW4JUQANBWtNQzwpCQkM2bN8fGxkpKSrq7u2/YsEFZWbmB7V1cXDAMq9mipKR0+fLlFooH2joajfb07XvXvr31PqU56asoSlCic0uvJ2QaKcoUMTmpxRUkIoHNw44d3NepWzcvb2+88wIAWq8WKYRBQUHe3t4WFhZLlizJzMw8cuTIs2fPXr9+LS1dz9RTCIWGhqqpqVlY1H4HGYD6GBoafvmRHREREXjlypHAC2wWy1pVrpjJvji8m7asBEKIi/HPx6TPnDTerndvLS2tFg3DZDJJJBKFQmnRswAAWoLwCyGTyZwzZ46Ojs7Lly+r3gy1tbUdN27c7t27V65c2cCOLi4uJ0+eFHoe0L717duXQqEkP7lL57ETChgho3tRSD9v+JOJhIk2ulIUksdAh/fxX1ri7Gw2e+WGrZduBLMJVMTnSZLQvBmTZv8zjVBnZnAAQKsl/GeEoaGhWVlZ48aNq54fwcfHR1FR8dy5c0I/F2hnsrOzr1y5snjlusNHj0VHRzdyLxkZmfxKdkpx+bwehtVVsNpIC63sHxnl5fXM/9kMLBar10D3/V8l0mc9yZnzJGduRMrUu/7BsV4TWtGaUACAPxJ+IXz9+jVCyMnp1zI0ZDLZwcHhy5cvhYWFDezIYDAePnx48+bNT58+1XpeCMTBuq07rR09Jt5I2VHccXYkd8C/awYM8WIwGA3vlZ2dnZeXl1XKLOfwzJQFzE1FQEhdita9o1WGsFd+2LE3IEHDodJxLiJTfzbRZRieO0LT2Hfu3hPuuQAALUf4t0ZTU1MRQmpqajUb1dXVq7oUFeudfT8oKCgoKKjq16ampidPnrSzsxN6PNA6BRw9sevex5L5EYhIRghxEcq3n/rs7WUPnwnhd68L3OXtu3djZ8wtochzVE35cprkgm9snuDvT2wexirOG2jXIy41nUgU2pe/k5eCKqberdte7Dh/38kd/fraC+tEAIAWJfxCWFZWhhCqNSyQ8wAAACAASURBVHV6Vf1r4Nv9lClT3N3dDQ0NS0pK7t69u2vXLhcXl6ioKBMTE4HbJyUlffz48fjx41U/UqnUixcvwlCFpsIwjMlkttz196tXrwLvPIiJ/6KuquLQo/OYUV4SEhICY6zfsbdk7s8qWI1j6xMTd+fp06ddunSptcunT5+GTZ1XMOEiUjFACKEhW7nzlJ6lFVjUuSis4PAKKjmvJvdd9TRhnPfIw6fOCOvTlTFZSODSgxpmX64nV/1DAEJXXl6OYRg8hRU6LpfL5XI5nPY2HwWdTv9jaRB+Iaw6JYvFqtlYWVmJEGpg+scjR45U/9rOzs7Y2HjSpEk7d+6s2V6Ttra2ubl5//79q36UlZVt4FoT1AfDMDKZLCUl1RJHHj3l38dJ+YXdp6CBsxAj92Fk2K7Dgx7fuFT3y018fDxf0wrRBMQoshgS/uJ1v379arX7LV9TMPbMzyqIEEKIseT50W3dPYzVNWXoNbdc8zTBTluBRCBscDAfcj2ysLBQT09PKJ+RhGovjvgTu1KCLiEtLQ3LCLQEAoEgJSUFhVDoqgohnU7/86btjvCfESopKSGECgp+e9O56ulgVVdjjB49mkKhvHv3rr4NJCUlraysBv5f9+7dmxEZCN/aLTvuFSoUTrqMLJ2QojbS61I5cFG69wmXEb5sdu2p0YqLi3mSgldf40sp5hYU12osKirKKuMi9d8LqqZ5/sh9QwJfn4tO+1JYllvOepZW4HP9XVJh2W6XjgghAkIjDJW2bN6UnJwslM+or6uDshLqthPiw/rZ9RDKKQAAIiD8K8KqdwFjY2M7d+5c3fj582c6nW5gYFD/fr8hkUgkEonH4wk9HhABDMOOnLlQNv9F7Q4ty3zDgbdvB48cOaJms7a2NqkgVeChyAUppt11ajVmZWUhhdqNCCGsz6QcPn/3tXnSH74TCQQVSdpQU42x1trZZcxLsT/eZRUXVLC+x90PivquK4HdOH9cV1f3rz8jQmjrysXDZs8rnHELkWvc6ijLV3688b/Ht5tzZACAKAn/itDV1ZVAINy8ebO6JSMj4927d05OTmRyY+tuSEgIk8m0srISejwgAqmpqUjN5NdYyhrKDB0eRb6p1aijo6OAlaK8lNpb8zGF9xeGebjXalZQUEBl+YLPLacuJafQUU0ufHyfqyNtx1prX0vI9Ax6E5aaZ6kiM8xEw1GeTUyI+EzQth80/I9DUhtm36fP+lljlfcNIL66gH58RmkfKM+OqB4adO7Adh0dAXUaANA6Cb8QGhkZeXp63rhx48yZM3w+v6CgYPLkyQihxYsXV28zduxYKpUaGxtb9ePx48f37t0bHx9fWVmZk5Nz4sSJyZMnk8nkOXPmCD0eEAEmk8kn1/OkgUIvr2TWbT59YIfyubEo79uvJnaF3MXpU0e6171u09DQkGQWIEZe3eMoxN7YsG3768ziMjYXIZSQz9jx8stcW8PgUT3/623yTzeDg4Ns7o7qrh1zNUu1y7a9AX/x6XJycq5fv755+66goKBhbi7vQgLXGue7J+zz/H5imy0x7kWoi9PAvzgsAAAvLTLF2tGjRzMzMydOnDhz5kwWi0UkEvft22dv/2s0edXYpOrBiikpKZs2bap5BCUlpUuXLsGTvzZKV1cX5SQJ7KJkxXbtJmAkcM8ePe6dCRg/c2ohj4KpmZIYOYS8rysWzPabMVXgcXat95+yYVrRpIuI+mtVQkrUVTNivq+vb2lR4b+7Nx0dZLX+WZKdttIoy9/mV1OXph8f3Mnn7o1rFVbr/ZfV7MrLyysrK9PX1xc4FoPP5y9cseb87YcVlu7lsnqS7xKl1u3x7G93cNcWIb6VAQAQMQKfX8/It+bh8XhhYWGxsbFSUlLOzs61xuklJSXl5+d36tRJUlISIcTn8+Pi4j58+JCXl0cikYyMjBwcHKq66uPr6+vu7j5mzJiWCC8+MAyrrKxs6qjRwsLChw8fvvoUpywva2fbxcHBoW4ZcBkx5pGeL9/C+bdWTqXKvv6fwoI1NDTqO3hxcXFSUpKGhsYf7y4eP31uxeadTEv3UiVTakW+3LenHVVp188dl5WVRQidOn5s16b1Rfn5e1ysemoJGFFsf+YZU0IxKzMdIcTj8dZs3n7i/BVMVh3RpPi5yX17dj+6Z2utt4CWr9kY8CaXMXwbIvz/8/L5UndWTzBCATs31zo+g8GAUaMtoaysrA2NGmWz2c+fP//0OU5Sgt61c6du3brhnahe4jxqtKUKYUuDQigUf1EIT527uHT9NkbnUUwNK1TJUEh7rpr76f61C/r6+jU3y8rK6uXkkWG/kNfF82fZyEtRDJy5afa4GZMnCCt/aWnp8+fPY+ITNVVVbG27mZmZ1ewtLy+31NU4O9img4KAD+h59Q2VQpmzda/3KB+3kWMi+EblTkurn2uSPtzQe77jfcQjOTm5lJSUmJgYNpv9z7I1BUveISLptwPx+Sp7+saGB6uoqNRshkLYQtpQIQx/GjH2n7nlhv1K1DqSuEz5jNdq5d/vXjkjrBd4hEucC2FLLcME2qXQ0LBFu08VzntafUOyyNarKPWdo4dXwrvnNd8T1dDQeB/xcP5/a0L37OQQqUQeW1NNeeeeVQ79+goxj6ysrJubm5ubm8BeKSkpCUmpnHKWwEJYVMm+5GYza8NaKWnZl0X08tG/zQjP6zw8nV0+a+Hy2ISkTJ5khXYXjM2qJMmjgJFo3EEkX+OKlkAot3CLiIgYMWJE7XMAMRYXF+f1z8L8qbeRnDqqmiwJTctPedvPfUTcm4iG73gBEYNCCJpg3sr1hT4naj6WQwgh/W45Zh5nz1+cNmVSzWZFRcUzh/chhFgsVgNzKbQod6/Rl+9f6aVd+9ZoclG5JJWsLkUryc5cuX5TsfPuuvtybH0uL1/F+/cq0u/6s8kDocSnKMATLXpc8/X/SkmVggbn0QViaO6K9fmee6qq4C8GtjkdfQ4ePbFo3myccgEB4Ak/aCwul5tTXI4UBTy6q7QcfPPR0/p2xKsKIoS27djxsYhzOfa36bZzylkz7nzso63kHx6P+Bg3NVbq1WkBOxPJPEkFpPf77G6m/VCvcSj8twmPZErStDQ1hR4etGmf4xOQgYDhfsxOntdCQkWfBzQArghBY5WXlxMl6nnoJalQXFIq2jiNQiQSPyQm21qYnviYZqetqCZF+5Bd/DmPQScTdeUlLJRlXA1Vo7KKz786VdHBnt+tzr1NIgnVfRbVczQ6NAq5LPj5I4dJjw/p129Fi38Y0KbwCCTBHTLKRXD/oJWBQggaS1ZWFisV8OoeQgjlfTXUa6WvkMvKygbevT/La4i9rlIpi+ugr1LBxU55dKaSiAghjM83kJdyM1IfdXlGnsUAJCn/a8+sBCQvaF17SQXE/P+b+EyG/IUpKxfOlpYWNPs2EGNkPg/xsV8DjKvlf9fSEvT3CuAHbo2CxiIQCHbduxFiH9TtUnx1bOqY1jtUxMbGxnG499200gEGyne/ZG/tb0ElEVOLK6be+TA08PXaiIRlYbEKZEx2Sw9UPYiaVUY8Pg4NWiTgcBXF5Mpi+eD/1M6N1TgwYPdML7/pU0T5cUCbMKBfH1L0nbrtMm/OTPIeIvo8oAFwRQia4OD2DW8HuGdRJPgm/x/8yeNI3VvvoC/dt68wh4MK3fqt248bGrksWUhCmLasxJfCstn3ozf3t+ysLle1QWEl+997n6J3OZa5r6ZmRstHXbCwNHhWmlN3ultaVOA035HD3Z319PQMDAzgVXog0I51/k8cBmVKq/INe1Y3Ul+eNimNHuOzDcdgoC4ohKAJNDU1Xz8KHvfvvLg7y5G6KYHFIOSnTp8wZtXSDXhH+7Op02cc2r2TV5SLEFrzNGGPs3XNFe0VJajnh3Xtf/7F0NJbTi49PA6Ecjiczv1cs1SNkbZ19WaEry/Vo05tevkE3hEEDVNTU4u8f8Nn6qyvIQxMqyOByySkfXC273nk3g348tTaQCEETaOtrf0kOIjNZicnJ8vKympra+OdqAkW+a9aMnNGdhmLj5BZnSV8KSTiJBudM3evH92zveo1r/DbV4aNnZpPVy/T6Ezg86Qz3ulQK2+E3IAqCBpDT0/v5aM7RUVF8fHxdDrd3Nxc4MLUAHdQCEFtXC732o2bIeEvvn1PtzQ18hw0wGlg7VmkqVRq1Xpbbcto37EnDwWcjUnTlxf8OrOxonQHWqF9105R8UkIIRMTk7g3EXFxcZ8/fyaRSDY2PkZGRqKNDNo8BQUFOzs7vFOAhkAhBL/Jy8vr5+6Zod6LYe6B7HWe5SZf2XSu055DIdcu4vg6oBAF3b1v2UHfRlHwZyljc7uoy91Myg4ODvbw8KhqtLCwaItVHwDQSHCrGvxmkNfYRPsVjCEbkXEfpKSHzAcUjTn+UrHf9LmL/7xzWyAnJ3fj/sP32SVsHla3Nyw1v5umvIOe0uG9AuaaAQC0S1AIwS9RUVHf+UqY+YBa7Uz7f0IiXpWUlOCSSuhsbW0dnFxXhMfXmm/+WVpBemlFT21FKQopPT0Nn3AAAJGDW6Pgl8hXrws6OAroIBC4Rn0+fPjg4OAg6kwt4+K16679HYZeeTW+o24HBcm8cvaz9IKkgrJDbjYEhOLyGUpKrXR+AACA0EEhBL+UlVfyKbICu3hkCSZTwMrybRSBQLgV8kBPVTGrjPkxp0RNkubSQXW9gzkBoaSCsrSSSt+x7nhnBACICNwaBb9YmhrJ5MUJ7KJnx7azAZN0On3RilXXE7JGW2rP7t7BXleJgNDz9IJZ96NLEbmivGzW5An79+798uUL3kkBAC0LrgjBL05OTlJLVjMc5iKp39ctyoxT5hW1s0KIEFq8dBmjpGTG4QAqgS9FJbG4GJ1CLmbzXEw1OiQ8UZakfrv3ceKhPf2Hjly/dTveYQEALYW0Zs0avDP8jevXr5uYmFhbW/95U1A/Pp/P5XKp1J/LslMoFFNDvYeb/6nU64lkfq63Tkh8qhY0887Fk6qqqvglbSmOAwYO9xkjpaCEyFRzm87ffmRv7m0wwVLDSFFKU4ZuqSIz0kj5+rM3WZU82x49/3y4Gthsdvt44aS1YbPZVCq1TaxQ37ZgGIZhGJksjldH4viZQQM83AY9VFebtWxlavoPjCpJ5jI7WZkffHhLT08P72gtRU9Pb8HiJQihR48elX1+baetULOXgNCq7jqee3f/4zcbZsYCoF2CQghq69Kly8uHtxFCFRUVVTONiYmIxw8d1ATMgEUnk3SkKcnJySYmJqJPBQBoafANF9RLrKogQqistESaKvirIYlZPniAQ+DlSyKOBAAQASiEAPykb2KaUir4FZG8ctY5Z+Oj61ZcvXJZxKkAAC0NCiFAP378CAkJuXHjRnJyMt5Z8DR8hFfgtxIuVmvCGRSTWypJIWnJSOzv12HDiuUYJmBuNgBA2wWFUKzl5eW5jhjTeegkn2ORYy997j1hiUX3vgkJCXjnwoeurq7P1Bmzn34tZnKqG6Oyihc//ry6rylCSIZGtlSS/PjxI34ZAQDCB4NlxBeLxbJ3HfrVcSVm4VzVUoFQ7o9Yx2Gjo8Luampq4hsPFwuWLtfRN/BcOE+Ky9SUof9gMDWk6ccGd9aR/TmIRp1Oys7OxjckAEC4oBCKr0PHTv4wGVJdBX/Sssx1Wb149aYLxw7glAtnXqN8GGXlqed2uRkqa0rTib+/r5bH4qmoqOCVDbQ0Lpd789bt+09ffv2ebmliONLdqd3MrwsaALdGxdfFm/cquo2p245ZDXoa+VL0eVqPgU5Oj7IqtGUkalXB1OKKp6m5zyOehoSEFBYW4hUPtJC8vLyOdo6Tz7w8QXMJ77EmgGvnuea4y3AfNpuNdzTQsqAQiq/8vFwkpy6gg0DkIrGetkNXV7fXQNcdUenVw2a4GH9lePyMe5+8jZQJD88+2rJ0gG2n3du24pkSCJvrSN/Evv6MIZuQaT+kaogsnYvGnnom2W3mwuV4RwMtC26Nii8lJaWU0hwkX+dZIB8jodojJ8XN9n0H/Bcv8gy+0UtDRoPKv56YY6cp92BMr+oN5thoLLl8gi5B/3f2XBxzAmF5+/btd4IKZta/Vnulw5zbu+wYDIaMjAwuwYAIwBWh+PIZ6ibxTsAb4oTYh/Y9u4s+T6tCIpE279od8vKd+4odJOfxJBp9ca/f5hwnEwmbe+sf3L2Tw+HUdxDQhjx78aqwQ+0lqRFCiEDgdbD79OmTyBMB0YFCKL5mzZiqGX+dkBD2W2t2omrIyh3rVuAUqnVRVlZ2cXFRVlIarCdft5dGInZVk42KihJ9MCB0jIpKPlXABHsIIR5VsrKyUsR5gCjBrVHxRafTI0Jueo6b+u3Zvgrd7jwSTTY7Wp7x/fr1C9ra2nina0Xyc7OVaCSBXUpUQkFBgYjzgJZgaWIo9fZjuaAuWnackdEMUQcCIgSFUKypq6uH3r6am5v76dMnNpttaelhaWmJd6hWR1NbN6GSK7ArsxITzxcu2x9XFxfp/9aVO8xBkr9f/WfGqfBLDQwMcMoFRAFujQJkYGAwbNgwb29vqIICObu4hKQzePzaA4hKWJy4ogobGxtcUgHhkpaWPrR9g9LRYSg7sbqRkBShdmFC4MmDOAYDIgBXhAD8gaqqqvfEyctuXNhgp08j/fzuWMTk+IV/XbdtNyxS2G4MH+Khqa42e/nStMxsRJchsss6WZgfateLcYIqUAgB+LOl/qsDZOSG7t1lqSyjKUFKLeOmMJjrtu8ZMnQY3tGAMPXo3v1N6F2EUFlZmbS0NN5xgIhAIQSgUWbNnTd95qykpKT09HRjY2MDAwO4FmzHoAqKFSiEYqqwsHDL7gMPwp8VFhaqq6sPde6/YPa/4rYSb1NRKBRLS0t4kgpAOwOFUBwlJSU5Dh2V23s2d+R5JKWYUZoTExV4updj5MPbampqeKcDAACRgns7YgfDMPdREzJ9TnF7jkNSigghJKvGcpydMnDtiAnT8U7XJn379m3SaO8uZkZdjPS6mBnNmDDux48feIcCADQWFEKx8/LlyyIVS6RV+/4eZj4gqYCZnp6OS6q26+2bN8MHOgzipN52Nbo92Py2q1HvkvhBfe3i4+PxjgYAaBQohGLnY3RMoabgqUSZut1jYmJEnKdN4/F4M8b7HnE0sNNWrG7sr6+8t7fOnGmTcAwGAGi8liqEGIa9e/cuJCSk8d+LeTzemzdvQkJCEhMT/7w1+Gt1XgyvRiAQ+PX