{
"cells": [
{
"cell_type": "markdown",
"id": "60ec1462-2648-4741-82d0-416ed9110998",
"metadata": {},
"source": [
"# 2025-09-17 Linear Algebra\n",
"\n",
"* Algebra of linear transformations\n",
"\n",
"* Polynomial evaluation and fitting\n",
"\n",
"* Orthogonality"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "6c1a2094-4530-4c30-b496-7f7bffae049c",
"metadata": {},
"outputs": [],
"source": [
"using Plots\n",
"default(lw=4, ms=5, legendfontsize=12, xtickfontsize=12, ytickfontsize=12)"
]
},
{
"cell_type": "markdown",
"id": "6bb5811e-f493-4616-a0bf-7bda080eb794",
"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": "28613d67-a16d-4593-8dda-376acc3dd17d",
"metadata": {},
"source": [
"## Math and Julia notation\n",
"\n",
"The notation $A_{i, j}$ corresponds to the Julia syntax `A[i,j]` and the colon `:` means the entire range (row or column).\n",
"So $A_{:, j}$ is the $j$th column and $A_{i, :}$ is the $i$th row.\n",
"The corresponding Julia syntax is `A[:,j]` and `A[i,:]`.\n",
"\n",
"
\n",
"\n",
"Julia has syntax for row vectors, column vectors, and arrays."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "b08291cd-577c-4762-b9b9-1c918e85aa86",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×3 Matrix{Float64}:\n",
" 1.0 2.0 3.0\n",
" 4.0 5.0 6.0"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"[1. 2 3; 4 5 6] # note - using . sets every value as a float\n",
"# Equivalent to Python's np.array([[1, 2, 3], [4, 5, 6]])"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "1eed26da-a552-4536-9771-c49757854eb9",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×2 Matrix{Int64}:\n",
" 1 2\n",
" 4 3"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"[1 2; 4 3] # Pay attention to the ordering"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "a57820e1-b40f-401c-8a92-561f27edadb8",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3×2 Matrix{Int64}:\n",
" 1 0\n",
" 0 2\n",
" 10 3"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"[1 0; 0 2; 10 3] # ; separate rows"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "d203c0b9-485b-4448-8d49-4c2ce3b8add2",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1×3 adjoint(::Vector{Complex{Int64}}) with eltype Complex{Int64}:\n",
" 1+0im 2-1im 3+0im"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"[1; 2 + 1im; 3]' # transpose"
]
},
{
"cell_type": "markdown",
"id": "004938d8-1455-4ce2-be7b-b28c3e687a3b",
"metadata": {},
"source": [
"## Implementing multiplication by row"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "4c535825-79b6-4c91-92e7-f063be96af2c",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A = reshape(1.0:12, 3, 4) = [1.0 4.0 7.0 10.0; 2.0 5.0 8.0 11.0; 3.0 6.0 9.0 12.0]\n",
"x = [10.0, 0, 0, 0] = [10.0, 0.0, 0.0, 0.0]\n",
"matmult1(A, x) = [10.0, 20.0, 30.0]\n"
]
}
],
"source": [
"# We can impl A * x by hand to check our understanding of the syntax\n",
"function matmult1(A, x)\n",
" m, n = size(A)\n",
" y = zeros(m)\n",
" for i in 1:m\n",
" for j in 1:n\n",
" y[i] += A[i,j] * x[j]\n",
" end\n",
" end\n",
" y\n",
"end\n",
"\n",
"@show A = reshape(1.:12, 3, 4) # 3x4 matrix\n",
"@show x = [10., 0, 0, 0]\n",
"@show matmult1(A, x);"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "53aaf7a9-8a4b-4fd5-ade8-29f254202742",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"size((A[2, :])') = (1, 4)\n",
"size(x) = (4,)\n",
"(A[2, :])' * x = 20.0\n"
]
}
],
"source": [
"# Julia has syntax for dot product\n",
"@show size(A[2, :]')\n",
"@show size(x)\n",
"@show A[2, :]' * x;\n",
"# What would happen if the multiplication was in the other order?"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "ca672af9-a36f-4c47-b804-de1bd5f7206e",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"matmult2(A, x) = [10.0, 20.0, 30.0]\n"
]
}
],
"source": [
"# Let's simplify with the dot product syntax\n",
"function matmult2(A, x)\n",
" m, n = size(A)\n",
" y = zeros(m)\n",
" for i in 1:m\n",
" y[i] = A[i,:]' * x\n",
" end\n",
" y\n",
"end\n",
"\n",
"@show matmult2(A, x);"
]
},
{
"cell_type": "markdown",
"id": "c5cd7dec-6d5a-49d2-a4b9-3702f4018804",
"metadata": {},
"source": [
"## Implementing multiplication by column"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "1f135474-6dff-4aad-a494-d582a7babfaa",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3-element Vector{Float64}:\n",
" 10.0\n",
" 20.0\n",
" 30.0"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# One more time, but by the columns\n",
"function matmult3(A, x)\n",
" m, n = size(A)\n",
" y = zeros(m)\n",
" for j in 1:n\n",
" y += A[:, j] * x[j]\n",
" end\n",
" y\n",
"end\n",
"\n",
"matmult3(A, x)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "74df1e68-703e-409d-a12e-35c4635945b7",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3-element Vector{Float64}:\n",
" 10.0\n",
" 20.0\n",
