math_linear_algebra.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Toán học - Đại số Tuyến tính (Linear Algebra)**\n",
    "\n",
    "*Đại số Tuyến tính là một nhánh của toán học, nghiên cứu về [không gian vector](https://vi.wikipedia.org/wiki/Không_gian_vectơ) và các phép biến đổi tuyến tính giữa các không gian vector, chẳng hạn như phép xoay, co giãn, tịnh tiến (dịch chuyển), v.v.*\n",
    "\n",
    "*Học Máy phụ thuộc rất nhiều vào Đại số Tuyến tính, vì vậy hiểu được định nghĩa về vector, ma trận, các phép toán, và cách áp dụng chúng là rất cần thiết.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<table align=\"left\">\n",
    "  <td>\n",
    "    <a href=\"https://colab.research.google.com/github/mlbvn/handson-ml2-vn/blob/main/math_linear_algebra.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>\n",
    "  </td>\n",
    "  <td>\n",
    "    <a target=\"_blank\" href=\"https://github.com/mlbvn/handson-ml2-vn/blob/main/math_linear_algebra.ipynb\"><img src=\"https://kaggle.com/static/images/open-in-kaggle.svg\" /></a>\n",
    "  </td>\n",
    "</table>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Vector\n",
    "## Định nghĩa\n",
    "Vector là một đại lượng được xác định bởi độ lớn và hướng. Ví dụ, vận tốc của tên lửa là một vector 3 chiều: độ lớn là tốc độ của tên lửa và nó (hy vọng là) hướng lên. Một vector có thể được đại diện bởi một mảng gồm các số được gọi là *số vô hướng* (*scalar*). Mỗi số vô hướng tương ứng với độ lớn của vector theo từng chiều.\n",
    "\n",
    "Ví dụ, giả sử một tên lửa đang bay lên với một góc nhỏ: tốc độ thẳng đứng là 5,000 m/s, và cũng có tốc độ chếch nhẹ về phía Đông là 10 m/s, và tốc độ chếch nhẹ về phía Bắc là 50 m/s. Vận tốc của tên lửa có thể được biểu diễn bởi vector sau:\n",
    "\n",
    "**vận tốc** $= \\begin{pmatrix}\n",
    "10 \\\\\n",
    "50 \\\\\n",
    "5000 \\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Lưu ý: theo quy ước, vector thường được trình bày dưới dạng cột. Ngoài ra, tên vector thường là chữ viết thường để phân biệt chúng với ma trận (sẽ được thảo luận bên dưới) và in đậm (khi có thể) để phân biệt chúng với các số vô hướng đơn giản như ${meters\\_per\\_second} = 5026$.\n",
    "\n",