3grpevHhho0jTlqk9R6WxorQcqzQ0NBSXVACAJmmRQhgTE2NhYWFra+vm5mZhYeHo6Jibm9vwLlFRUaampj169HBzczMzM3NxcYEpHFtIR2srhcy3ArvoaW+trKxEnKdNS0xIMJUUvHZjJyWJqb6jQh8/FnEkAEBTCb8QMhgMNze3wsLC4ODgjIyMo0ePvnjxwtvbu4FdCgsL3dzcKioq7t+/n5GRceDAgbCwsLFjxwo9G0AI2dnZyedEo6yEWu2EpKcGskSYRKNJKBQKt55LaIzPn99Jc/6Myfn5+aIN1W7l5uZu2rRpxOhxk6ZMDQoKwjsOaD+EXwiPHDmSkZGxffv2wYMHa2lpTZs2zc/P7+nTpw3cJgoICMjNzd27d6+Li4uWltasWbOmTZt2//79Fy9eCD0eIJFIwZdOaVwYT3p7BTEZCCFUUUR9dkQvZPmNc8fxTtfGdO3W7VUBS2DXq4yiXtqKYw0Vzp0+JeJU7dKCpf9pmHf1D899qD78DLPjqHXHZbQMExJqf58D4C8IvxDevn2bSqUOG/Zr6qlRo0YhhG7dutXALpKSku7u7tUtVVeQDewCmsPCwuLTs4cz5eLNz3tq7urZ8arvEuOymJdPNDQ08I7WxlhZWbFlVZ5nFNVqv/81R4FO0ZCmd1OT+fAqEpds7cnuPfv2BD7A1nzge29HPUaj/jOxOcFlXns793XmcgUvDAJA4wn/hfqYmBhjY2M5Obnqlq5duxKJxOjoaIHbYxj2+fNna2vrmtOa2NraVh1K6PFAFRUVlX3bNmIYVllZKSUlhXecNux80A33Af2ep+V7GKvrykqkllTc/ZIdlVVy0qMzQohIJHC4XA6Hg2EYjUbDO2xb9d+W3fz5j5GE3G+tVi4sa4/l/63Yvm0rTrlAOyHkQshisYqLi2stTEMikeTk5HJycgTuUlJSwmQylZSUajZKSUnR6fT6dkEI8Xi83Nzcb9++VW8PU6IAXKiqqj5/99FUVyu/gvWjlKknJ9FLW3FRL2MykYAQisllhEd/1+nSFyECDXEnjh65cskCMhlmdGqCwsJCNomOlHTrdvG7+wTeXbx9m+hDgXZF+IUQIVR3ykoZGZnKysoGdql7USItLV3fLgih+Pj4sLCwvXv3Vv1Ip9MjIyMpFMpfJxdPGIYxmUwMw/AO0uZ5+YyWjX26Y+CvRXrzKlibI79EphdKy6tUFGWxpVTS3dZsfXTj5JnO7oNcbTua93d0VFZWxjFzW5GYmPhzCqS6ZFRKy8oZDIZoE7VPXC6Xy+VyOBy8gwgZnU7/Y2kQciGsKoElJSW12ouLi7W1tRu/C5/PLykp0dfXr+9EVlZWS5cuHTNmTDMDizkMw8hkMtwabb51W7Z5OPVnfMiYYKEmR6NkMCon3Hq/vLfJLqefr6NkMCpnXJmQKG+Z0W/pESR/+mGs7NYRqxf5zZo2Gd/krZ+NjQ0qzhLcV5CqqiQvIyMj2kTtU1UhpNPpeAfBgZAHy5DJZDU1tVrjxVksFoPBqK8QysjIyMrK5uXl1WwsKSnhcDj17QKEpeoLB94p2gMJCYn74c+0hkyc+a7I426i982PW/pbDjRQqd5Anka5MqyTckkKMnNAVi6sgQvy5j1deez67Tt3cYzdJkhKSspIUFGKgJdfCaEBfuMbejULgMYQ/qjRLl26JCcnZ2X9+gYXERHB5/O7du0qcHsCgdClS5fY2NjCwsKau1QdSujxQJWIZ88793XWsu7ZefA4DYuuriN9v3//jneoto1MJs9duOjp2/dhb97rqCrZasojhMo5vA3PEt0uvZwS/GHszShJVhH92tKfO1DoRT6HF67ehGfoNiLw2D7i0TEoM+5XEx8jBG9QKUuZPXs2frlAOyH8Qujt7Y1h2Pnz56tbzp49SyAQqt+pxzAsKCio5muFXl5eXC730qVLNXchEoleXl5CjwcQQhevXB0+a8XHQQdyFrzInXk/e8Grhx0m93AeCvNEC0VqamoHeUmEUBmb63vjnYmS9B2fnldG2N707nHLq4dz4TOZSzN/biqrVsollZWV1XeowsJCeIKLEHJ2dr55cj89YAhxYy/CuZnEwz7E5SY2BZEp0W/wjgbaA+GPXvP19T1w4IC/vz+TyezSpcu9e/fOnz8/YcKETp06VW3A4XC8vLw6d+78/v37qpbJkycfOnRo0aJFDAbD2tr6xo0b165d++eff8zMzIQeD5SUlMxftalwThii/VqDm2/SN2f0ydHT/D4+h+kxm4tOp1dyMYRQwLsUH0stbwut6i4ZGvmAi4XXtRtvwm34DjMQQkhSrrS0tNZ66Onp6TMW/vchJo4vrYQqSlTkpHav9x84oL9oP0cT5Ofn5+XlGRkZtdyANQ8Pj8ocj6SkpDt37ujp6bm4nIFF5IGwCL8QUiiUe/fu+fn5rV+/nsPhSEtLz5s3b8uWLdUbEAgEXV3dmu9u0+n0Bw8e+Pn5rVy5ksvlysrKLlmyZP369ULPBhBCd+/eY9iMqFkFf9K2zuLQ0tPTdXR08MjVfhgbGyfkl3Exfnhq/p3RPWv1EhBa3NNw1q3FOYY9kE6nsoxkFRWVmht8+fKlr4d3zuCtfGfHqpacwoxRi6dvmZ0xbdJ4EX2GxuHz+fsPH9u2/whXWoUvrYJykoy01c8d2t2hQ4cWOqOJicn06dOlpKQIBMFTvALwF1rkfSZVVdXAwEAmk1lYWKiiolLrSyKVSq37OEpTU/P69etVu6iqqsKLVi0nJvFrpYq5wC6OmtnXr1+hEDYTmUweM3HS9odX6WQiSdD/10aKUnqyErwjnvnjz2AcVq1/IL4z5mZ7HUZ6NR6QK2oXTru2YqvjcA+3VvXGxewlK859zCv1C0PUn29M5aa86T3I8/m9a4aGhvhmA6DxWnA9Qjqdrqmp2aRbJVW7QBVsUTJSEohdLrCLxKmo+w4o+AvLVq5mGXQqYAp+JauSw1OWpGoRytEFP6nf7+/l5uZ+L2H/VgWrUCRKu465cTu4qUkePHhoO8Bdy9JWw6KbmW3f7Xv283i8ph5EoLi4uCthb0u9A6qrIEIIGXTPHnFg0uzFQjkFAKIBC/OKnX69eyp9FfQgkI8Rvr6ytrYWeaJ2iEgk7j50RElDO7uMWbf3aVpBFw15W0055LKQTPnta19KSgqmZiLwmCxVs+iEr02K4b9+y+jVB985782c/yJ7wcvECTfXPsuzdxnCZrObdByBzl25XtBjKqp7yduhR2JaZkVFRfNPAYBoQCEUO3Z2djrsDGLik1rtko+2+Qxzl5CovcYs+Gtrtu6Y8+AzF/ttoaYfDObZT2k+lloYIiAeV09Ls2avhIQEgSX4eh2xKmSkmvCn8+HDh0M3HhdNvoyU/r+0Fl2mfNDKTwo9tuza16QPIlD8t+98ZcHPAglKejVfoAKglYNCKHYIBMKD65dMn66XvbUMxYeh7EQUfVfx9Bgn8tddm9bina5dcR88WELXeNCll9cTMuPzGVFZxUfep04Jfr9toKUcjfIqp1I+6tyONctq7mJubo7SPiCegHuqCsmPBtr3avzZdx85Xdh/CSKSarVX9J934kLgX3yc2nlkZVBFseC+imJZWdnmnwIA0YBCKI5UVVVjXoafmtRnGi+0f9TGhbKfgrfNv3nxNDydFbrbDx8Xsnhx+YzjH77fSsxSoFOuefWwUZMLS83LKaucmXw2ywAAIABJREFUNKgPxuMVFBRUb0+hUKb4jpIO2VDrOMTk55qliQ4ODo0/dXRcPNKxEdBBkWDyUPNfTxzq3E824Z6AjooiOrOw1lBYAFoz+I9PTJFIJE/P4cOGDYVlmFqUlJTUvScRw50HzOys7dpBVVmSmlXG3PP6263kHHN1Zc7re5df3Y3OK9c2tTh48kzVGiwbVy3/Pt3vwTHPwm4TkLopKs6U/RKq8SPyYfBVIrEJ31wpFAri1vMskI81//UDj8GDNTdsL0uKwEz6/mrlceQvz9y4AgbLgLYECiEALatr166fklP27dw+J+ReUXGxppZm8g/GQVfrbuq/Vtd7lJrj3r/f0zdRNBqNSCRePH7w7du3N+49fB99W19b03liryFDNjX1et2+e9eoL8/53UbW7mDkyUtLNL8