" 30.0"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Naturally though, Julia has this built in\n",
"A * x"
]
},
{
"cell_type": "markdown",
"id": "aea4c19f-1108-45c9-830a-f3e23a1449a8",
"metadata": {},
"source": [
"## Polynomial evaluation is (continuous) linear algebra\n",
"\n",
"We can evaluate polynomials using matrix-vector multiplication.\n",
"\n",
"For example,\n",
"\n",
"$$ -3 x + 5 x^3 = \\left[ 1 \\vert x \\vert x^2 \\vert x^3 \\right] \\begin{bmatrix} 0\\\\ -3\\\\ 0\\\\ 5\\\\ \\end{bmatrix} $$"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "f399d870-2281-4e34-aa4b-d6288b89fd1c",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"f = Polynomial(1 - x + 3*x^2 + 9*x^3)\n",
"P(p + q) = Polynomial(1 - x + 3*x^2 + 9*x^3)\n"
]
},
{
"data": {
"text/plain": [
"3-element Vector{Float64}:\n",
" 1.0\n",
" 12.0\n",
" 83.0"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Polynomials\n",
"P(x) = Polynomial(x)\n",
"\n",
"# Let's manipulate some polynomial coefficients\n",
"p = [0, -3, 0, 5]\n",
"q = [1, 2, 3, 4]\n",
"f = P(p) + P(q)\n",
"\n",
"# Either representation should be the same\n",
"@show f\n",
"@show P(p+q)\n",
"\n",
"# And lets evaluatethis polynomial at some list of points\n",
"x = [0., 1, 2]\n",
"f.(x)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "87d35bba-47ea-45c9-909b-c61aa6c2121d",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAlgAAAGQCAIAAAD9V4nPAAAABmJLR0QA/wD/AP+gvaeTAAAgAElEQVR4nO3dd3zM9x8H8Pf3+73snciQiCBiEztqb43RqtWW2rRGW3vV6tKW0qGUBkVrU2r1F3vVliK2SAgigkT2uvt+P78/Lu4uU8Ttez0f/ePue5/c93P51r3ymV+OMUYAAACWijd0BQAAAAwJQQgAABYNQQgAABYNQQgAABYNQQgAABYNQQgAABYNQQgAABYNQQgAABYNQQgAABYNQQgAABbNNIIwKirq119/NXQtQE2hUBi6CqCGy2FUcDmMSmkuh2kE4c2bN//55x9D1wLyMMZycnIMXQvIg8thVHA5jIokSaW5HKYRhAAAADqCIAQAAIuGIAQAAIuGIAQAAIuGIAQAAIuGIAQAAIuGIAQAAPOUI1Ku9PJiCEIAADBDx+KZ3ybxZgr30pIIQgAAMDcSozEnxeel29sAQQgAAOZmZ6x0PZmVsjCCEAAAzM38yFKMDb6AIAQAALNyJJ6dfVLa5iAhCAEAwMx8e0l8pfIIQgAAMB8XnrEDcermoIPs5T+CIAQAAPPx7SX16GCb8lyg08v7SBGEAABgJq4ns79j1UE4vV6pMg5BCAAAZuLbS5L0ogXYqBzX2a9UP4UgBAAAc3AnlW2KUTcHZ9YvbcAhCAEAwBx8e0lSvMjBWq7c2wEIQgAAsBgPMti6O+rm4OwGPP/yTUbzIAgBAMDkfXtJUt1ooqoz17fKK6RbKVZYvKKLFy/u37+/8HE7O7tPP/1U+XjPnj3Xrl0rUKB///7+/v5arw8AAJi3+ExaE6VuDs5qwAulbg6SLoLwzJkz06dPL3y8du3aqiDcsmXLn3/+WaBAs2bNEIQAAPCqFkSKWYq8x1WcuAGBr9bZqf2u0dGjR7P8/vjjDyIaNGiQZjFbW9sCxdq0aaP1ygAAgHl7lk0rb2msHQzmZa+YbPoYI1y1apVMJhs4cKAezgUAABZl4RUxXZ732N+BG1ztlXNN50EYExNz/Pjxbt26lS9fvvCr2dnZuq4AAACYq6fZtPS6ujk4NZi3fvVY03kQrlq1ijE2bNiwAsdzc3M9PT3t7OwcHBy6det24cIFXdcEAADMzMJIdXOwvD0Nf/XmIOlisowmURT//PNPb2/v0NBQzeO2trZvv/12cHCwtbX15cuXt23bdvDgwf/973/t27cv8n1SU1Ojo6O/+eYb1ZHevXtXqVJFp5WH4jDG5HK5XC5/eVHQPVwOo4LLoU/PsmnZDfXTKXVIxvL97iVJYuzlm27rNgjDw8MfPHgwbdo0KysrzePLly/neXVuHzp0qEuXLh9//PH169eLfB+FQiGXy58/f646kpWVJUmvcANi0CLGmCRJ+P0bCVwOo4LLoU/fX+HS5HnrJHzs2NBAVuAXX8oLodsgXLVqFRENHjy4wHHNFCSiDh06dO7c+X//+9+DBw+KXEHh7u5eo0aN77//XndVhdJjjImiaGNjY+iKABEuh5HB5dCbZ9m0Ikrd+vusvszVoWC/qCRJWVlZL30rHY4RPnnyZM+ePS1btqxZs+ZLCyvzLzk5WXf1AQAAs/Hd5XyjgyOqlz3OdBiEf/75p1wuLzxNpjDG2IULF3ie9/Mr3T0zAADAgj3Npt9uqrs9P6sv2L1G/6YOg/D33393dHTs06dPgeOPHz8ODw9Xdd3m5OTMmDHjv//+e/PNN93d3XVXHwAAMA8FmoNlmyyqoqsxwtOnT1+/fn3EiBFOTk4FXkpISAgNDfXw8AgMDLSxsblx48azZ8+qV68eFhamo8oAAIDZiM+k5Tc07zv4Ws1B0l0QpqWlTZs2rcjdZKpWrbpmzZqTJ08+evQoJyenS5cubdu2HTBggJ2dnY4qAwAAZmPeJTHzxc6i/g7c64wOKukqCDt37ty5c+ciX3JwcBg8eHDhqaQAAAAli01nmjuLzmzA2wiv+564HyEAAJiML/6TcsS8x5WcuKGvNzqohCAEAADTEJXC/tS4Df2Xjcqys2hhCEIAADANsyMkxYscrO7Cvf8qt6EvAYIQAABMwJUktvWuujk4r/Er33ewOAhCAAAwATMvSNKLDbQbluN6VdZafiEIAQDA2J15wnbfVzcHv2okcNp7cwQhAAAYu+nnRdXj5t5cV38t5iCCEAAAjNs/D9ixePVtBb9p/NorB/NDEAIAgPGSGM28oG4OdvXn2pTXZnOQEIQAAGDM1kdLlxLzmoM8R19ruzlICEIAADBauRJ9HqGeIzMgkG/goeXmICEIAQDAaP16XYpJy2sOWvP0eSOdZBaCEAAAjFG6nL69rB4dHF2Lr+Kk/eYgIQgBAMA4zY8Un2TlPXayopn1tT86qIQgBAAAoxOXwX64oh4dnFxP8LTV1bkQhAAAYHRmR0iqu+/62NHEOjpMKwQhAAAYl8tJbG2U5u2WBEcrHZ4OQQgAAMZlyllRtb92TVft3H23BAhCAAAwIv88YAfi1BuqLQoRtHW7peIgCAEAwFiIjKadUy+ZaFeeC9Xq/tpFQhACAICxWH1buvpcvaHaD810tWRCE4IQAACMQpqcZmvsrz2wKl9fBxuqFYYgBAAAozDvkvj4xQp6exl93VhPCYUgBAAAw4tJYz9dVS+ZmFKPr+Cgj+YgIQgBAMAYTD4r5bzoFvVz4KbU08fooBKCEAAADOxoPNtxT90cnN+Ed5Dp7+wIQgAAMCSJ0eSz6jkyzby4/lX1mk0IQgAAMKSVt6SIZ3lLJjiihSGCnsYGX0AQAgCAwSTn0uwIdXNwQFW+hbeecxBBCAAAhjM3Qn3TQQcZfdvEAKmEIAQAAMO4ksR+vaGeIzM9WNDbkglNCEIAADCMT0+Lihc5GOjMTa5nmEhCEAIAgAFsjJaOxqvvMvFTM8FWf0sH80EQAgCAvmUqaPp5dadoZz+ue0UDdIoqIQgBAEDfvrwo3k/Paw5a8/RLcwM1BokIQQgAAHp2M5n9eEXdHJxcj6/mYrDmICEIAQBAnxjRmJNi7osc9HfgPqtvyOYgIQgBAECf1t+RjmjMkfmhmV63FS0SghAAAPQkJZemnlPvI9PZj+tT2fAxZPgaAACAhZh+XozPzHtsIxh4jowKghAAAPThwjO24qZ6jsys+oJh58ioIAgBAEDnREYf/SuKLwYHg1wMto9MYcZSDwAAMGO/XJP+e6aeI7OshcH2kSkMQQgAALp1P51p3mupfyDfwdcoOkWVEIQAAKBbY06K6fK8x242tKiZ0TQGiYhI+8s3rl69um7dugIHW7Ro0aNHD80jkZGRv/zyy40bN9zc3N56663hw4fzPFIZAMDcbI6R9j5Qd4ouaCr42BmwOkXQfhDevn17/vz5jo6OVlZWqoM8z2sG4cGDB7t16+bq6vrmm2/GxMR8+OGHBw4c2Lx5M8cZUWMZAABeU0ouTTyjnina2ocbXt3o2jy6WtC/ZcuW0NDQIl/KyckZPny4h4fHxYsXvb29iWjy5MmLFi3q3bv3u+++q6P6AACA/k06Kz7KzGsO2gi0vKVghM0dAyRzeHj4/fv3R44cqUxBIpoxY4ZMJgsLC9N/ZQAAQEeOxLPfb6mbgzPrCzVdjTAHdRaEoijevXs3NjZWkqQCL/37779EpNle9PDwaNq06alTp0RRJAAAMH0ZChpxXLVukGq7cdOCja5TVElX1XrnnXeqVKlSqVIlb2/vL774QqFQqF6Kjo4mIn9/f83y/v7+2dnZcXFxOqoPAADo08wLYkxaXg7yHK1oJVgbaQ7qYIzQzs6uT58+bdq08fLyun///ooVKz7//POoqCjVVNLU1FQicnd31/wpDw8PIkpJSSnyPZ89e3blypURI0YonwqCMGLEiLp162q98lAajLHs7GxBMK4J0BYLl8Oo4HIonXvGLbmmzr2x1aUGzjnZ2fquhiRJhXslC9N+EIaGhmp2e37yySfNmzdfv379hAkTGjVqRETKqaEFKqfsFC1uBYWtra2zs7Pyx5U8PDzwv5qhMMYEQcDv30jgchgVXA4iyhFp1GlS9YpWcqQvGvIG+ZVwHGeYICzAxsbm008/HTJkyLFjx5RJ5urqSkSJiYkODg6qYomJiaqXCnN0dAwICBg9erSuawulwRizsrLSXB4DBoTLYVRwOYho9iXxRkpe/HBEv7eRudkbZo6MJEmaA3PF0UeXbbly5YgoIyND+bRatWpEFBMTo1kmOjra0dHR19dXD/UBAAAdOfuELYxUN8I+rMG3K2+MM0U16SMIjx49SkSBgYHKp+3btyei3bt3qwo8fPjw0qVL7du3x4J6AADTlaWgwcfUt5jwd+AWhJhAL7H2g/CXX35RtfZEUQwLC/vpp5+8vLxUO8u0a9euTp06K1euvHr1KhEpFIrJkyczxj7++GOtVwYAAPTmswvirZS8GOSIwloJzqbQSaz9McKffvrp008/dXZ29vLyevjwYXZ2tre3919//eXk5KQswPP8+vXrO3bs2LBhw8aNG8fGxj569GjatGmdOnXSemUAAEA/TiawX66pO0U/qsm/WcE0Ovm0H4Q7d+48evTonTt3nj9/7uTk1KhRoz59+qhSUKlevXpXr15du3bt9evXmzVr9vbbb7dp00brNQEAAP3IUNAQjU7RSk7cgqYm0CmqpP0grFOnTp06dV5azMvLa8qUKVo/OwAA6N+kM+KdVPXy+TWtBSdT6BRVMtaF/gAAYCL23GdhN9Wdop/U5tsY/UxRTQhCAAAou6fZNOKEQrWnaHUX7pvGJtMpqoQgBACAsht9UkzIynss4+mPtoK9zndq0TIEIQAAlNGKm9Jfd9WdonMbCE09TalTVAlBCAAAZXEnlU08q7533hte3HRjvdFSyUyy0gAAYFi5Er1/WEyX5z11tKI/2woy04wU06w1AAAY1KwL4oVnqiky9GMzIdDZ9DpFlRCEAADwavbHsUVX1EODvSrxI6qbcJqYcNUBAED/nmbTkGMK6UVrsIIDt6KVia2XKABBCAAApcWIhh1XxGfmPeU5+rOt4G5j0Dq9NgQhAACU1o9XpD331UODcxsKbU1qE5kiIQgBAKBUzj9lM86r10u08uFm1jeHEDGHzwAAALqWlEN9D4m5L6bIuNvQ+naCYPKtQSIEIQAAvBQjGnxMEZuuvunuqtaCv4NZxCCCEAAAXmrB5XxDgxPr8j0DzCc+zOeTAACALpx+wmZHqIcGQ7y4b5qY9nqJAhCEAABQrCdZ1PeQKNcYGtzUTrA2r+gwr08DAADao5Do/SOKuAz1refXtZVVcjKToUEVBCEAABRtxnnx8CP10OD0YD7U39xSkBCEAABQpC0xkuaGou3Kc182MquhQRUEIQAAFHQzmY08Iaoag/4O3Kb2MvNYNVgYghAAAPJJzqUe+8XUF/catBFoW0fBy86gddIlBCEAAKhJjD44oriTqh4aXNJcaOpppo1BIkIQAgCApjkR4t4H6hT8qIZp32uwNMz84wEAQOltjJa+uaSeINPMi/v5DfOcIKMJQQgAAERE55+y4RoTZMrb07YOgo355yCCEAAAiOIzqddBMUuR99RWoO0dZX7msq12yRCEAACWLlukdw4qHmaohwZXtBKaeVlEChKCEADAwjGiocfFs0/y7SDzQVULSgcL+qgAAFDY9HPipmj1BJlQf+7rxhYwMKgBQQgAYLl+uyktiFSnYE1XbmM7s91BpjgIQgAAC/V3rDT2pPpGg952tLeL4GJtwBoZBoIQAMASnXvKPjgiqlZL2Mvo706yymZ3i6XSQBACAFicO6msx35FxovFEgJHG9tZ0DTRAhCEAACW5Wk2hYaLT7LynnJEYa2EtwIsNw4s95MDAFigTAX12J9vT+3PGwnDqll0Flj0hwcAsChyiXofVGguGRxenZ/TwNKDwNI/PwCAhWBEo/4Vwx+qU7CrP7e8hWUtGSwSghAAwCJ8HiH+flu9ZLBxOW5LB5kMIYAgBACwBD9flb68qE7Byk7cni4yB5kBa2REEIQAAGZu6XVpwhn1wvlytrTvTcHbzoA1Mi4IQgAAc7bipvTJKfVdBh2taHdnWZCLhS4ZLBKCEADAbP0RJY06qU5BOxnt6iyz2IXzxUEQAgCYp213peEnROlFDFrztK2DrF15pGBBCEIAADO04570/hFR8WJ+jDVP2zoKXf2RgkXQ/pyhlJSUXbt27d69+9atW1lZWZUrVw4NDR07dqyVlZWqzJIlS44fP17gBz///PNatWppvT4AAJYm/CHTTEGBoz/bCj0qouVTNO0H4dKlS2fOnFmxYsVGjRq5uLj8+++/EyZM2LVr1/79+2WyvNOdO3fur7/+atCggeYPZmdna70yAACW5kAce+eAIufFLFGBoz/aCv2qIAWLpf0gDA4O3rt3b2hoKMdxRCSXy/v06bNr164tW7b0799fVcza2vrChQtaPzsAgCU78Zi9c0CR/SIFeY5WtxH6ByIFS6L93063bt26du2qTEEisrKymjhxIhGdP39e6+cCAACVfQ9ZaLj65kocUVhLYWBVpOBL6GNfAWWfp4uLS4HjUVFRsbGxHh4edevWVfWaAgBAGWy7Kw04Iua+GBfkiJa2EIZXRwq+nM5/R5IkLVq0yMrKqm/fvprHs7Ozq1Wr1qlTp4YNG5YvX37ZsmW6rgkAgLladUt677A6BYnoh2bC6JpIwVLReTvs+++/P3DgwGeffVa7dm3VwTp16ixcuLBu3bp2dnYXL16cP3/+mDFjiGj06NFFvkl8fHxERESvXr1UR8aOHRsSEqLrykORGGNZWVmMsZcXBd3D5TAqBrkcy28LU/+TqU7JEX3TQDGiUk56uj5rYYwkSVKN05WA0+kF+/PPP4cMGdKxY8ddu3bZ2NgUV+zevXv169e3traOj48XhCLuCbJt27bvvvtu6tSpyqc8z7dt29bd3V1X9YYSMcYyMzMdHBwMXREgwuUwMvq/HAsi2YwL6pagwFFYS35IENYLEhFJkpSTk/PSy6HDFuGWLVuGDh3atm3bv//+u4QUJKJKlSp17tx569atMTExQUFBhQvY2Nj4+Pj069dPZ5WFV8AY43me59HrYhRwOYyKPi8HI5p4RvzpqjoFbQTa0E7oVQn/M7waXQXh33///cEHHzRv3nzXrl12di/f5FyZ2Lm5uTqqDwCAOREZjTwhrta4v6CDjHZ0knXyQ1vwlekkCMPDw997771GjRrt3bu3NF0EmZmZR44csbW1rVy5si7qAwBgTnJE6n9E3H5PnYJuNrS3i+wN7KZdJtoPwgMHDvTs2dPDw2PevHm3b99WHXdzc6tSpQoRRUdHr1279v333w8MDBQE4fr165MnT46NjR07dqy9vb3W6wMAYE6e51Cvg4qj8erpHb723P5QobYbUrCMtB+Ee/fuzcnJefToUYcOHTSP9+vXb/PmzUSUnp7+1VdfffXVVxzHWVtb5+TkENH777+/cOFCrVcGAMCc3Ell3feJt1LUKRjozB0IFSo7IQXLTvtBOHr06O7duxc+7u3trXwQHBx869ats2fP3r9/Pysry9fXt23btthuGwCgZCcT2DsHFE81dmWu5crtDxX8HJCCr0X7QVi9evXq1auXXKZatWrVqlXT+qkBAMzV6tvSqH/zLZlv5cP93UnmXtKUfCgVzLIFADBqjOjz/8Rhx/Ol4HuB/P5QpKB2YIdPAADjlS3S0OPipmh1BnJEcxrynzcsYu8RKBsEIQCAkXqUyd7eL154pp4aYyvQ762F93FbJa1CEAIAGKP/nrG3DohxGfmWSezsLDQuh6kxWoYgBAAwOn9ESaNPipkK9ZF67tzuzkJFR6Sg9iEIAQCMSKaCxpwU10ZJmgffDuDXtRUcrQxVKTOHIAQAMBa3Ulifg+LV5/luCjS5Lj+/qcCjKagzCEIAAKOw45409LiYonHrAUcrCmuJqTE6hyAEADCwHJGmnhMXX8vXHVrDldvaQaiDHUR1D0EIAGBI99PZu4fFM0/ydYd+UJVf3lJwwDe0XuDXDABgMFvvSh/9Kz7PUR+xFWjxG8LIGugO1R8EIQCAASTn0rRzYtjNfN2hAY7clg5CU090h+oVghAAQN/2PWQjTogPM/J1h74VwK9pLbhh+1C9QxACAOhPmpwmnRVX3pQ0M9BGoO+aCOPqYImEYSAIAQD05FQCG3xMvJOaryFY1537o41Q3wMhaDAIQgAAncsW6fP/xIWRkqgRgjKeJtXlv2wkWGNmjEEhCAEAdOtIPBtzUryZnK8hWMOVW9sG82KMAoIQAEBX4jNp8llxY3S+EUGeo3G1+XmNBTt8ARsHXAcAAO2TGK28JU09l2/LNCIKcORWtxHalUdD0IggCAEAtOxUAht7SryUmK8vlOdoZHX++xDBCTeRMDIIQgAArUnKodkRwm9RinydoUT1PbilzYXm3mgIGiMEIQCAFkiMVtySPjsvJuUImsddremrxsLomryAEDRWCEIAgNd1MI5NOVewL5SIulfklrUQKjggA40aghAAoOyOxrPPzounnxSMwNpu3NLmQhtMijEFCEIAgLK4+px9+Z+09a5U4Li9jKbU4z+rj2XyJgNBCADwam6nsDkR0paYAhNiSODog6r8Z7Wyq3k6GKZmUCYIQgCA0nqQwb6+KP1+W1LkbwdyRG8H8POa8DVdKDPTQJWDskIQAgC83J1U9ss1KeymlC0WfKmFN/dtE6GVD0dEjBUcLATjhyAEACjJ6Sds/mVp932pYE8oUaNy3DdNhM5+mBFj2hCEAABF+/cxmx8p7rlfRCOvugv3WX3+g6o8biFoBhCEAAD5pMlp/R1p6XXp6vOiI3BqMD+oKi/DpFBzgSAEAMhzK4Wtvi2F3ZSe5xTxagMPbnwdfkBV7BFjbhCEAGDpciXaGSuF3ZQOxRU916WFNzctmO9REW1A84QgBADLdTGRrbsjbbgjPc4q4lUZT30q81Pr8Q080AY0ZwhCALA4DzLY9rtsbZR0sdDuoEo+djS4Gj+6Jh/giAg0fwhCALAUiTm0JUZaf0c6lVB0FyhH1LY8N7oW3zOAt0I/qMVAEAKAmUvOpf89kDZGs/CHkrzgzqB5XKxpUBA/uiZf0xVNQIuDIAQA8xSTxnbHsj0PpOPxLLeY/BM4aufLDazK967MO+Dr0FLhygOA+cgR6fhjtue+tPcBi04tdrczjqiFDzcgkO9bhfew0WcFwRghCAHA5N1IZscfs/0P2f44KV1eUslartyAqnz/QK6SE7pAIQ+CEABMj8ToynN2LJ4df8xOPJaeFLX4QVMdN657Re7dKnx9LISAQhCEAGAa0uR0MZGde8qOxUv/PmbJuS8pbytQ2/Jc94p8N3+0/6AkCEIAMFJpcrqcyCKe5f13M4UVvv9DYV521MWP7xHAdanAO1vpvpZg+hCEAGAUMhV0K4XdSmY3U9iNZPrvGYtOLe3N/ZytqJUP18qH7+THNSjHofUHrwRBCAD6livRg3QWm063U9iNZHYzmd1Kofvpr3ZP23K21NKbb1Oea+3DBXtw2AgbysxgQbh69eqFCxfeunXLycmpZ8+eCxYs8PT0NFRlAEDr0uSUmM2e5VB8JruXRvfT2f2MvPyLzyzLfdwFjmq4cg09uGZeXJvyXC03tPxAOwwThIsXLx43blyrVq0WL158//79n376KSIi4syZM/b29gapD1gskVFqLqXKWbqcskXKFilLQUT0PJcRUaaCckTtnMjFmpR3cLUTOFuBiEjgSTWCVfhVOxkpHxib5zmUK1FSDkvMpsQclphDSTn0LJs9y6akHErMVh5hidlU3Br20pPxVMuVa1iOa1SOa+jBBXtwWPMOumCA/62SkpJmzZoVHBx86NAhKysrIqpaterIkSOXLVs2adIk/dcHzM+zbErMYc+y81okz7LpaRZ7lkOJ2fQ8l6XL84WfkbMVyE5GRGTDk72MIyIbgexlRETWPDlYERHxTOZiU/CT2MvIpqgodZARI8pUEBFlKChXJCJKyWUSkcQoJZeIKFeiDDkRUZbIlL+iNDkpXjvYSiZwVNmJq+FKNVy46q5cXTeunjtnh+QD3TPA/2W7d+9OS0sbMmSIMgWJaMCAAePHj1+/fj2CEEqJET3OpIcZ7GEGe5BBD9LZw0x6kM7up1N8FtP1V7Y+KdupLxTXocgTmdJnFjgqb88FOFJlJ66mK1fdhaq7ctVcOGvscw2GYIAgPHfuHBF16NBBdcTOzu6NN944cuRIdna2ra2t/qsExkxi9CCDRaVQVCqLSmG3UlhUKsWmFbt7JBgJexm523AeNlTenio6chUduYqOVMmR83cgPwcO93YA42GAILx//z4ReXl5aR708fERRTEuLi4wMFD/VQKj8iiTXUmiy0ksMoldSWK3U5juOjB5jlysycmKc5SRnUzd2ehqzXEa3ZKvLzmXlPNDMhUsRyIikkuk2gxM9WqGIi/gtTg8qV0u1iTjyMOW87AhD9u8qFM9LWfLuduQhw152KBXE0yGAf5XzcjIICI3NzfNg8qn6enpRf7I/fv3T5w40bBhQ9WRL774ol27drqsJhSLMZaVlcXKMu+vaLEZ3Jmn3KXn/NVk7koyl5jzupMBXaxYOVvysGbuNszDhjxsqJwNlbNl7tbM3ZrsZczFmhxl5CAw4/+yzlJQjsQRUZbIlA9UM3pyJS5TQYyxtOxcSVZw6+hMBVdkozldThxHylkndgJTjiM6ykjGERG5WjMikvHkZEVEZMMzO4EjIgcZe4U2HCMxm4r+x2zutP6vA16HJElcKSYXG+BrwNramoiysrKUD5SU6WhjU/Q+8BUqVKhXr95PP/2kOlKrVi07Ozsd1xSKxhjjed7BwaHM75AtUsQzdvoJO5XATidIj1+2UWSR3G3Iz4ELcKQKDpyfPVfRkfwdOT97qujIGed8y7JxfFkBxlhmJjk4YMa1UXj9fx2gRZIkZWW9/PvFAEFYrlw5IkpMTHRxcVEdTEpKUr1UGM/zLi4ujRo10k8NQQh/F4UAACAASURBVBeeZNHReOn0E3b6Cbv47NVG+DxsKMiFq+bCVXPhgpwpyIULcuYcsXsWAGiDAYKwdu3aRBQZGVmlShXVwcjISG9v7+KCEEyUXKJTCWzfQ2lfHLv4rLS9RQ4yqu3GBXtw9dy5eu5cHTfOHXeMAwCdMUAQdu3a9bPPPtu+fXvPnj2VRy5evBgTEzNs2DD9VwZ0ISaN7XvI9j1khx9JaSXeHE7JQUZNPLk3vLiG5bhgdy7QmeOxZQgA6IsBgjA4OLhbt24bNmxo167dwIEDY2NjR4wYYW1tPXnyZP1XBrRFIdGReLYrVtoXx6JSXt72q+zENffmmnlxzb24eu6cDJPpAcBADDNnbs2aNW+//fawYcOUrUBHR8f169fXrFnTIJWB1yEyOhrPtsRI2+9Jz7JLKslz1Lgc19qHa+HDNfPifTDVCQCMg2GCsFy5cidOnDh9+vStW7ecnZ07dOhQYDUFGDmR0eFHefn3tMT8K29PXSrwXfy4jn58OWyWAADGx2CrqHieb9GiRYsWLQxVASibM0/YH1HStrtWT7MVxZWxEailN9elAt+5AlfPHbcIAACjZvTLicE4JObQuihp5S3p6nPl+F8R6eZhQz0r8e9U4tuWx10CAMBk4OsKSsKIDj9iq25J2+9Jxe345W5DPQP4vlX4Dr7YQBIATA+CEIr2KJOtuc1W3ZJi0oqeAuqmzL/KfEc/5B8AmDAEIRQUmcQWRkqbYiR5UZu/yHjq5s8PCMh5u6o9bpoDAGYAQQhqB+LYwkjxQFzRW8BUdeaGV+cHB/E+diwzkyEFAcA8IAiBJEZ7H0hfX5TOPS0iAW0Eeqsi/2ENvoNf3vxPbKwPAOYEQWjR0uS04qb001XpQUYR4VbDlRtTk/+gKu+GrT4BwHwhCC3U8xxaeEVcel1KyS3i1VY+3OS6fPeKPPb8BACzhyC0OJkK+uWaND9SfJ5T8CWeo67+3Ixgobk3AhAALAWC0ILIJVp5S/rqohifWfAlexkNrcZPqMMHOiMCAcCyIAgtgsRoY7Q09z8pOrXgWKCXHX1cSxhdExuBAoCFQhCav4NxbOo58WJiwQh0sqIxtfiZ9QUn3OodACwYgtCcnXnCJp0VTyUUjEB7GY2rzU+pJ2A6KAAAgtA8JeXQF/+JS65LUv4QtOJpaDV+bkPe1x5jgQAARAhC8yMyWn5Dmh1RcFIoz1H/QP6LRnwVJ0QgAIAagtCsnEpgY0+JlwoNB/aoyM9rzNd1RwQCABSEIDQTCVk07Zz4R1SBrlCq48b90lxoWx4RCABQNAShyWNEK29KU8+Jyfn3iHG2os8bCZ/U4mXYHRsAoHgIQtMWm85GnhAPxOVrB3JEA6ryC5oK5e0NVS8AAJOBIDRVjGjFTWnyWTFNnu94DVfulzeEjn7oCwUAKBUEoUm6l8ZGnBAPPcrXEHS0oq8bCWPRFwoA8CoQhCamuIZgKx9uVSshyAUNQQCAV4MgNCUPM9igo+KR+HwNQScrWtBU+Kgm7pgEAFAWCEKTcSCODTyqSMjKd7C1D7eqtVAVt4wAACgrBKEJyBFp+nnx56v51gg6W9HCEGFEDTQEAQBeC4LQ2EWlsPePiBHP8nWHdvLjVrUW/B0QggAArwtBaNS23ZVGnsi3Ul7G08z6/JwGAlqCAABagSA0UlkKmn5eXHxN0jxY0ZHb0E5o4Y0MBADQGgShMbr2nPU5JN5Mztcd2qcyv6KV4GptqEoBAJgnBKHR+ecB639EkaLRHWor0HdNhHF1sE4eAED7EIRGhBF9fVH6/D9Rc3poLVducwehjhu6QwEAdAJBaCzS5TTkuPjX3XyDgsOq8b80F+xxlQAAdAZfsUbhQQZ750C+NRIynr5uJEwLRncoAIBuIQgN78Rj1ueQ4onGljHlbGlze1l7X3SHAgDoHILQwMJuSh+fEuUaHaLB7tzfnYRKTkhBAAB9QBAajEKi0SfFlbfyDQq+W4X/vTUGBQEA9AffuIaRJqd+hxThD9WDgjxHXzcWpgdjxxgAAL1CEBpAfCZ136/4T2NqjJMV/dlWeDsAU2MAAPQNQahv156zrvvE++nqFKzixO3tItRwRVMQAMAAEIR6dTSevXNAobmJdhNPbndnmbed4eoEAGDZEIT6s/u+9O5hMUuhPtKlAre1g8zJynB1AgCweBiU0pO1UVKvg/lScHRNfm8XpCAAgIEhCPXhp6vS0GOi4sVCCY5oflPh1xaCgGFBAABDQ9eozs2/LE0/L6qeChz91lIYXh1/ggAAGAUdBmF6enpGRoa3t7