    "Danh sách N số cũng có thể biểu diễn toạ độ của một điểm trong không gian N chiều, vì vậy vector thường xuyên được biểu diễn dưới dạng điểm thay vì mũi tên. Vector với một phần tử có thể được biểu diễn như một mũi tên hoặc một điểm trên trục, một vector với 2 phần tử là một mũi tên hoặc một điểm trên mặt phẳng, một vector với 3 phần tử là một mũi tên hoặc một điểm trong không gian, và một vector với N phần tử là mũi tên hoặc một điểm trong không gian N chiều v.v., mà hầu hết các chiều trong số đó con người khó hình dung được.\n",
    "\n",
    "\n",
    "##  Mục tiêu\n",
    "Vector giữ nhiều vai trò trong Học Máy, đáng chú ý nhất là để biễu diễn các quan sát và dự đoán. Ví dụ, giả sử ta xây dựng một hệ thống Học Máy để phân loại video thành 3 loại dựa trên những gì ta biết về chúng: (video tốt, video rác (_spam_), và mồi nhử nhấp chuột (_clickbait_)). Với mỗi video, ta sẽ có một vector đại diện cho những gì ta biết về video đó, chẳng hạn:\n",
    "\n",
    "**video** $= \\begin{pmatrix}\n",
    "10.5 \\\\\n",
    "5.2 \\\\\n",
    "3.25 \\\\\n",
    "7.0\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Vector này có thể đại diện cho một video kéo dài 10.5 phút, nhưng chỉ 5.2% người xem hơn một phút, trung bình video này nhận được 3,25 lượt xem mỗi ngày và bị gắn cờ 7 lần là spam. Như bạn thấy, mỗi chiều có thể có một ý nghĩa khác nhau.\n",
    "\n",
    "Dựa trên vector này, hệ thống Học Máy của chúng ta có thể dự đoán rằng 80% đó là một video rác, 18% đó là một trò nhử nhấp chuột, và 2% đó là một video hay. Điều này có thể được biểu diễn dưới dạng vector sau:\n",
    "\n",
    "**class_probabilities** $= \\begin{pmatrix}\n",
    "0.80 \\\\\n",
    "0.18 \\\\\n",
    "0.02\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vector trong python\n",
    "Trong python, một vector có thể được biểu diễn bằng nhiều cách, đơn giản nhất là một danh sách các phần tử:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[10.5, 5.2, 3.25, 7.0]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[10.5, 5.2, 3.25, 7.0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vì ta sẽ thực hiện khá nhiều phép tính khoa học, nên sử dụng `ndarray` của NumPy thì tốt hơn nhiều, bởi thư viện này cung cấp sẵn các hàm thuận tiện và được tối ưu hoá cho các phép toán thiết yếu trên vector (để biết thêm chi tiết về NumPy, hãy xem [Hướng dẫn sử dụng NumPy](tools_numpy.ipynb)). Ví dụ:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 10.5 ,   5.2 ,   3.25,   7.  ])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "video = np.array([10.5, 5.2, 3.25, 7.0])\n",