QkkiksNtXnT1H//hoUmTggKSVKDnxclEXFk0f7+sD83+CtoTA5/P/vFXr4+vr6+7uDqtPNBMszNtyGAyGwFURls6bo/vl+VBj1Vrth6Mz1Twmzp6/QFgB0tPTuwwcmj/rAZKU/9XK58udnxLw72Bfn1FCOQufzw8PD3/++t2P3ILu1maDXF1qzpXREsrKyuCF+pYgzqtPwBUhACIV+uD+1YH6ddtHGimP279nwpSpwhpmoqOjc3zn+mmLXAscFmIdeiG6FEr7pBixZ4xDJ2FVQYQQgUBwdHR0dHQU1gEBED0ohGKNz+enpKTk5ubq6ekZGBg06REU+DsYl0MhCfh9VpakVpYUWhgaTPabP2/2TEXFepaibYqhg9272HTce+TEyydBZWWMTtZW/+5b2bNn7VnfABBzUAjF16XAoEWrNnBVjNiKBrSi76TcL5v8l0wYOxrvXO0cRiDy+Py6U6+VsDgsDqYrh/YdOnbsYlDgiQD7Pr2bfzodHZ0dG9Y0/zgAtGNQCMXUqXMXF+67UDTrMZL4/5CNytJ5B6ZUMJn/Tp2Ea7R2rlfvPuHfYwfo154yNDgpe2xHnUGGqgsfx8V2GD5y8szYF6GtamZRANoruBUmjlgs1vINW4smXvhVBRFCErLFE86u2b63vLyemU2AMKxYv3FnTF5q8W8zkEXnlF6IyZjQUaeDgtTVEd20Y64WWI0IOHoSr5AAiBW4IhRHkZGRbGNHRKszWJQiwTJ3iYiIGDRoEB65xIK2tva567e93F2tZAid1eUwPvqYXZJTzjo6uJM0lYwQIhMJ/3VVn5Px4TGLtBrvtACIA7giFEeZmZnlsoLXWmLI6P7IzBRxHnFjY2Nz+OwFFol2+lO6Ap3ybzeDwBG2OrK/5hG1VJFRSA7NSorGMSQA4gMKoThSUFCgV+QL7JJg5ikJY7wiaFi/fv1yEV2GSvY007RQrv26IRfjWylLWlKYZ07B3VEAWhwUQnHUp08favwDhNVZl46PScTes7e3xyOUeCGRSBeu38qt5DBY3Lq9r34UWavIbnUw2bdts+izASBuoBCKIzk5uUmjhsvcWo74NWZe5vOl76zyGewEIxVFw8TEZMX6TWte1F5isIjJOfEhdZSlljSVLEXgl5SU4BIPAPEBhVBMbV3rP9laWnmPg+SDLej1ZclH21T29x9nRNi7ZT3e0cTIzNlz1O2cxt7++OZHURmbm8lgBidl+95492+3DiHJOf+FxRUUF585fQrG8QLQomCuUbGWmZkZHh7+/Ue2SQe9Xr16aWpq/nkf0Dj1zTVa18SxvlkvH5dzuDJUiqWKjCyNfC0h09dKx0pFtoLL/ZBXeSej7Oj5Sz1gRhiEEMw12mJgrlEgptTV1YcOHQqTbuNrmf/KKUMjLw42JREIr38U7X3z9bpXD9r/p2HrqYWGGjKnjPF+8uY93LUGoCXArVEAcGZmZjbYd+Ls8K/5FeyD71I2OlrQfp+MVFOGPtVU8dC+PXglBKB9g0IIAP6Wr1ozcdUWv7f5X4rKDeQl627QT0fx+ZMw0QcDQBxAIQSgVfAcMfL5h+j6bn5KUkiVFRUCuwAAzQSFECCEEI9X551CgAsShcPD6jZ/LSo3MOwg+jgAiAMohGKtqKjo34XLda27a3a007TsZucy9NXr13iHEmuDh3teTsyp234yoWDM5OmizwOAOIBCKL4yMzO79HW6iHqkz3mWO/951vyXL/tscJ+2+PylQLyjia+lK1ffzuNfTczB/v9eUyWXt+ltmrRZJxdXV3yzAdBewesT4mv8zPnprpv45gN+NWmYFc64tXDtAFen/jBSHxdSUlKPnr1YuXTR4Dv3lSUobB5WySdOnPGP37wFXC43NjY2OTlZQ0PD2tq6kS8pAgD+CAqhmCotLY35lsEfPKB2B026pNu4oOs3/5k+FY9cAElLS+8OOIwQKigooNPpVW953rh+bc2SRabydDrGzqzkZpSxvXzHrd20hUiEmzoANBf8KxJTKSkpfHVTgV0sdauo2CQR5wF1KSkpVVXB4Nu39qxY1FuekJiRSeHzbJXoNrKks4cPjvBwxzsjAO0BXBGKKSqVSuCwBPdxWRJ0mmjjgHrx+fzVSxZa0XgkIiV4VE/i/6cWK2Fxxt58dfLE8clT4NodgGaBK0IxZWRkhDJiECZgDSCZr+H97bqJPhIQ6NSp09RKRkpRxVI7Y2KNCTblaJTTQzpvWe2PYzYA2gcohGKKQqGMG+Up+Whb7Y4fscrfI9zd3PAIBWqbOnvhvD1neYg4zFSjuvFjdsnK8PhR197Ovh9NYFYcOnQIx4QAtANwa1R8bVnjnzBmYuS5icW2E5GmOSrJlkgKU46+cv/GJQqFgnc6gC4FXg36nM8YuZdxeKCa9M+b1fvffHuTWTTbtoOFnQyPz/+QXbJ+5dKol5HHz57HNy0AbRcUQvFFJpNvXz579969mw8eRN/do6WpObCf7aTjEbAYRSuxee/hEs9TSFY9h4nllbMQQk9S82PySs8M7VJ9j9RBT7mXtuLgy7dv3bo1dOhQXPMC0FZBIRR3/R0dPQYPxjsFECCvqATJaSCEypyWnn+22ctC69Sn7xsdLYi/L8VHIxHX9TPb+N9SKIQA/B14RghAa/X/yWW4TvO+yZvveJVcUMnWkZWou2E3DfmsrEzRhgOg/YBCCEArpSgngxi5Vb8unvvopGS/ggqOwC2JRML/iyYAoMmgEALQSi2eNU323pqfP5AojDGHuQRyfgW77pZxeQxVFRVRZgOgPYFCCEArNWHsGDddsvyFqSjnC0IIYbxSw37bXn6ptRkfofXPkmYvW4FDRADaBSiEALRSBALh0olDJ+Z62r9YpbPHTj/AoZ+RQlQJb/Hj2Nzyn7MCpZVUTrj1XsncZtLkyQUFBcnJybC0JABNBaNGAfry5curV6+/pv3oaG7ct29fWHeiVfEcNtRz2K/hoHw+f8HcOd6Bl1hMJkJIWkZmxoLlZALfxriDEo0kT6ekFJWbWFnvPnRUU1MTv9QAtCVwRSjWKisrvSdO7+07d+r93LXpeuMuxZrbu27YvhvvXKBeBAJh9779ydn56cVl6cVl8elZyf9r774Dmrr6PoCfJBB2WIJsREBlOVBxIThQVLROVEBfF8466tPap7ZaV7Xap3WPOqqlWqptHXUXFQUUJwruIoIgyAqBkISRdd8/YhEhImqSC8n381dzzr03X1PIj3vvuec8vJ92aPfvA1pHupk6MKraGMr0M+/27uqfn59Pd1iA5kFdZ4RcLnfv3r337983MzMbPHjw0Lc9qbZ7927q9XFvxsbGEyZMUFM8UAifPCPepGf1jJdLn1cQUhHy6fexMy04e+bOxFTOzcDNmzefXk9YGeAUeTSlu5NloLO1jTH7Sakoq+x5r46+GfnFWKcJ4K0YlBqGXaenp/fp06e4uDggICA/Pz8rKys6Onr37t0N7KKnp1fn3oaDg0NeXt6bto+KigoLC4uMjFRZaN1z//79vtGLudFH6naIK+w2Bec+TGGxWHTk0hICgUADa+d+OneOf97NTTcyvgps29neonbXfy88qHL1O3zytLozaJhQKDQxMWG8PqsAfDipVCqVSg0NDekOQgO1/LU4adKksrKya9euXbly5enTpzNnztyzZ8/vv//e8F6RkZG8Wh4+fKiObFDjzLkLJT4jlXSwjWVOHR48eKDxRPDOcjIzCkVVfrbmdaogIWRVH6+riZekUiULjABAbaovhLdu3bp27VpkZGTnzp0JIQwGY+3atfr6+lu3bm14RwMDA8tazM3NVZ4NaissKaVMrJR2SU2sS0tLNZwH3oOpGed2AT/Y1bp+F5vFdLcwnjFlMmohQMNUXwgvXrxICKl9U9DCwqJ3797JycmVlZVv3V0ul6s8Eijl7uKoX5qttEufl+3o6KjhPPAegkKHZJVVGOsrv4jtxDES30+Onog7CAANUX0hTE9PJ4S4u7vXbnR3d5fJZJmZmQ3seOzYMSMjIxaL5ezs/Nlnn5WXl6s8G9T2UdgQ89sHCVXvL4+yFyaCXA8PDzpCwbuJjIrKEkozeCKlvZlloi+7t+Y/TktKStJwMIBmRPWjRsvKyggh1tavXatRvFR0KWVnZzd48GBXV1ehUBgXF/fDDz/ExcVduXLlTcMNioqK/vrrr2fPnilecjic6dOnY4DcO7G1tZ02JmxH7MzysZuJ/r9TOZfmWf0cuXPTGolE+bSW0EgSiUQDnyGTydy+9+eFU6LGeTvqs177+U/JLzPSY90t4gfasA/G7O3evbu6w2iG4oPFYBmVUwyW0b4hciwW662lobGFcOTIkQ0/lrR9+3Z/f39CiGIYap0fU0WOBi575uTk1GRds2bNrFmzdu/evWnTpiVLlijdXiKRiEQiHo+neCkSibTyf6G6ff3f/3BMf9y8PpCyaysxd2RzM4wqinZuXNM7sBcmKPlAMplMM5/hoEGDDvUbEHE0afvg9rYmL9fvTcrhfnruQRsr06SckgJh9c1bh51cWn2y6HMtqB+KD1YL/iFNjexfdAdRscacIDW2EFpbWzf8AbHZbMV/cDgcQgiPx7O3t6/pLSkpIYQ0MP6ldlYmk7l27do9e/bExcW9qRA6Ojri8YkPJ5fLF8yZ+cWnn2RlZb148cLT09POzo7uUFpCIpFobCT6b4ePLVv61dCN6+1MDSwM9fOF1ZUS2U/DOvrZvvyNk8qplSdjV5Ty1m7YpJlI6qMY4o9CqHK6/PhEYwvhnj17Grml4u5gTk6Oj49PTWNOTg6DwWjdunUjD2JlZWVqalpzwgdqxWQy3d3d69zWheZlxarVl86eWeNrasrWiz5xJ+YjfwezV99oekzGip5uYQf3Dxw2vF+/fjTmBGiCVH9TLSgoiBBy5syZmpaKioqEhITOnTubmpo28iCPHz8WCATOzs4qjwegrZav+/7La89LqyQWhvq1q6ACg5DJfo5TI8YWFhbSEg+gyVJ9IezZs6e3t/eBAweysrIULf/73/8qKiqmT59es82GDRtGjx6dk5OjeKl47r6m9/nz51OmTCGEREREqDwegLbq26/f7CWros8/sTY2ULqBg5lhewv2mmVLNRwMoIlTfSFksVj79u2TSCT+/v6RkZFBQUHLly8fMmTI1KlTa7a5fv36kSNH+Hy+4uWJEyc8PT3btm3br1+/gICAtm3bXrt2bcqUKRMnTlR5PAAtNj5qwqHjp8oo5bc8SirEPjZmly/FazgVQBOnlkm3AwIC7ty5s2XLlrt371paWu7YsSM6OlpP79V7DR8+vFWrVra2toqXkydPNjc3T01NLSws5HA4kyZNGj16dEhIiDqyAWg3f3//fDFDUC01M6j72306o3Bu19Z/F2JVCoDXqGv1CQ8Pj02b3jg+LSIiovZlz3bt2i1evFhNSQB0CoPB+GL5ytlLPv1paAeDWk8W/no/l8VkeNuYUUzcIwR4DRbmBdA2EVETjh85HPrruTBPOw9Lk0JR1ZXnPBtj9voBvvHZJT17B9EdEKBpQSEEaE4oirp48WLC1ZvZLwoD2rcbPHCAm5tb/c1+2v9rz04dzAxYJZViRzOj1X29XcyNnvCEG+4Vn01arfnYAE0ZCiGQM2fOrtu+N/PZMwYh7q1bL54bPWAAbtA2RYWFhQNGjs81a1vaui/hBBxIfmy5dUL0mLBvl31ZZ0tjY+NT8ZemRoxllhV3ZMrzs3mPBLJcCevgiTMODg60hAdoslAIdd38/y45kvqibNDXZHhbQlE5+Y9Sl62YFJ+44duVdEeD11AUFTJi3MNei+XtXj4RL/MM5PaasuPgLOddP82ZMa3O9o6Ojn8nXsnIyEhLS5NKpWN9fHx9fTWeGqAZQCHUacdPnDqc+oI/aT9RTFjFYBAH79IpB2N+Ght2/kJISH+6A8IrcXFxuebeNVXwJSaLP3rD6vX9Zk+fqnTWMQ8PDywkAtAwLNeg09Zu3cUPW0XqfIEyGKVDVq7evJOmUKDcqfiksnaDlXQYmMqsW2VnK19aEgDeCoVQp+W+yCc2SoZaEAevp5lZGo8DDeHxBcSQo7RLbsQRCAQazgOgNVAIdRtFvakHc/s3Nd7urqziDKVdjOIsTMwL8N5QCHWaq4szKfhHScfztDZtPDUeBxoSMWak5Y19RF5vNbScO61amFlYWNARCkAboBDqtKUL51ic+JJQry+YLJdZnlqydOFsmkKBcm5ubjPHDjU/MJVU8l+1Zt1seWjGL9vX05cLoNnDqFGdNnBAyJSEKzG7R/MGLCbOHQmhSPYdq7hv5o0ZENS7N93poK5vln7h7vrr8u8GV7PNKDNbws30dHaIOX0Ya0kCfAgG9ea7RE1ZVFQUVqj/cHK5vLKy8s6dO//bse/Bw0eEwejg6/P5x9O6detGd7RmTyAQmJmZqengfD6/uLi4VatWteeyf29SqfTSpUsJ8RdEQmGfkAGDBg1is9kfflg1EQqFJiYmWKFe5bBCPei0wMDAwMBAulPAOzA3Nzc3N1fJoe7cuRMeFtrV2rCXkyWLwfgz4ficSbyNu/aOCQ9XyfEBmj4UQgDdVVRUNHJgv1+H+rmaGytahrWxyxdWjZo+uaWdXW9cHgfdgMEyALpr8acLF3VxramCCvamhmv7es2ZFEVXKgANQyEE0F2J8edD3W3qtwe6WPNKSsRiseYjAWgeCiGA7pJUVxvqseq3sxgMfSajqKhI85EANA+FEEB3GRoaciuUnPaJJDIZRVQ1HgegiUMhBNBdwaFDYu7m1G//42Gekamp+h7/AGhSMGoUQHdt3LrNw6FlKwvj0e1erdb799OiHbezt+yNoTEYgCahEALoLiMjo4Sbt4cE99p4PdPP1ozNZN4rLq+imF+uWj1q9Bi60wFoCAohgE7z8PBIzytMSEj4+8wZHrdoef8BoaGhVlZWdOcC0BwUQniNTCZjsZQMIwTtFhwcHBwcTHcKAHpgsAwQQgifz58+/zMXvwCHDr0cvLt06z80Meky3aEAADQBZ4RAioqKuvUPy+05Tzp/NWEwCSH5RU9HzpuzdsHk6VP+j+50AADqhTNCIFPnfZbTb4m0a4SiChJCiK07b+axr77bnJ+fT2s0AAC1QyHUdRUVFTfv/yP3G1K3Q9+orNu03/44TEcoaEKEQuGKpV/5e7Vxs7PxcHIM/2hYVlYW3aEAVAmFUNfl5eURWw+lXRJ735T76RrOA01KYWFht45+v+7Y7MmsmOFtO8XTXPJPSq+Ovgd/+43uaAAqg3uEuk5fX59IqpT3SaqMDA00GwealqmR4yWl3F1DO3pYmihaJnVwecwVTJw5LSg42MHBoeHdAZoFnBHqOhcXF0ZRBpEqmXDS5GlC/55dNB8Jmggul/vkQdqiHh41VVChXQuzz3u4fzrvY7qCAagWCqGuYzKZMyZFmpz5pm5HwT/W6WdGjRxBRyhoEv755x+JWDKgtW39rqGedrdvXNN8JAB1QCEEsnzxosEWXKufo8ijC4SfT56nGcZvcor9vzN/7DcwwKVR3cVkMhmE6DEZ9buM9FhymUzzkQDUAfcIgTCZzD9idickJBw4eirtzFYHe/v+vbtM2ZVoampKdzR4DUVRKSkpd1LTxFJpRz/f7t27q3UaIG9v72qZvFIqM6q3ZmF5tdSMg0WaQEugEMJLmGSricvIyBgWMYXLaV3mGEAx9SyO7TfN++TPn3/s0rmzmt7R3NzcuZXbkcf5Ub5Odbp+e5A7dsJENb0vgIahEAI0A3w+v89HY/PCdxHnDoqWEkJKuM/CIsfePPeXi4uLmt73yJm4Lt5tW5kb9XK2rmmMf1Z8IL3EuvTQ7q2bza1aBPUPWb12nYmJSQPHAWjKUAgBmoHvNm0r7j6rpgq+1KJV8ZBV/12x9reftqvpfR0cHK7fezgmbNDqq5kdbEwJg3GXW8EVVhgyyUBzadvWrtxK8aW4w20P/HI64XL79u3VFANArVAIAZqBY6fjxFF/1m+nvAdc3rhcrW/t7Ox8/e4DLpd77949Fos1b2a0N5vsDOvIZr0cajfO2/FSNjesb1B2MY/JxPg7aH7wUwvQDAiFImKkbHAKgymlNBGgRYsWffv2tbOzK81/sX6gX00VVOjj2iLQgbNz505NRAFQNRRCgGbA2tqK8JVNgC6tNqg3pFN9EhMT7U3Zlob69buGt2n59/GjGksCoEIohADNwIQxw42v/1K/XT/lz6Gh/TUWQygUmiurgoQQC0P9CqFAY0kAVAiFEKAZmDsz2iXrrF7qsdqNzPRE++Qtq75cpLEY3bt3z+FXKu3KLqv0aOetsSQAKoTBMgDNAJvNTo47MXHWghsb18td/CkWW+/5nTb2FgfjjltaWmosRvfu3asI615RuZ8tp3Y7Rciu1Jx938ZoLAmACqEQAjQPlpaWJw/9wufzHz16JBaLfX2XWllZaT7GzgMHo8eN+mloR+8WZooWsUz+1aXHHfqG4vEJaKZUXwgFAkF8fPytW7dSUlKKiorGjx//2WefvXWv8vLyJUuW/PHHH8XFxe7u7vPmzfv4448ZDCWTHALoMnNz8+7du9MYIDQ09Pcz56eND5dXCjwtTXhVkrwK6fSP53+5bDmNqQA+hOoL4eXLl0eMGEEIadGiBZfLDQwMfOsuMpls2LBhly9fnjVrlq+v7/Hjx+fNm1dcXLxixQqVxwOAD9SjR4+H2bkikSg9Pd3a2lp989oAaIbqC6GPj8/p06c7d+5cVFTk5+fXmF1iY2MTExNXrly5dOlSQsisWbNCQkK+/fbbqVOnurq6qjwhvJVcLo+Pj798PaWQV9ajg/egQaG2tkrW4gFdZmJi0qlTJ7pTAKiA6keNuri4DB48+J2+N2NjY5lMZnR0tOIlg8GYPn26RCL5/fffVR4P3io7O9u7a+/w9cdWPHf9kQRPP1vo03vQtt176c4FAKAWTWKwzI0bN7y8vOzt7Wta+vfvr2inL5SOEovFfYeFPxu2kXLr+rLFqz+398ylu8a7OtoPHTKY3ngAACpH/3OEQqGQx+PVOYNs0aKFnp5eTk4OXal0VuzBQ0Weg2qq4Ev6hqXjd3y2bA1NoQAA1KhRZ4RVVVX5+cqmd/qXvr6+k1PdFcsaSSQSEULqPAvFYDDMzc2FQuGb9nr8+PGVK1e+//57xUtDQ8NTp07p6yuf8wLeRC6XV1VVUdSr2Sr/PBMv8pmtZFNze16llMvlGhoaai5fcyYSiTDsWR1EIhFFUfhsVU76L7qDqJihoaGe3lsqXaMK4dWrV/v169fABq6urs+ePWt8stoU1auysu50FRUVFWw2+017ubm5jRo1atCgQYqXhoaGmnysWGvI5XIWi1V7JTmhqEL55M6EME0sKIrCsvWNhM9KfUxMTFAIVU5RBXXzL91GFUIfH5/Y2NgGNviQX3gOh8Nms3k8Xu3GysrKyspKGxubN+1lYGDg5ubWWW1rc+usNm4uCUUZxKZ1/S6qrMDa2rp+OwBAs9aoQmhraxsREaGuBHp6bdu2ffz4sUQiqbm2effuXUKItzemLtS0aZFjDi9Yw/MZWKed8eh8145+WG0OALQPPd9rYrG49pXowYMH8/n8+Pj4mpajR48SQsLCwmgIp9u6des2pL2z6dFFRFpd08hIT7Q//eXO9d/SGAwAQE1U//iE4llsQojirmFOTs758+cJIW3btnV2diaEVFdXGxoadurU6fbt24pd5s+fv3379vnz5x89etTLy+vEiRObN2/u2rVrSEiIyuPBW/2yc8va9Zu3bAiUWzhQxpak6ImPu+sv5044OjrSHQ0AQPUYtUcMqkRlZaWxsXH99k2bNs2fP58oK4SEkL///jsyMpLH47FYLJlM5ufnd/LkyQamboqKigoLC4uMjFRteF0jl8srKytrD5aprbCwsLS01MPD461jrqA+gUBgZmZGdwotJBQKMVhGHTBYRpUMDAxu3bpVv11xOkgIYbPZqampdYplaGhoVlbW+fPnuVyuu7t7cHAwvnxp17Jly5YtW9KdAgBAvVRfbJhMZsODORkMRocOHeq3czicUaNGqTwPAABAAzAIEAAAdBoKIQAA6DQUQgAA0GkohAAAoNMwMhOgabl27dqmPQfS7t/X09ML8O/46expXl5edIcC0GY4IwRoQhYtXRE2b+XBFmMeTTx6b2zsT8w+QeHRO/fG0J0LQJvhjBCgqfjrxMk9F+6VzThGap4W9w3ltg3+av2gwG5dfHx8aE0HoLVwRgjQVKz4fkvZiO9InTlT9A1LBi1bvXEHTaEAtB8KIUBTUVBcQqyclXR49Lx1O1XjcQB0BS6NwjsTi8WPHj2SSqVeXl5K55WF9/PGiX+ZLJlMptksADoEhRDegUAgmLHg8wtXrhOXjhRLj/H8bqe2bvt/3GRra0t3NG1gyTErEBQTs3rrUeekenu1oyMRgE5AIYTGkkqlgaEfPfKZJFm0uabx/MO4bv3D0q5c4HA4NGbTDp/MnPzZH98KRq9/rZWSW/79zecbFtMUCkD74R4hNNbPWMsMAAAPVElEQVSuvT8/te8t6RZVu1HuPfBFt5nLvv2erlTaJHry//W1KOf8+Qnh579sKnpquS9yamjX3oGBtEYD0GY4I4TG2nfwiGiIkrGL4i7jj23vv+HblZqPpGWYTOax2J9jDsRu2h1dVFLKJMTZyfHrr2cPGhRKdzQAbYZCCI1VVFRMLByUdOgZVEkxlEM1GAzG5IlRkydGvX1TAFARXBqFxjIxMSGVfCUdFMUibxjuCADQ5KEQQmMNHdhPL+2v+u2Mxxe7dfHXfB4AAJVAIYTG+mLhXNvLW8mLR6+1lr1ocWrxd8u+oCkUAMCHwj1CaCwrK6vzR2OHRUzhtezIdwqgWGzz/BSTpwkHf9rq7u5OdzoAgPeEQgjvwMvLK/12clJS0u20e1XVZV3HhgUFrWOz2XTnAgB4fyiE8G6YTGZwcHBwcDDdQQAAVAP3CAEAQKehEAIAgE5DIQQAAJ2GQggAADoNhRAAAHQaCiEAAOg0FEIAANBpeI4QQI3EYvFPMftPnE/KzMx0a+32Uf/e0yb/H6YgAGhSUAhBXUpLS/f8vD/xZmoZv7xLe+/J40d36NCB7lAaVVxc3HvQiOduAyra/4cEu/7DzU6MO7JxZ5/LZ/+ysbGhOx0AvIRCCGpx+Upy+NQ5Jd2nS9rOI4acy7n39k/5dOrQoO9Wfk13NM0ZETX1SfBXcu8BL187eFU4fJXxqNtHkVOunjtJazQAeAX3CEH1SkpKxkydXTD9uKT3DOLoS6xdSIewktmnd13659eDv9OdTkMyMzOflJNXVfBfcq+Qp0Jmeno6LakAoD4UQlC9HXv2lfSYTSzsX2tlMPijflj+3SaaQmlaamqqoFUvpV0C1163b9/WcB4AeBMUQlC9c5dvSNv2VdJhal1eLZVKpRpPRAOpVCpnKr/1IGPp68iHANAsoBCC6lVXVxN9Q+V9emyJRKLZOPRo06YNp+Cu0i7zgrR27dppOA8AvAkKIaiebztPkndfSYdcxqoWGhkZaTwRDTp06GBR+oQU/FO3oyDdvOSfzp070xEKAJRAIQTVmzt1gnXCRkLJ67Szk/eOHjaYlkiax2Awju7fbbd/AuPeKSKXEUIIJWfcO91yf9TR/bsZDAbdAQHgJRRCUL2OHTtOGxpk8fNEwi942SSTGCbu8Hx0cO3yr2iNplG+vr4pF0+PF51z2RLUcn0P1y1B40VxKfGn/Pz86I4GAK/gOUJQi3Urlwb4//X12oklfAHRMzAg0vCPhqza+beOXBet4eDgELtnG90pAKAhKISgLqNHDB89YjghRCwWY1IxAGiycGkU1A5VEACaMhRCAADQaWq8NCqRSHJzcxkMRqtWrd66cWZmZp0WFovl6uqqlmQAAAD/UkshXLRoUVJSUlpaWlVVlYWFRWlp6Vt3adOmjUwmq93i4OCQl5enjnjQxD158uSLb/53O/VutURqZmI8YvCAr//7HxMTE1rCiESiuLi4m2kPjY0MenTp1KdPHxaLRUsSAFATtRTC2NhYGxubyMjIs2fPVlRUNHKvPn36zJkzp+alrg0vBIXzF+Ij535ePHQt+XgzYbLyxRUbr+8/3LPf1fMnNb900YnTZ6Yv/ELoO1xk14FIqywTjln+58tTh2IwLwyANlFLIXz27Jm+vj4hpGvXrhkZGY3cy83NLTw8XB15oLkQiUQTP/60eOZJYmb7soltLO49M8vCeeKsBWcPx2oyTGpq6pRFK0vmxBETK0VLadexpS8ehoyMeHQj0czMTJNhAEB91DJYRlEFAd7V8RMnBX4jXlXBf8n9htxJzy4rK9NkmIVLV5eM3lxTBV9y8C4OmLZpxy5NJgEAtWpCo0Zv3749bty4sLCwuXPnJiYm0h0HaHDz7kORQyelXXInvydPnmgyzOOMp8SlY/12sd9Hx+MuajIJAKhVE3qgvqCgwNjYmM/nx8XFbdu2bd68eZs3b37TxhKJpKioqGasqYGBgaOjo6aSgrrosVgvp+WshyGXvtMoFaFQeOiPPxNvppULRb38fcePGeXk5PROYeRv+jPRxKKcX/5OhwKApqyxhXDbtm18Pr+BDSZMmODi4vLeOZKTk7t27aqYiTgjI2PcuHFbtmzp2bPn+PHjlW6fnp5+6dKlTZtervJqbm5+8eJFXJJ9V3K5vKqqiqIouoO81Nmvndn+ZEGHoXU7KIp6dtvJyUkoFDbmONdv3Jww+5PSTpHVrYaTFsbH76at/XHEV3OnTZ88sfFhWJSUyKWk/pqCxVkODvZvTSISiTCztjqIRCKKovDZqpz0X3QHUTFDQ0M9vbdUOkYjvwRdXV1zcnIa2ODixYt9+vSp06gYLNOYxyfquH//vp+fX1hY2MmTJ5VuEBUVFRYWFhkZ+a5HhtrkcnllZSVdTybUJxaLPTv1yIk6QGw9arcbJO8do3fvwK6tjTlIcXGxb6+QohnHiYXDq1ZptdWuUX+u/7JvvZ/SN5k+/7OfqztIu9b9U8z88MJd0/qOHTO64d0FAgEG1KiDUCg0MTFBIVQ5RRU0NHzDSqJarbFnhNnZ2WrNUYevr6+pqamG3xRox2azTx78OTQ8kttzjqRdCLGwI4UZZjdi2gnv7zp1pJEHWb9tZ0nQJ69VQUKIngEvfMvnKxfebHQhXLf8y3NBA5+bWMm9B75souSGl7b6yXPCR49q5EEAoOlrQvcIaysrKxOJRBwOh+4goGl+fn73ky+s37YzPv4/hYUFnh6eEeNCJ0Ssa/wNwriEK7Jhu5V02LR+/qJASfsbWFlZXb9watKchbfPLCOOPkxJFZX/OHx42A97/8DpCIA2oacQJiYmZmVlDR8+3MLCghBSWlpqaWlZ00tR1LJlyyiKCgkJoSUe0MvKyuqbpYvfe/fKikpioPxiL8VkvdPtpZYtW549HCsWi9PT042MjNzc3JjMJjTQGgBUQi2FcOPGjcnJyYSQp0+fikSisWPHEkLc3NzWrVun2GD79u2HDh26e/euohCuW7fu1KlToaGhLi4u5eXlp0+fvnr1aps2bRYuXKiOeKDdPNxbP8p/TFp1qdshFRswyXuczLHZbF9fX9WEA4CmRy2FMDMzMyUlhRBiaWlpaWmp+O+qqqqaDZycnLy8vGruynbr1i0xMXHbtm2KbRwcHObNm7d8+XJFmQR4J/OmRl5euaW0VUyddoPkvWOHh9ESCQCassaOGtUMgUDAZDIbM4gRo0ZVoqmNGlWVcZNnnCk2FgxZRgxMCSGEkrOvxrjf++Xmpb819o/FqFE1wahRNcGo0aYCXxygEgf37dy47ceNP4ZWM9hE35BVUTostP8PGqyCANCM4M6/TktKSlq0aBHdKVSPwWAsnDs7+96NrOvnHl/4M+9Ryo8bv9NkFZTL5UFBQRp7O50yd+7cW7du0Z1CCx05cmTt2rV0p6BH0zojBA0rLi5+8eIF3SnUyMjIiJb1vGQy2YMHDzT/vrrgxYsXXC6X7hRaqKioqKDgHZ4v0iY4IwQAAJ2GQggAADqtuV4ara6uzsrKUjyYAe/t6dOnZWVl+BhVTiqVUhSFD1YdysvLMzIy8NmqXE5OTnFxsfZ9sE5OTi1btmx4m6b1+ETjDR8+PD09HYMAP1BFRQWfz7e3t6c7iBbKzMxs3bo13Sm00IsXL6ysrHRzlL9alZeXV1dX29jY0B1ExcaMGfPFF180vE1zLYQAAAAqgXuEAACg01AIAQBAp6EQAgCATkMhBAAAnYZCCAAAOg2FEAAAdFpzfaAeVE4mk92/fz8lJeXWrVtcLrdXr14LFiygO1TzQ1HUrl27YmJiCgoKHB0dp02bNnnyZLpDNXsURT1+/DglJSUlJSUvL69NmzbffPMN3aG0RFpa2uHDhy9fvpybm2tubu7j4zN//nx/f3+6c2kUCiG8lJqa2qVLF0IIi8WSyWR6evjZeB/z58/funVrv379IiIikpKSpkyZkp6evmbNGrpzNW88Hs/b25sQwmQyKYrq3r073Ym0R3BwcEVFhb+/f8+ePUtLS2NjY3/99dcjR44MGzaM7mgaRAFQFEVRubm527Ztu3btWnZ2NiEkIiKC7kTNz40bNxgMxtixY+VyOUVRMpksLCyMxWI9ePCA7mjNm0gk2rBhQ2JiokAgMDY27tGjB92JtMeqVatyc3NrXl6/ft3AwMDZ2VnxM6wjcI8QXnJ0dJwzZ063bt0wedV7i4mJoShq/vz5ivXTmUzmggULZDLZ/v376Y7WvBkbG3/yySe9e/c2NTWlO4u2WbJkiaOjY83LgICAfv36PX/+PD8/n8ZUGoZCCKAyV69e5XA4tS/cBQcHGxgYJCcn05gK4J2IxWImk2lmZkZ3EM1BIQRQmaysrJYtW7JYrJoWNpvdokWLrKwsGlMBNN7NmzcTEhKGDh2qU4UQAyK0nEQiaaCXwWBgUIwKCQQCT0/POo1WVlbPnz+nJQ/AOykrK4uIiDAyMtqyZQvdWTQKX4La7MGDB76+vg1sYGRkVFFRobE8Wo/FYonF4jqN1dXV+GsDmr6Kiorhw4fn5OQcOXLExcWF7jgahd9PbWZnZ/fdd981sIG+vr7GwugCKysrHo9Xp5HH41lZWdGSB6CRKisrw8LCkpOTY2Njhw4dSnccTUMh1GbW1taLFi2iO4UO8fLySkpK4vP55ubmipaioiIej9ezZ096gwE0oKqqasSIEUlJSTExMeHh4XTHoQEGywCozMCBAyUSyZkzZ2paTp48KZfLQ0NDaUwF0ACxWBweHn7u3Lnt27dHRUXRHYceWKEeXklNTZXJZDweb+DAgaGhoatXryaE2NvbOzg40B2teeByuR4eHjY2NufOnWvVqtWTJ09CQkKqqqoyMjJ0agyeOjx8+LCyspIQEhgY6OnpuW/fPkKIlZWVm5sb3dGaMYqiRowYcfz48VmzZkVHR9fu8vLyMjY2piuYhqEQwiscDkcgENRpXLJkyapVq2jJ0xydPXt27NixlZWVjo6Oubm5HA7n2LFjQUFBdOdq9tq3b3/v3r06jRMmTMBkBR+iurr6TRNo3Lx5UzHnoi5AIYRXLl68KJPJ6jS6ubm5u7vTkqeZKigoOHr0aF5enouLy8iRI21sbOhOpA2uXbsmFArrNNrZ2TU8LhoaJpfL4+PjlXYFBARwOBwN56ELCiEAAOg0DJYBAACdhkIIAAA6DYUQAAB0GgohAADoNBRCAADQaSiEAACg01AIAQBAp6EQAgCATkMhBAAAnYZCCAAAOu3/AWxNwus9cJMYAAAAAElFTkSuQmCC",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Vector transformations are \"nice\" if norm(v) = 1\n",
"# Let's build a transformation from an angle\n",
"θ = π * .7\n",
"v = [cos(θ), sin(θ)]\n",
"@show norm(v)\n",
"\n",
"tplot(X, v * v' * X)"
]
},
{
"cell_type": "markdown",
"id": "f0eb59ea-fcc6-4c83-a503-70f3169314f1",
"metadata": {},
"source": [
"## A related projector\n",
"\n",
"Compare to the operation $I - v v^T$.\n",
"Note that $\\left( I - v v^T \\right) X = X - v v^T X$."
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "baebad37-e3ee-4657-9314-29d91fc67346",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Here's ther related projector\n",