fuTmHkGNEnp8Sl19VL5m0F2the6IllEgAARkP738jXrl0bNWpUhQoVnJycfHx8nJyc+vXr9+DBA80yo0aNci/k5MmTWq+MAUmMPvo3Xwo6WdHeLjKkIACAUdF+i3Dnzp1r1qzp1atXgwYN3N3dDx8+vHHjxnPnzl2+fNnFxUVZJjMzMzU1dfjw4Zo/6OnpqfXKGIrEaMQJcfVtdQq62dDeLrI3vDAqCABgXLQfhN26dRs+fLiqR3T48OF+fn7ff//9H3/88cknn6iKWVlZ/fbbb1o/uzEQGQ05Jq67o07B8vZ0sKusFpbMAwAYH+130wUHBxcYF+zVqxcR3blzR+vnMkKFU9DHDikIAGC89DFrNDY2loj8/f01D0qStHr16ri4OFdX13bt2tWuXVsPNdE1hUT9j4hbNe4yX9GRO9RVqOqMFAQAMFI6D8L09PS5c+e6u7sPHDhQ83hubu6wYcNUTz/44INVq1ZZW1vruj66IzEacjxfClZy4g53FSrjzoIAAEastEF4+vTp7OzsEgo0aNDA1dW1wEHG2KhRo27durV27VrN/tJ33nln1KhR9evXl8lkV65cmTJlyrp168qVK/fjjz8W+eb37t3bv3+/m5tbXqVlsuXLl3fp0qWUldcDRjTuvGx9tKA6UtGB7Wmb68mx9HQD1ksnGGNZWVmMsZcXBd3D5TAquBxGRZIkjnt5U4Qr5QULCAi4f/9+CQWOHDnStm3bAgcnT568aNGi2bNnf/nllyX8bHp6eu3atZ8+fZqUlGRra1u4wO7du5csWbJp0ybVERcXF543lnUIjGjsSXHZDXVbsIoTd6y7UMHBPNuCjLHMzEwHBwdDVwSIcDmMDC6HUZEkKSsr66WXo7Qtwh07duTk5JRQoPAg35QpUxYtWjRx4sSSU5CIHB0d27dvv2bNmpiYmFq1ahVZxsrKStUiNDbTz+VLwQoO3MGuZpuCAABmprRB2LBhw1d631mzZi1cuPCTTz5ZuHBhacpnZmYSkUxmelu+zbwgLohUp6CvPXekG8YFAQBMhk56F+fOnTtv3rxhw4b9/PPPpemfffDgwf79+8uVKxcYGKiL+ujO0uvSN5fUKehpSwcwRxQAwKRovwUWFhb25Zdf+vv7N2nSZMWKFarjVapU6dixIxFduXJl/Pjx77//flBQkLW1dWRk5DfffJOcnLx48WJBEIp/Y6OzMVr69LR6N21Xawp/E+sFAQBMjPaD8Pr160T04MGD0aNHax7v16+fMghtbW2vXr06cuRI1Uuenp6LFy/W3HfG+O17yAYfE6UXM42crOhgV1nDckhBAAATo/0g/Oabb+bOnVv4uGqNYFBQUEJCwu3bt+Pj47Oysry9vevWrWtao4Pnn7I+hxTyF32i1jxt6yhrhBQEADBB2o8fe3t7e3v7lxarVq1atWrVtH52PbiTynrsV6TL857yHP3ZVujshxQEADBJxrIUz1Q8ymSd/icmZKmP/BAi9KuCXyMAgKnCN/grSMmlbvvEe2nqLQhmN+DH1cHvEADAhOFLvLSyFNRjv+JSojoFR9bgv2xkStNcAQCgMARhqTCi4SfEE4/VKfhWAP9rc6QgAIDJQxCWytwIcWO0euF8ax9uc3tBhl8eAIDpw3f5y62/I319UZ2Cddy4nZ1ltmgNAgCYBQThS5x/ykb+K6q6RMvZ0t+dBFcTvm0iAADkgyAsSWw667FfkaXIe2or0M5OskBsJQoAYEYQhMVKk9Nb+9VLBjmiFa2E5t5IQQAAs4IgLJrIaMARMTIp35LBD6ri1wUAYG7wzV60iWfE3ffVE2R6V+Y/x5JBAABzhCAswvIb0uJr6hRs6sn92UZAlygAgFlCEBZ0+gkbp3GXQV97bntHwc6U7o0BAACvAEGYz+Ms6nNQzH3RGnS0or1dBD8HtAYBAMwWglBNLtG7hxSPMvMmyHBEf7YV6nsgBQEAzBmCUG3qOfG4xm6in9Xnewbg9wMAYObwRZ9nU7T001X1BJkOvtwXmCYKAGABEIRERDeT2Yf/qifIVHTkNrWXYZ4oAIAlQBBSmpx6HRTT5HlPbQXa1kEoZ2vQOgEAgL5YehAyoqHHxRvJ6qHBJc2FJp5oDAIAWApLD8JvL0l/3VUPDY6swQ+vbum/EwAAi2LRX/qHHrE5Eeqhwfoe3M/NMEEGAMCyWG4QPspk7x9WqO40qLzRIHaQAQCwNBYahBKjQUfFp9l5TwWONrSTBThiaBAAwOJYaBDOj5QOPVJPkPmikdDJDykIAGCJLDEIzz9lczWGBtuU56YHW+LvAQAAyAKDMF1OA46K8hcTRT1taUM7AWvnAQAslsVNDvnoXzEqRb2t9spWgq89YhBMw4QJE9auXWvoWgAYhq+v79WrV3XxzpYVhL/fljZEq1cNjqvDv4VttcF0PHr0aNGiRT179jR0RQD0LTMzs3r16jp6cwsKwjupbLzGHXfrunPfNsGqQTAxDg4Obm5uhq4FgL5ZW1vr7s0tpT2UI1K/Q+oNRR1ktKW9YIscBACweJYShNPPixcT1eslFjcXarhiaBAAACwjCMMfsp817jXYpzI/rJpFfHAAAHgp88+DJ1k0+JhC1Ris7MStbIUuUQAAyGP+QTj6pPgkK++xjKcN7QQXHY65AgCAiTHzIFx/R9p+T90p+nlDoZkXhgYBAEDNnIPwUSb7VGO9RDMvbKUGAAAFmXMwjDwhJuXkPbYVaFVrbKUGAAAFmW0Qrrwl/fNAvV7i2yZCLayXAACAQswzCB9msCln1Z2iLby5T2ub5ycFAIDXZIZbrDGiESfE5Ny8pw4yWtNG4NEaBNCBu3fvRkREREdHC4IwefJkQ1fnlSUnJ4eHh58/fz4+Pl6hUAQEBHTp0qVjx46FS3744YczZ84MCAgo24lycnKWLVuWk5MzbNgwT0/PImuye/fu8+fPx8XF+fj41K9fv3///g4ODmU7XSnt3bt3x44dSUlJ5cuXHzBgQPPmzQuXmT17dvfu3UNCQsp2CsZYWFhYcnJy7969q1atWrhATEzMgQMHIiMj4+PjHR0dg4KChgwZ4u/vX7bTlREzBbt27erWrVspC/9yTaQVuar/ll0XdVo3CyRJUnp6uqFrYYn69eu3efNmQ9dCrV69eqpvEmtra0NXpyw2b95MRG5ubnXr1q1bt66dnR0RDRs2TFUgLS1N+aBu3bpXr15ljOXm5mZnZ7/SWXJycnr06EFEHMcFBwcnJiYWKJCUlKTcS9PZ2bl+/fre3t5EFBQU9ODBg9f7fMWSJGnEiBFEVKFChS5duvj6+nIct3DhQlUB1Qfv0aPHnj17lD+SkZHxqmcZPXq08oP7+/vHxMQULvPRRx8Rka+vb8OGDatXr85xnLW19erVqwsUS09Pd3BweKWzM8ZEUSzNl5W5dRjeTWMzzqs7RTv4ch/VNLfPCGAkWrdu/e233+7fv79JkyaGrovahx9+uG7dulIWDgkJuXTpUmJiYmRkZGRkZGxsbIsWLX7//ff9+/cT0fPnz2vUqLFo0SK5PG+f4mPHjjVo0OCvv/4qfX1EURw8ePCePXvmzp37999/3759u0OHDklJSZplGGM9e/Y8ffp0SkrKxYsX4+Li5syZExUVNX78+NKfaObMmT/88EMpC2/atGnlypW9e/e+c+dOeHj4nTt3unbtOnXq1MuXLxORQqFo0qTJjBkzMjIylOUvX77cqlWrn376qfT1IaJp06YtW7bsk08+OXnyZHZ2dtu2be/du1egzIcffnj//v24uLiIiIibN2/+999/zs7OY8eOTU1NfaVzvZZXDViDKGWLUJRY691yVVvQZW1ubJqkh+pZGrQIDcXYWoQqrVu31laL8Pjx44cOHZIk9b9cURQPHjx44sSJUr6Dl5fXpEmTylyB7du3E9F3332nfBoXF/fhhx8qW2nt27dv1qzZsWPHGGOSJB0+fFj5WCUnJ+fAgQNnz55VHVEoFP3795fJZKtWrVIeOXPmTLly5UJCQlJSUkqohiRJFSpUcHZ2Ln3NmzZt2rt371IWVrZQ7927pzpy/fp1ZSwpnz5//nzatGk1atSoWrVqmzZt6tatu2PHDuVLJ0+ePHjwoEKh0Kzt4cOHjx49qnmK6dOncxw3d+5c1fsHBARUrVo1Li6u5Lp9/PHHRKT5a2Q6bhEacozw2bNnUVFRrq6uNWrU4DgtDOL9cFU6/lg9U/SnZkJFR4wNgnl6lMnGnJT2PZSyxZcX1go7GXXz55c2F7zsdHiWyMjIjz/+eP78+VOnTlUe+fbbb2fNmrVs2bKWLVvq8MQvPH78mIgqVaqkfOrr6/vzzz+PGzcuLCzMxcUlPDy8cuXKRMRx3LFjx7744ovff/996NChysJTp079+eefN2zY0LRpUyKSJGno0KG7du3atWtXaGioskxISMjx48fffPPN0NDQ8PBwJyenIqvBcZyLi0t6erqOPmZsbKy9vb3mkGf16tVlMtmhQ4eUT11dXefNm2dvbz937ty0tLTw8PD69esrX7p9+/bQoUPnzJnzxRdfKI/8/PPPEyZMWLhwYZs2bZRHZs+evXDhwrCwMGUHLBHVrFnz9OnTXbt2bdeu3dGjR8uXL19c3RISEgRBKPNwbBlov9tw+/btXCETJkzQLJOTkzNixAgfH5/mzZvXqlWrZs2a58+ff83z3k5hsy+ovxK6V+SGYGdtMFOMqPs+cWes/lKQiLIUtO2u9M5BhU7PMnbs2Pfff3/mzJknT54kohMnTnz++ed9+vQZNWqU7k76/PnzmJiYc+fOLVmyZObMma1bt+7VqxcRSZL0xx9/1K1b183NLSgoaNCgQZ07d540aVJiYiIRzZkzp2PHjmPGjImMjCSivXv3Ll68WFl