    "video"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Kích thước của một vector có thể thu được bằng cách sử dụng thuộc tính `size`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "video.size"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Phần tử thứ $i$ (hay còn gọi là *entry* hoặc *item*) của một vector $\\textbf{v}$ được ký hiệu là $\\textbf{v}_i$.\n",
    "\n",
    "Lưu ý rằng chỉ số trong toán học thường bắt đầu từ 1, nhưng trong ngôn ngữ lập trình, chúng thường bắt đầu từ 0. Vì vậy, để truy cập $\\textbf{video}_3$ theo ngôn ngữ lập trình, ta có thể viết:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.25"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "video[2]  # 3rd element"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Minh họa vector\n",
    "Để minh họa các vector, chúng ta sẽ sử dụng matplotlib. Hãy bắt đầu bằng cách nhập thư viện (để biết thêm chi tiết về matplotlib, hãy xem [hướng dẫn sử dụng matplotlib](tools_matplotlib.ipynb)):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vector 2D\n",
    "Hãy tạo một vài vector 2D đơn giản để minh họa:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "u = np.array([2, 5])\n",
    "v = np.array([3, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mỗi vector này có 2 phần tử, vì thế chúng có thể dễ dàng được biểu diễn hình học trên đồ thị 2 chiều, ví dụ như các điểm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e352e72e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_coords, y_coords = zip(u, v)\n",
    "plt.scatter(x_coords, y_coords, color=[\"r\",\"b\"])\n",
    "plt.axis([0, 9, 0, 6])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vector cũng có thể được biểu diễn như mũi tên. Hãy tạo một hàm đơn giản để vẽ các mũi tên:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_vector2d(vector2d, origin=[0, 0], **options):\n",
    "    return plt.arrow(origin[0], origin[1], vector2d[0], vector2d[1],\n",
    "              head_width=0.2, head_length=0.3, length_includes_head=True,\n",