"tplot(X, X - v * v' * X)\n",
"# This also works since I is the automatically-sized identity.\n",
"# tplot(X, (I - v * v') * X)"
]
},
{
"cell_type": "markdown",
"id": "53a2519d-7e34-42c5-9b63-0ba9d3b3ab8a",
"metadata": {},
"source": [
"## Discussion\n",
"\n",
"Discuss with your group what the vector $v$ looks like in these diagrams.\n",
"\n",
"* What is the **nullspace** of each transformation?\n",
"\n",
"* What is the **range** of each transformation?\n",
"\n",
"* For each figure, what happens if you apply the transformation $v v^T$ to the red points?\n",
"\n",
"* What about $I - v v^T$?"
]
},
{
"cell_type": "markdown",
"id": "c610ae2c-1583-4aec-b22b-14a33303ad61",
"metadata": {},
"source": [
"## Givens Rotation\n",
"\n",
"We can rotate the input using a $2 \\times 2$ matrix, parametrized by $\\theta$."
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "0f57eb78-dc04-489d-a6b6-555130adc36d",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"our_givens(0) = [1.0 -0.0; 0.0 1.0]\n",
"our_givens(π) = [-1.0 -0.0; 0.0 -1.0]\n"
]
}
],
"source": [
"# Create a Givens rotation matrix from θ\n",
"function our_givens(θ)\n",
" s = sin(θ)\n",
" c = cos(θ)\n",
" A = [c -s;\n",
" s c]\n",
"end\n",
"\n",
"@show our_givens(0)\n",
"@show our_givens(π);"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "6c4a4024-2caa-4ffb-9586-a7c8f3b1a816",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Let's plot what happens when we apply our Givens matrix\n",
"tplot(X, our_givens(.8) * X)"
]
},
{
"cell_type": "markdown",
"id": "80af860c-1f82-46c7-b6c8-897229c05459",
"metadata": {},
"source": [
"* What is the **nullspace** and **range** of the Givens matrix?\n",
"\n",
"* Is the Givens matrix **orthogonal**?\n",
"What does that mean?\n",
"\n",
"\n",
"(An [**orthogonal** matrix](https://en.wikipedia.org/wiki/Orthogonal_matrix) is square and has [**orthonormal**](https://en.wikipedia.org/wiki/Orthonormality) columns and rows.)"
]
},
{
"cell_type": "markdown",
"id": "89dc5f64-f799-4508-b57e-c73b745afc3d",
"metadata": {},
"source": [
"## Reflection\n",
"\n",
"Using a construct of the form $I - \\alpha v v^T$, create a reflection across the plane defined by the unit normal vector $v = \\left[ \\cos \\left( \\theta \\right), \\sin \\left( \\theta \\right) \\right]$.\n",
"\n",
"Try different values of $\\alpha$ and convince yourself of why it works or sketch a diagram for the projections $v v^T$ and $I - v v^T$ and decide how to modify it to make a reflection."
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "f90160ca-c8ba-42c6-ba58-ce3f405c0590",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function reflect(θ)\n",
" v = [cos(θ), sin(θ)]\n",
" A = I - 0 * v * v' # Replace 0 with the correct constant\n",
"end\n",
"\n",
"tplot(X, reflect(π / 2) * X)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "75b16123-9dc6-4737-800a-42aafdff76fd",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Test\n",
"@test mapslices(norm, X; dims=1) ≈ mapslices(norm, reflect(0.3) * X; dims=1)\n",
"@test X ≈ reflect(0.5) * reflect(0.5) * X"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "fd8e5a1d-e4da-44a3-8388-54d3647fc39a",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# It may help to think in terms of the projections\n",
"function reflect(θ)\n",
" v = [cos(θ), sin(θ)]\n",
" A = (I - v * v') + (v * v') # Multiply one 'direction' by -1 to flip\n",
"end\n",
"\n",
"tplot(X, reflect(π / 2) * X)"
]
},
{
"cell_type": "markdown",
"id": "8b1cffa1-bcb8-4f27-a1b4-1da38532a2a5",
"metadata": {},
"source": [
"* Where is the vector $v$ on the figure above?\n",
"\n",
"* Does the reflection matrix change if you replace $v \\left( \\theta \\right)$ with $v \\left( \\theta + \\pi \\right)$?\n",
"Why or why not?\n",
"\n",
"* What happens if you reflect twice?\n",
"\n",
"* What is the **nullspace** and **range** of the reflection matrix?\n",
"\n",
"* Is the reflection matrix **orthogonal**?"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.11.6",
"language": "julia",
"name": "julia-1.11"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.11.6"
}
},
"nbformat": 4,
"nbformat_minor": 5
}