/5dsmJCT06NHj3LlzqhRUUkbC+PHjHz16VFyVzp49e/369QI/qEWOjo5ZWVlZWVmqIykpKQqF4u7du8p+4D179jRo0ODBgwctW7YcMmTIoEGDhg4d+uDBAyIaMmTI4MGDv/766wMHDhDRhQsXpk2bFhoaOnHiROVbpaam1qtX78yZM6oUVCpfvvyRI0e+/vrr+Pj4AvV59OjRnTt3Tpw4MXny5G3bts2YMUM5UKofuoqKoUOHfqehe/fumq9OmDBh1apV48ePv3Llys6dOzMyMkJDQwv/akqPEY36V1R9KbjbUFhLM5wQC6AUmcQ0byumT6cSWFSKbk+9bNmyihUrvv/++1FRUQMHDgwICFi5cqVOz7hkyZLAwMCQkJBPPvmkefPmO3futLKyIqK0tLS9e/fu2rXru+++s7W17dmz56VLl6ysrE6dOkVEPM+vW7fOzc2tf//+t27dGjx4cJ06db7//nvV25YvX75v3741a9YsfEZfX9++ffsWd8v19PT0QYMGOTk5LViwQDefmFq2bMkY27Bhg+qIcmBVkqTU1FSFQrFx48awsLAVK1a4ubm1atUqIiKiVq1a//zzj7Lw0qVLa9SoMXDgwJs3b7777rteXl5//PGHqmPP2dm5b9++jRo1KnxeV1fXvn37NmzYsMDxAQMGBAUFtW7d+scff5w5c+aXX36pk49dDF2lRd++fYv7W+bu3bthYWGhoaELFy4kojp16tjZ2XXu3HnRokXKI2Ww5rZ0JF7jdoNvCOXty/ZOAKArmn/s2tnZubq6FlnMxcVl8+bNLVq0CA4OFkXx5MmTLi4uJbztsGHDzp07p3qamJi4Zs2a8PBw1ZGvvvrqnXfeKeEdPvjggzfeeCMpKens2bPLly9v2bLlkSNHPD09lTXRLOng4PDdd9+pnnp7e69du/bNN99s0KCBTCbbtm2bct7p68jKyurZs2d0dPTWrVsrVKhQQsnZs2fv2LFD9TQmJubmzZt16tRRHRk9evTYsWOL/Nlx48atXr167NixMTExDRo0OH/+/I8//uju7p6UlMRxnEwmW79+vWZ5KyurKVOmqJ46ODhs3rw5JCSkYcOGcrn8yJEj5cqVK+MHJiKihQsXJiUlxcfH79q16+uvv46JiVm3bp1WhsxKwwDNpl27domi+N5776mOdOjQwcvLa/v27WULwmfZNPWcuoeoSwVuQFV0ioI5q+fO1ffgLhmiUfiGFxfkUsavJ19fX9Xjd955RzktpUiNGzfu1KnT3r17Bw8e3Lhx45LfNjAwMDs7W/U0Ojra09NTc2mHu7t7ye9QuXJl5chfv379mjdv3qdPn/nz5xf4OpozZ45m/VU6derUuHHjc+fOTZ48uVq1aiWf6KVyc3N79+595MiR33//veTwJqIKFSpofszHjx/b29trHimhd9HX1/fEiRMTJkz4/vvv5XK5n5/fr7/+GhYWlpGRUeDPjrFjxxbZqK1Tp0737t23bNnSp0+f1x++VTUfBw0aNHbs2F9//XXAgAFdu3Z9zbctJV0F4V9//bV7925BEBo0aNC3b1/NAWHl9NxmzZqpjvA837Rp0z179qSmpjo7O7/qucadFp+9+FdgL6NfW+B2g2DmOKI9nYXRJ6X9cVKOvoYJbQXq6s+/zr+vbdu2qR77+fmVUHLr1q179+719PTcuHHjmDFjlHNPijNz5kzNp97e3t26dStz99Jbb70lk8k0m5hKffr0KbL8r7/+eu7cOU9Pz+XLl48YMaK43s7SkMvl7777bnh4+PLlywcPHvzS8h999JFyEZ5SSEiIv7+/Zm9nyapXr/7PP//I5fLMzEwXFxe5XD5hwoT69esLQr5L3KVLlyJ/fO/evVu3bvX09NyxY8fx48dbt25dyvO+VO/evZW/VZMPwg0bNnh6esbHx8vl8pkzZ+7cuVP1v7JyUlaBdrTyaUJCQnFBqFAonj9/rnzMcZyqU2XfQ7YhOt+Nlqo4YaYomD8/B25XZ4HIlP7s6927d2mKxcTEjBw5skmTJrt3727WrFnfvn0vXrz40ladtiQkJCgUiuImcxZw5cqVyZMnv/nmm2FhYQ0bNuzbt+/Zs2fL1jsqiuKgQYN27ty5ePHiDz/8sAzvUDZWVlbKJuDWrVvT09Pffffd0vzUw4cPhwwZUrt27cOHD7dq1ap///6XLl16zd5RzTcnIkdHR628W2mUKggTEhL27dtXQgFHR0flJCsiqlev3tmzZ5WxJ5fLf/vttwkTJvTs2fP27dvKD6bsxCjwIZX5p1q8WcC9e/cOHz5cpUoV1ZGwsLA333wzS+RGnbAmyku+Oq5seEBGWlppPhOUHWMsKytLkqSXFwWtUih0O2PTGMjl8gEDBnAct3nzZm9v702bNrVq1eqjjz7aunWrLk63bdu2pk2bVqxYUfk0Pj5++PDhRPTSbkkiysjI6Nevn7Oz8+rVq318fNauXdu9e/fJkycvXbr0VauhXHS/adOm7777TrmKTg8uXLig6nb+999/x40bV6VKFc0mZnEkSRo8eHBWVtaWLVs8PT23bNkSEhIyaNCgvXv3lmFUb8WKFe+9957qL4///vtv1qxZMplMudKxgLRX/H6XJKlUVSrNmsTDhw+X/CYBAQEl/LhyTu3GjRuVT5WTaJ4+fapZZtCgQUR0+/btIt+huAX1k88oVMvn+ZW5pxOwfF4fsKDeUIxtQf2QIUMKfxvUrFnzdd5TudRK9XXBGJs3bx4RLVu2rJTv8EoL6jt37iwIQsWKFZVLuZT7nA0ZMkRztXhxBg4cyPP8wYMHC1R+w4YNpTy7yrVr14r7dk1KSirlm7zSgnrlH7K+vr4tWrRQDm1WqlTp5s2bpfnZWbNmEdHatWtVR5Q7zmju0FZ6giDY2NhUq1atefPmgYGBRGRtbf3bb78VKGb4BfXNmzdXLh8pjkxW0vu0a9fuhx9+uHPnjvKpcmwgMTFRsx2dmJjIcVwJSywLi0xiP19TN0rG1eZx93kAferbt2+NGjUKHPTw8CjzG6alpfn4+Pzxxx+ak+mmT5/u5OSUk5MjimKB4asizZgxo27duqU849KlS8PDwy9evJiQkFClSpVevXqFhoYWufd0AU+ePKldu/aWLVs6dOigOvjdd9/5+fmVYW8wLy8vzcmomkrf0Tpq1KjSz7HgOG79+vXHjh2Li4vz9/f/9NNPBw8eXJreyKysLGdn51WrVilbL0qffvqptbV1VlaWXC5XrjwpvRMnThw8eDAqKioxMbFWrVojR47s16+fcu6S/rxqwJbB2rVrSeOPhSVLlhCRasMhxphcLvfw8AgKCiruHQq3CEWJNdup3k2t4kZ5Wq6Oqg8FoUVoKMbWIgTQGxPbdFu1O63qqTL5WrVqpTzy1ltvWVlZaU5t2rdvX2JiYnGTsor08zXpzBP13PFf3uAdX+2vEAAAACJdzBoNCQkJCQlp0KCBl5dXbGxsWFjY9evX3333XdWsUX9//48//vjHH38cMWLEkCFDoqOjp0yZ4uPjU2AbthLcT2dzItRzxvtV4d8KwMJBAAAoC+0HYc2aNdetW7d8+XLlU09Pz7lz53722WeaZRYsWMDz/K+//rpq1SoiatKkyerVq4u8WWWRPjktpb9odrpY0w8hSEEAACgj7Qfh+vXrFQpFQkLCs2fP3N3dK1SoUHj2qkwmW7hw4Zdffnn37l13d/dXmiOz/Z60K1Y9R2ZBU8HPAXNkAACgjHSyoF4mk/n5+ZW8cwQR2dvb165d+5XeOUNBE86om5exbgAAB1RJREFUU7CVDzeyBpqDAABQdiaWIl9dFO+n582RseJpeUsBjUEAAHgdphSEN5PZj1fUzcEJdfharshBAAB4LaYUhJ+cFnNf5GAFB252A1PaZREAAIyTydy9Nt73jf/i1AsHf2iGhYMAAKAFJtMivFv1bdXjTn5c38omU3MAADBmJtMilPi8qlrztPgNdIqChcrIyFDdjwzAcmRmZuruzU0mCFXG1+FrYI4MWCRfX99JkyZNmjTJ0BUBMICAgAAdvTPHGHt5KUPbvXv3wFtBKa6Bvvbczb4yJ4wOGhRjLDMz08HBwdAVASJcDiODy2FUJEnKysp66eUwsZG2+U15pCAAAGiRKQVhr0r8gKqmVGEAADB+JpMrjc4t+Ksj9pEBAAAtM5kgtMt8YugqAACAGTKZIAQAANAF0wjCW7duXb161dC1gDxnzpwZP368oWsBeU6fPl36m1qDrp06dWrixImGrgXkOXnyZGkuh2kEYUpKSnZ2tqFrAXmePn366NEjQ9cC8uByGBVcDqPy7Nmz+Pj4lxYzjSAEAADQEQQhAABYNNPYYk2SJLlcHhERYeiKABFRdHR0SkoKLoeRwOUwKtHR0cnJybgcRiI6OjonJ+elxUxji7Wff/55zpw5QUFBhq4IEBFlZWU9f/7c19fX0BUBIlwOI5OVlZWcnFy+fHlDVwSIiDIzM/38/A4cOFByMdMIQgAAAB3BGCEAAFg0BCEAAFg0BCEAAFg0BCEAAFg0BCEAAFg0BCEAAFg001hQr+nRo0c7duzYt2/fnTt3eJ4PCgrq379/3759DV0vCyWK4rVr1yIiIi5cuPD06dOQkJBJkyYZulKWIiUl5euvvw4PD8/Ozq5Tp8706dNDQkIMXSkLlZiYeOHChYiIiKtXryoUigULFlSqVMnQlbJct2/f/uuvv44cORIbG+vg4FC9evXRo0e3bt262B9gpkaZebVq1erXr1///v09PT2JaPz48Yaul4W6fPmy8n8kQRCIqHfv3oaukaVIT08PDg6WyWSDBw+eOnVqQECAlZXVoUOHDF0vC6X6W1z5D+HixYuGrpFFq1mzJs/zjRo1+uCDD/r06WNvb89x3IoVK4orb3pBuHLlynPnzqmeJiUl1ahRg+O4a9euGbBWFis+Pn7JkiWnT59W7riPINSbr776iohWrVqlfPrkyRMvL6+goCCFQmHYilmmbdu2/fXXX/fu3VP2iCAIDev777+PiopSPb19+7abm5ujo2NGRkaR5U0vCAv78ccfiWjt2rWGrohFe/78OYJQnwIDA93d3eVyueqI8q6ER48eNWCtAEFonEaMGEFEFy5cKPJVc5gso9xT1dnZ2dAVAdCThISE6Ojo9u3by2TqYf4uXboQ0alTpwxXLwAjVXJMmHwQpqam/vbbb97e3u3btzd0XQD05O7du0RUYGdn5b7bypcAQCU6OnrHjh1NmjQp7s4NxjJrVC6Xl1zAysqq8EHG2EcffXT37t3NmzejRahFJV8OjuM0GyKgf2lpaUTk7u6ueVD5NDU11TB1AjBKOTk57733Xk5OzooVK4orYxRfZ0lJSR4eHiWXSU5OdnFxKXBwxowZmzZtGjduXL9+/XRWO4sTFRVVrVq1EgoIgqBQKPRWHyhM+YdIbm6u5kFl50+RfzICWCa5XN6vX7+IiIiwsLDg4ODiihlFENrb2y9YsKDkMnZ2dgWOzJkzZ/78+WPGjFFOlgFt8fT0LPly8LzJ96ibOmXjLykpSfOg8mmBZiKAxRJFccCAAbt37168eLFyskxxjCIIbW1tp0yZ8ko/Mm/evK+++mrEiBFLlizhOE5HFbNMrq6ur3o5QM+CgoIEQbh586bmwRs3bhBRzZo1DVQpACMiiuLgwYO3bt26aNGijz/+uOTCJvmn/Q8//DBr1qxBgwb99ttvSEGwQPb29q1atTp58mRiYqLq4K5duziO69y5swErBmAMGGNjxoxZv379vHnzJk6c+NLyRtEifCWLFy+eNGlS48aNx44de/HiRdVxPz8/Hx8fA1bMYl2+fFmhUCinbzx//jwiIoKIfHx8/Pz8DF01czZx4sSjR4+OGTNmzZo1dnZ227dv37FjR8+ePatUqWLoqlmip0+f3r9/n4gSEhKI6MaNG6IoElH9+vWVe82APo0aNSosLKxPnz5dunRRfiMpBQYGurq6FvEDel3TqA3F7Rf3zTffGLpqFqrIiU5Tp041dL3M36xZsziOc3R0VK6jqF+//tOnTw1dKQu1dOnSIr+XUlNTDV01S1Tc7MudO3cWWZ5jjL0sXI3LhQsXkpOTCx+vWrUqdrk1iGPHjhVebhEQEFDckh3QosuXL4eHh2dmZtarV69Hjx7W1taGrpGFevjwYYEhW6V27dqhRah/RX4pEVFwcLBye+oCTC8IAQAAtMgkJ8sAAABoC4IQAAAsGoIQAAAsGoIQAAAsGoIQAAAsGoIQAAAsGoIQAAAsGoIQAAAsGoIQAAAsGoIQAAAs2v8BKfdhaX9xl98AAAAASUVORK5CYII=",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(f, legend=:bottomright, xlim=(-2, 2)) # Julia got the label right"
]
},
{
"cell_type": "markdown",
"id": "1625b12b-e518-4d2c-8a33-40e21e2742fa",
"metadata": {},
"source": [
"## Polynomial evaluation is (discrete) linear algebra"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "ddb5cb97-de43-41b5-926c-ecda7fcf4f8b",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3×4 Matrix{Float64}:\n",
" 1.0 0.0 0.0 0.0\n",
" 1.0 1.0 1.0 1.0\n",
" 1.0 2.0 4.0 8.0"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Let's make a matrix of monomial values evaluated at points\n",
"V = [one.(x) x x.^2 x.^3]"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "03436512-f276-423c-8f65-e17d0d7b9aa9",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3-element Vector{Float64}:\n",
" 1.0\n",
" 12.0\n",
" 83.0"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# And lets multiply that by our coeffs\n",
"V * p + V * q;\n",
"V * (p + q)"
]
},
{
"cell_type": "markdown",
"id": "8f956e12-978a-4f89-8ada-c20a613d0701",
"metadata": {},
"source": [
"## Vandermonde matrices\n",
"\n",
"A Vandermonde matrix is one whose columns are functions evaluated at discrete points.\n",
"\n",
"$$ V \\left( x \\right) = \\left[ 1 \\vert x \\vert x^2 \\vert x^3 \\vert \\cdots \\right] $$"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "43a53d18-301e-4e51-9889-9f6074dfcf5d",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"vander (generic function with 2 methods)"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Let's populate a Vandermonde matrix ourselves\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": "code",
"execution_count": 16,
"id": "a1c639a9-db7c-4f80-87d9-2113e99b3d79",
"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": [
"x = LinRange(-1, 1, 50)\n",
"V = vander(x, 4)\n",
"scatter(x, V, legend=:bottomright, label=[\"\\$1\\$\" \"\\$x\\$\" \"\\$x^2\\$\" \"\\$x^3\\$\"])"
]
},
{
"cell_type": "markdown",
"id": "a11f0cfb-38bd-4567-ab2c-4819253d8bbc",
"metadata": {},
"source": [
"## Fitting is linear algebra\n",
"\n",
"$$ \\left[ 1 \\vert x \\vert x^2 \\vert x^3 \\vert \\cdots \\right] \\left[ p \\right] = \\left[ y \\right] $$"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "b3e63737-3410-4147-94f0-f5aaff582c1a",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Let's pick some points to fit a curve to\n",
"x1 = [-.9, 0.1, .5, .8]\n",
"y1 = [1, 2.4, -.2, 1.3]\n",
"scatter(x1, y1, markersize=8, label=\"data\")"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "0fdb264d-49de-4f58-a309-133ec2ee8a7b",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"size(V) = (4, 4)\n"
]
},
{
"data": {
"image/png": "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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"V = vander(x1)\n",
"@show size(V)\n",
"\n",
"# Write y1 in the polynomial basis\n",
"p = V \\ y1\n",
"\n",
"# And plot the result\n",
"scatter(x1, y1, markersize=8, xlims=(-1, 1), label=\"data\")\n",
"plot!(x, vander(x, 4) * p, label=\"\\$ V(x) p\\$\", linestyle=:dash)"
]
},
{
"cell_type": "markdown",
"id": "4c6ec205-f64c-450d-b684-f36fe79dba9f",
"metadata": {},
"source": [
"## Some common terminology\n",
"\n",
"* The **range** of $A$ is the space spanned by its columns.\n",
"This definition coincides with the range of a function $f \\left( x \\right)$ when $f \\left( x \\right) = A x$.\n",
"\n",