    "              **options)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bây giờ hãy vẽ các mũi tên cho hai vector tương ứng **u** và **v**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32fc3c50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_vector2d(u, color=\"r\")\n",
    "plot_vector2d(v, color=\"b\")\n",
    "plt.axis([0, 9, 0, 6])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vector 3D\n",
    "Vẽ vector 3D cũng tương đối đơn giản. Đầu tiên hãy tạo các vector 3D:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.array([1, 2, 8])\n",
    "b = np.array([5, 6, 3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bây giờ hãy vẽ chúng bằng cách sử dụng `Axes3D` của matplotlib:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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1NFt6MRsaGtDY2KhanayCJ0wrka0Z0mXHp+ic7ctqB1acDqye5kYkxS63aSnJR4SUFBVVYVVrMBoF8slhwNnVQTVA97ilpQWjR4/G1KlTsWLFCmHjF4tFVW9Pp9MYNmyY7THrjnTZ5TYRAUWF/P5jIiIBGp8nQydkBD1NWAToY2RFZql2RGlGoiBTP68VV4uERH6sWDLmm+FQ0tINS1u1pAyRJcAAMGzYMNx4440YOXIk4vE4Zs6ciQsuuMD2uHVDukS2bIUYS7ZamqTdBBZ77GPHjlkm23LzMEK2lEizMm+qrisWi7aTb7WmG1r11dLP14OVi/0gSZKkFnvQuZG3WmTzdDevGXuszs5OoaR77NgxPP/889izZw+am5sxd+5cPPPMM7jiiitsjet50tUiW9IkaUsZvWWyHdJlI1sAtiJPLdJ0MrLlx5YkCfF43BLh8i6GWofRqJgiYq0iDy8uzXkitGrhqjU3CU+6IkuAV69ejdGjR6u9rr/0pS/hzTffPH5JtxLZGukFazWBxWuqtGuDCBAhUmGGSE2Ylpk9PT0lYx87dsz2nKk4wquFDnxUTM6NeDyuq5N6zcplBHp6sV7Jr9Z1cAv8M9/V1YWTTz5Z2PgjR47EunXrkMlkEIlE8Oqrrx6frR3pBU8mkwgGg2hoaOi3JY7Zjl9GlkNaZCvK7UAfDIo+RTe6YcmWypnZaiE7VXVk7Gf/G4D60WBLgL2IcjqpKCuXFz5UZixtANRnzY0EJo3Z1dUltMPY1KlTMXfuXJx55pkIh8M488wzcd1119ke13Oky0YbRCZ6+49VAj0E5UiXyLAc2dqVKQqFAvL5PBRFsdQvV+/4rJ0IgLDNKtlxZVlGPB4vGZN2smATOKIjQ6eljHLPhBeX5k6Qu9Z1oNVmJBIxVGVm54PM3yPRiTQAWLp0KZYuXSp0TM+RbiAQKLGCxWIx02TLQo+wjJBtpTHKgY0+gd5sOr9PmVXwZMt2RbMLnmwpMmeTTpLU27uWXkazkaEXIj8tWFmaA33kUQ92NsCapc3KB5knXdGarlPwHOnmcjk1kmK9mlbBEybrVTUqI5j12fJLfQAqQdo9ByJF+iAZIVsj86em7cVisWQ7IkokVhrfSGRYr70Yyi3N6b473bPXLUdBpRWC1f3djFjaRPfSdQqeI91wOIzm5mZks9mSnRysggjHCtnyY5SDFtnSUl+EX1iWe7fcoV4SVrYN0gLb5MbuqoKHUUsXHxXTz7F1914D+yEKh8Oqfn+8fIgI5T7IlSxthEKhgEAggGQyabvs1w14jnTZr50oXS+TyaBQKNhKkJXz2bJL/Xg8rhl9Wj0Xdtsgq2SrdS3LNbmp9Lt2YCQqpo9UOp2uOzIy6y2266kVDVERtZnn4MUXX8SiRYswcOBALFq0CJMnT0ZbWxtaW1stH3/nzp247LLL1Od79+7duPvuu7Fw4UK7p+Y90iXYfdkpsi0UCgiFQrbcCFoPmRld1cpDSmRbKBQQiURQLBZLdjmwCqtNbtj74YRnlyWjQCCAbDaLWCwmzEVQyxCRuOPLgZ2C0zIG/1EqFouYM2cOpk+fjnnz5mHMmDHYsGEDduzYYYt0x44diy1btgDofSeGDx+OSy+9VMg5eI507Zrx+f4C4XAY4XBYWP8FK0ksM+ei1dOWlqF2QJGjEztMOAUzZFQsFmsyKrbzcSqXsOJ3PibLo5ZW7MUPEdBH8JIkoaWlBcFgEEuWLBF+nNWrV2PMmDFCNqUEPEi6QF/TEzOlrzzZUmSbSqVsR2Xks2XdCGYcA0ZI10hPWysggkqlUsJ3mKgWRCzR3ayuE0165B5hkUwm1ZWQk7t5uJWw44/lpL7/m9/8Bpdffrmw8TxJuoBxotEjW7Pj6IFe5Hw+ryabypHt6tWrcfPNy5FKJfGVr1yMu+66rSQ5xP+e1eW+kXnz2wXFYjHT41TyOdcKzEbFhGw2WzNRsR2wUX497ubR1dXlSBItn8/jhRdewL333itszLol3Upka3QcPVBRA9moqKy23EO5adMmfOUrHejpeRzAcDzyyBLk83nce+/3ys6/Etnasaw1NTUhm83afplIS6zVl1IPelExFaxIklTTiSu7MLsqoJ/XO3+3I116J7q6uoT53Fn87ne/Q2trK4YMGSJsTE+SLntT+ZtslGzZscx26GILBGgDRiOJilWrnkdPz/8D0CvIp9OP45lnvoh77/2eSpykvVnRVss98FrVaeSVpN1brYAiZr67G+mIXiQlmq8kSSW7BJRLXFltFVkL/lkeVhN3JPmRPOP0ebHHcKow4tlnnxUqLQAeJV2gfwkvT7ZWsu6VwJMt2bOMeoZjsShCoYP4hJ8AHCppxG11995KD3c+n0c6nf5kDtryh9lon653Pp9XN/GkjxcVZ9RSGawIiI4KvYZyiTs6d/qTSqUclyh40hVdAtzT04PVq1fjpz/9qdBxPUm6bAKJsvl8g3IzYxmpxkqn07peWKPE3dFxNR5++P9DV1cExeJIxGIPYNmy/1SjTxE9bc3Mm52/UdDHIZPJoKGhAQ0NDWqRBxsh8YZ/PpteL6RkJSpkz5lI20k4PT5LxrIsIxgMIhQK9fPU6lXcibjvTpBuLBbDwYMHhY4JeJR0gT4Nsbu721aCqRxhspptNBpFJBLRJS0jD/awYcPw1luv4ZFHHsOxYzvxxS8+jOnTp6sVNbFYzJZXmL78xWIR6XQahUIBsVhMd95mwCbe2CIS6jBWyRLHZ9O9ZOuyAr2okO/DAPRGVG58gNyUMawkLs1KNGyk65USYMCjpJvJZJBMJgFA3aTPKrQIkyfbSqWvZiSKk046CUuX3qG2PozFYgiFQuomkHbOg8iW9fDataxpJd5EtJw0ulTXsjXRz3oRvB0rmUwiHo+XnLeXZZlK98WKRMNfA15aBHpJd/jw4Y6emyh4knRJRiDdyA5YwmGrvMz0GTBCujx5ieppC/RFT+TFFGUrY7VgvZaTdi13/FhGoiPa/47d1UHkHl9ugyUWFvw5W5Vl3HaVmD2W1cQd0PuMbty4EUeOHMHEiROFnUNnZycWLFiAv/71rwgEAvjFL36BadOmCRnbk6RLrQTNFkhogZJw3d3dpsnWCFjXgCRJQsmLTR7S2Gy23Sh4B4dRLdgt8NERFYqQc4QlJDYq5qMjr0GrSMFM6W81zlskwZdL3NEzkMlkcOedd2L79u34+c9/jra2NkyePBl33XWXrQ6EixYtwkUXXYTf/va36vsgCp4kXTaRZifKouU4ADUDb+WB0ZqHlXJgo+CTWQMGDEAymbQd3fIdxcxqwaye50byppzZn7LoRpapXkIlItKKiukcKXfgxfNmQfdSkiQMGDAAr776Kr75zW/iyiuvRE9PD7Zv325Lcuzu7sZf/vIX/PKXvwQAtTeLKHiSdAlWX26+fwH90+qDyM6DjPWVLFrlxtCDXjLL6O/rgSKmSh3FrM7bTZTTDPXaBGpFh275TGnOdlEuWUmFHm5ExW76jll0d3dj4sSJGDZsGObMmWNr7N27d+OEE05AR0cHtm7dira2NqxcudJSxaYWarujSQWYfeGLxSKSySS6uroQDAYxcOBAxGIxYcSRz+fR3d2NVCqFaDSKAQMGGF6aV0pmZbNZdHZ2Ip/Po6mpCYlEwra1TJZ7t1ahRuTNzc3q9agnsDa2SCSCWCyGxsZGxONx9f4Ui0Vks1mkUik1GclanbwI9gNE7pjGxkZVhuLPO5VKqfZL1uJlFG5/fNnnVGRxRKFQwObNm3H99ddj8+bNiMfjfhmwWXmBj2z5XYJFyBRAXyZalAZKEkU6ndZMvrEwcw68PBGLxVAsFm3JE274TUWjnGZKVqZ6sbKx89SLisutBsxo5G5a0wjZbNb2LjKE4cOHY8SIEWhrawMAzJ07F/fdd5+QsQGPki6hEtGwnbnKZfWtki7rdgBgy07FSxRGGp+bBUXMZFcjeYIiG6tj5vN5ZLPZkheB/t1rGiJFhxTdRqPRfpYmPaM/66DwGso5CPS0Yv683YSWjCHqure0tGDEiBHYuXMnxo4di1dffRXjx48XMjbgUdKtFOkaJVt2PLMyBb+FTWdnp62bTnPgN34U0R6Sj5hFbO9OBM46JygiJLlCRG+CWoARS5PeRovlrGxu6p928hVmomIA6kfdrfvsxArroYcewpVXXol8Po/Ro0fjySefFDa2J0kXKO29QLDaBtGqTCGqpy3NnZJZIm1arNe2HImbkSZYAieLGkWHRDSS1NcshvebahU8eDFKZDVTrY0WtaxsLBl5FXofoUKhoDZKt7PBpBE4GekCwBlnnIG33npL2HgsPEu6QJ+/1CrZsuOUIx0j49t1UuRyOXXJL8K2ZsZra/R47E7DROBGmv1U8ptqRYletXaxhMRb2bSW6QD67egg+nzd0trpvPnObKK0YhYs6VLfFa/A06QL9F78zs5OW1vM6BGmGTK3IlGwO0HE43EUCgXbtjW7Xlu9uabT6X7br7PHtTJfvShR6wXlI0SvJe20lul0r2iV4KQcUy0Zo5I0o+enNhMVO9Hsxkl4lnR7enrURJPdzlxaMoXZNpFGyUePyGnJbRW07M9kMsK8tuxczfRysAqtF1SrLp/KgEni8Ko8AUC1srHQK//lSZhkHC+inJ+6XJUhnbcsy+pHrLOz05EG5k7Bs6QbCoXQ3NxsO4EF9JGOVqWXqDaLiqKgp6dH+MaP7AciGAwKGZedq8heDlagRcS0SohEIo7JE273K2BhpPxXb1sdrc0mvZCwA8pLM3SfWfdIsVjEww8/jL1796KzsxM7duzAZz7zGdvv7CmnnKI+8+FwGBs2bLA1Hg/Pki5tO243gUUg2xNf6WUUevNgibzc2GbPgx+XZAQ78kq5irdKv+s27MgTXnNPAMa6c+ltNklRslehde6pVArhcBiTJ0/Gv/71L+zYsQOf//zn8fHHH2PVqlW48MILLR8vEAhgzZo1GDRokIjp94NnSZdg56VnfauSJNmKbPl5sAQWCoUqeniNnoei9HUro33ZQqEQMpmMod0r9MYkbTwQsNa+kSWwahKxEXmintwTRqxs9Id6LziVpHRzdUBE/G//9m/o6urCuHHjsGTJEnR2duoWEBkFXT+ncFySLl8kEI1GUSgUhOjCLClaJTC9ORutTjMDGlNRFCQSCSFj1hKsemwJfNMYkXCKpPhVALkjQqGQqf4TtQz22rGJNBEJNUmSMGvWLEiShOuuuw7XXnut7TFZeJZ0rWTPeUKkIgFqBmIXhUIBXV1dAPT7z+qh3HloWbW0ssRmPj58o/Z0Om2LcInAav1lJVSSJ2hjTbru9SBP6GnForZScjPSZY/V1dWFYcOGCRv7zTffxNChQ3Hw4EFceOGFGDduHGbMmCFsfM+SLsEI2VSKEu0uh9kyWBrb7MOnNQeWGEUVTFCTG7aajsjFKorFIrq7uwGUtnfM5/OeIig2KiYCbmho8Lw8UY4MtaxsfNKu1vpP8O+J6K16hg4dCgAYMmQILr30UmzYsMEnXaA00tXTX4hs2W3HaRNFfiwrpMsWIITDYciyLMSkrVVmXOnBrnQOrMtBy5FgRaLJ5XJqtVsikVBfzGw2Wzd2J6vyhFeLOwDrWynJsqxuMODG+WrJC3ZB73MikUAqlcIf/vAHLF26VMjYBM+SLoGiEhaKYq6nrZXCBn7jR4p2rYLmkEqlLPW1BbSJk7fBibB/saXF1I84GAyWREOSJKmNpCvZnbxIUCLcE25auezCzMcnn89bKnIwCv66iYx0Dxw4gEsvvRSSJKFQKODKK6/EzJkzhYxN8Czp6kW6bMMYo0tyo6TL915gI1C7Lgp2eW+FGPlz1HM5VJpHuWulVZlGEQ8/F/aelIucKhGUlu/UDVitsjPjnqBzohJwJ5fqTo5L97ZQKCAcDqvRbrkiBzsNcfh7I7KX7qhRo/D2228LGUsPniVdApGdiH299EjHqd4LvC8W6G1KY+UFYY9vZENJ/nfLwWhlmpl5GyEoLd8pEbcbljQRRFUuQszlcv1EirDUAAAgAElEQVQM/16O/lk9n87Dqa2U2L