"* The (right) **nullspace** of $A$ is the space of vectors $x$ such that $A x = 0$.\n",
"\n",
"* The **rank** of $A$ is the dimension of its range.\n",
"\n",
"* A matrix has **full rank** if the nullspace of either $A$ or $A^T$ is empty (only the $0$ vector). Equivalently, if all the columns of $A$ (or $A^T$) are linearly independent.\n",
"\n",
"* A **nonsingular** (or **invertible**) matrix is a square matrix of full rank. We call the inverse $A^{-1}$ and it satisfies $A^{-1} A = A A^{-1} = I$.\n",
"\n",
"
\n",
"\n",
"If $A \\in \\mathcal{R}^{m \\times m}$, which of these doesn't belong?\n",
"\n",
"1. $A$ has an inverse $A^{-1}$\n",
"\n",
"2. $rank \\left( A \\right) = m$\n",
"\n",
"3. $null \\left( A \\right) = \\left\\lbrace 0 \\right\\rbrace$\n",
"\n",
"4. $A A^T = A^T A$\n",
"\n",
"5. $det \\left( A \\right) \\neq 0$\n",
"\n",
"6. $A x = 0$ implies that $x = 0$"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "f10c9c24-37a6-4a7a-96fd-a006cb1dc348",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"B = [0.4786659078573863 -0.6058243282505108 -0.08431381808611738 0.17379748782488802; -0.6058243282505108 -0.3523217832765442 -0.17822736587576016 0.21136858880960374; -0.08431381808611738 -0.17822736587576016 -0.12151179661611133 0.5425185862055443; 0.17379748782488802 0.21136858880960374 0.5425185862055443 -0.004832327964730654]\n"
]
},
{
"data": {
"text/plain": [
"0.07104701982881283"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using LinearAlgebra\n",
"\n",
"# Let's test the suspicious one\n",
"A = rand(4, 4)\n",
"B = A' * A - A * A'\n",
"@show B\n",
"det(A)"
]
},
{
"cell_type": "markdown",
"id": "5a829661-844f-4f06-bcd2-50034734de0f",
"metadata": {},
"source": [
"## What is an inverse\n",
"\n",
"When we write $x = A^{-1} y$, we mean that $x$ is the unique vector such that $A x = y$.\n",
"(It is rare that we explicitly compute a matrix $A^{-1}$, though [it's not as \"bad\"](https://arxiv.org/abs/1201.603https://arxiv.org/abs/1201.6035) as people may have told you.)\n",
"A vector $y$ is equivalent to $\\sum_i e_i y_i$ where $e_i$ are columns of the identity. Meanwhile, $x = A^{-1} y$ means that we are expressing that same vector $y$ in the basis of the columns of $A$, i.e., $\\sum_i A_{:, i} x_i$."
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "70fa0665-40f2-4f68-ac9d-5a2fc11066cb",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4×4 Matrix{Float64}:\n",
" 0.626678 0.673801 0.656624 0.0796999\n",
" 0.948769 0.105608 0.530424 0.930987\n",
" 0.681706 0.24679 0.400818 0.572381\n",
" 0.00619773 0.905357 0.750307 0.139434"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Let's grab a matrix\n",
"A = rand(4, 4)"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "eb3262b5-e635-4895-a2bf-b23dd2314ce7",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4×4 Matrix{Float64}:\n",
" 1.0 7.31358e-17 -1.32838e-17 -1.22347e-16\n",
" -4.21113e-16 1.0 1.83308e-16 -5.62053e-16\n",
" 5.27068e-16 -2.03387e-16 1.0 7.20475e-16\n",
" -9.9441e-17 3.77455e-17 1.50873e-16 1.0"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# And take its inverse via backslash\n",
"A \\ A"
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "2a2144cf-6a3a-4e23-a527-6ad3e22ffb3d",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4×4 Matrix{Float64}:\n",
" 1.0 2.6886e-17 -1.07425e-16 -3.06659e-16\n",
" -8.06184e-16 1.0 5.78853e-16 -4.69639e-16\n",
" 5.14419e-16 -3.06496e-16 1.0 -9.91941e-17\n",
" -3.70712e-16 2.41317e-16 2.29375e-16 1.0"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# And via inv()\n",
"inv(A) * A"
]
},
{
"cell_type": "markdown",
"id": "71e7ba1f-c2c1-47f4-931e-b79c47091041",
"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": "4f95ddeb-af05-4e5c-8db9-d66c46e12af0",
"metadata": {},
"source": [
"## Examples with inner products"
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "976a156f-f86c-4454-9463-633c585b295f",
"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": 24,
"id": "fd3d26bc-6b90-446e-9441-572faf179d4f",
"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": "d742c165-22a6-4f61-8398-1c000b9ea225",
"metadata": {},
"source": [
"## Polynomials can be orthogonal too"
]
},
{
"cell_type": "code",
"execution_count": 25,
"id": "290eb33d-168e-4a80-87d9-d4e947f7f9e8",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 25,
"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": "9ac85a44-c656-4717-86a5-4a1f3d44d6a6",
"metadata": {},
"source": [
"Which inner products will be zero?\n",
"\n",
"Which functions are even and odd?"
]
},
{
"cell_type": "markdown",
"id": "8e31caf4-09c0-4497-a041-1fcd41d41a01",
"metadata": {},
"source": [
"## Polynomial inner products"
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "9f36cf85-520e-4595-a835-fc61b912445f",
"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];"
]
}
],
"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
}