9LJpN+GbCboCipUCggGo1a6jWg9/NmyoGtuijYKPTo0aO2lpyKoqCrq8t0W8hy45E04UZlWqUlLOs5VZTSMmAv6cRA37kGg8ESGcaJ4g43XQXlYHbFo2dl48+nWCwKsWW6Be/MVAPJZFLVj5qbm209WOzN5HVQUeXAbGJPkvrvDGxVoiDZQ5ZlWx8e9vzNVqY5Cf5lpfOlMuB60onNyBO15p6wQu56H9py/ScIhw4dstx8vxxkWUZbWxuGDx+OF154Qfj4niVdSert1xqJRJBOp20/bKRDUgRqlmwqESbrtRW1MzC77Cc5xc6WJWwELkmSIR2YPW83s9Ze1ImtQIR7olYiXTPQ8xSzFrY1a9ZgyZIlKBQKmD17Ns444wxcdNFFOP/8820de+XKlRg/frzquRcN7xZko7f/gpZ7wSzoZe3u7ra88aMe6RYKBXWzykgkUnazSqORLiXeOjs7AfRtKGkXtCljLBazXQpcDRDhsBtQxuPxEqmFvM90rplMRn2J3dCJRYE+OuFwWG0N2tjYiFgshlAo1M8uSQ2MnDxXp8mdzpm04rlz52L37t2YMGECvvnNb2LAgAH48MMPbR1j7969ePnll7FgwQJBs+4Pz0a6QF8UYLVOmi2aoAjUKnnRw0YPnhNeW14LZiNxq9eA5qkoCkKhkOXdgNnfsePkEA2jOjHvJ6Z/d5JIRI+tda5UDBMOh9VVgBPyhJv3m71u6XQaAwcOxMUXX4yLL77Y9tg33HAD7r//fjWgcQKeJl2gP9kZAR8FxONxtQ+D3bnQltbkdBg4cKDtF4vXgrX2ODNLdHyxhCRJtrRgIi6yC9U69OQJXktMpVKe14kB9HOwOFXc4fY1EdlL96WXXkJLSwumTJmCNWvWOPYc1wXpamU09aC38aOIJuIA0N3dbbkIQYs4yQpnVAuudB34JBnNk3oNmAVppt3d3QgEAurLC/RtQ1MLSR4jYImYCIiNEOtFJwbEt8Z0UzemKB0QWxjxxhtv4IUXXsDLL7+Mnp4edHd3Y968eXjqqaeEjE/wNOnSTTYS5VXqY2B1ScySmKIoaGpqstw4hp0DW/UWj8cr+o4rPfCslBII2O9+xp430Lt7B203RA2EqEqNjaK0iKoWoeU5Zf+unJ+4ljp2mSFDO+4JNyUl9pxElgAvX74cy5cvBwC89tpreOCBB4QTLuBx0iWUu+FGtVWzD42W15aWonYgy3JJKbCILXLYaJkInIeZJB6/nXsymSx5UenlDYfDJVGU0VJgJ73AZqB33a3qxPzHho3YahVG3RPUyzmVSjkuxbDPqchqNLfgadItF+nyGz9W6mNglnTYJjpELHa+9mySw2ohAi+zsB8cI9FyJWiRN0VCRuZmxOLFd7MSXbfvJIzoxOzHhv6eztcL50jg5Ql632KxmCs7d9DvdnV1OVKNdt555+G8884TPi7gcdIlsGRnpGRXb4xKDgA9PVhrHkbBLtMpmdXY2GhqDP74fNmu0cY5enPn20FWSrgZvQ6VlrN83T5LwPRztU5S5T42mUwGAEq2Y9eKiO2co1vXiJcb2P9fSZ4w24eBlxeoFaNXUDekS8RgdePHShKF3hbkRsfgoaWx8o1jzIII3Mo10Dof8gOTw0GEE8PIPIiIWc2ZdxUA2kvZamuoRsCeI0WKXtKJzcCoPGGmeTpLul1dXRg7dqyr52QXniZd1q5EVVlWS1b1JAq9rmJ6MEK6eg1prDoISF9WlN7GP3bLdmk88ncaGc/pJAobQVEkH4/HK2qoteycYMnDjk5c7hzdjHTNHMeMewIoXQGwx/M1XZdRKBRw7NgxSFJvSXAikbA8Fi9RlGv4XW6McqgUMRuROHiwBB4IBBCPx219dGg8STJeBlwtVNJQveqcYGFEJ6am8VpJSS94pglGnSJAb9+FuXPnYuDAgWhsbEQgEMDkyZPR1NRk+fjZbBbnnnsucrkcCoUC5s6dK7yBOQBIFW5KTd8xingKhQKKxaJlLRToJfBkMolIJKI2uonFYqYkCiIrvqqN11ij0ajmC09lmkY+HloE3t3drXp5zaK7u1vVTs12KDty5AgGDhyoNh+iApF4PG56HkZA19Po/dYqeqjknKBiGSc36iSvtKiNS9kokZVhgN7iCCf9xLTSoo5pToHufUNDAzZu3IgHH3wQgwcPxq5du3DkyBH84x//sDV+Op1GPB5HsVjEOeecg4ceeghTp061MpTuBfZ0pCtJkrrDqd3CBsoqF4tFYRKF1Yi5HIz2tjU7Xj6fRzgcFmJRqzVYcU7QisMrRQ96USK1PZUkqaxOfOzYMezevQ8AMGbMcAwePNjU8d2OqMPhMM4++2zcf//9eOyxx9DU1CRkDhQoZLNZ9bqJhqdJl2DVqsV6belhtStRULRopTViufOgbHe5bXfMJvLY8RoaGjT3jzOKXC6HbDaraot0HWqVrCotZWnJrrUbsCgidvr6sOfIerPZj02xWMSBAwfw4ovbEAyeBkDC1q0bMWfOmTjhhBNM67ROg79mpO2LOr4sy2htbcV7772H66+/Hu3t7bbH5OFp0i3n0y0HLa8tNRG3i0KhgM7OTsNb5LDQOg8rSa1yYHVbtmlOKpWyNB5pbJlMRj1X1l3gpVJZlqQo8g+FQiXShNdcBVrETitEwp49HyMen4LBg0+Coig4eDCEbdv+gWnTYpoSjNY9dKvQQ+t8RB43EAhgy5Yt6OrqwiWXXILt27dj/PjxwsYHPE66QGnvBSNgt/VhtUv6favRRz6fRyaTgaIoSCQSQrRAO71ttcAWN9htoM56d4HeDxeREfUuiEajauToNbJioVUlZ9dVUEuQZeUT+aX3HCORCBoaev3ilYpX6DzdkhfY99PJYw4YMADnn38+fv/73/ukqwUjhKG1gy/vHKBxzLwk/DbsxWLRdu8FvQ+D0d/nIbIyjZUlSKfu7OxEJpNRxyQiphdUkiQ1kcMXP3iViOvJOTF27HD8/e9/U+eUTv8NY8eOM+wmoHtI/8/teyjqGIcOHVKbQPX09GD16tW45ZZbhIzNwvOkWynSNZN4MhPt8T0dIpGI2lTHKiiqIBeClVaLLNhEXqXKtErnzssSTU1NqlMhFouhUCiUdBij6IddirLZdFrish8ovajRzs6xbkLPe6pXBkwfWLY6SzSMBBEtLS24+GIF7777PgBgwoTT0dLSovmzen5iei7omXAq8mfPp1AoCN1G6qOPPsLXvvY19V5ddtlluOiii4SNT/A86QJ9hMHeECvOASOky4/LVmnZSehR5RdgbQt29vh8Is+ua4LvuUDyAetrJg2UPhTsJpK0PNUiT9aXTETMvpRaREznWSwWPUXEBLpHdE1roefE0KFDbZfThkIhw5G/EZ1YC3wJsMi+C5MmTcLmzZuFjaeHuiFdoE/jqZTlLzeOW+4BGpMlx6amJrUvrRXQQ97V1WVYB2bnzhdm8D0XGhoa1JcI6GsqFAgE0NjYWPLChUKhfttvaxEx+/Kxx2XnxRMx+bLJYeA1HZUIRpJ6i3q0ZBdRfXvdSnBpQS/yN6oTVzpPp5rdOA3Pky4bZVIEajXLX849oLVFjpHf1wK7VA8E+nrb0gNpBYVCQS0jpiSZSN2WXhQ6x0wmoybLjFjNiDzNEDErHWldF9aI7zUdlQe7bNfrOVHLyUijuRCzOjF/nrIsq9dHdKTrFjxPuhQtEpHZac7NuxjKbZduFfxSXYsczSTzWM06FAr182SaARt587ot+/f5fP6TDLe9VpFGiJhIlC1rJVmBjYgVRelH/lo6ai0RsZH7rOecYH225TRwt2DHSaCnE+udpyzL+NnPfob9+/cjlUqhq6vL9pY9e/fuxbx587B//34Eg0Fce+21WLhwoa0x9eB50qWILBDo7Ttgp6RSlHtA62Uy4iAwQwBa0WgulytJVpkBu9TT0m3z+by6ikgkEo690HpETMUXAFTCzefzZTVioDehR2PRB4R+lxwXWlFjrRd28M85r5+yzWLoPjr9oRE9rtZ5plIphMNhnHjiiVi/fj02bdqEYcOGoaWlBS+88AImTJhg6VihUAgrVqzAlClTkEwm0draipkzZ+L0008XcSqlxxI+osugXgPJZNK2b4+N9Ky4B7R+1oyDgMYo98KzckcoFLJdLMHqtiR10MsrSb2lo3q6rRug61coFBCNRkt81axPlnRereU2T8TsEp4nYlZCIbnGzYSWVejpp1RtSc9NpZ4TVuCWRxfoO88vf/nLSKVSmDVrFubPn49//OMfOPnkky2PyyYSE4kExo0bh3379vmkqwU2KWH15vNLdDt9Y9l5WE3o6UGvJaTWsSuBdUxEIhHEYjF1F2Oye1HDdqO6rUiw0W1DQ4MqdRD0XAFGiFhLJ+aJmKx/Wu0FRTgL3CAqmls4HFavk5FEltXzc+P5YAOS7u5ujBkzBsFgUCg5vv/++3j77bcxbdo0YWOyqAvSpX+afZB5RwJVUNl9eKiReKXEmxa0zsNoE3Uj4CNlIrNCoYBAIKAmyACo+rDbER4t/c1G12aImNV3eSJmJSJ2KybRzgIa101USmRZOT83pRj2WMeOHROeSEsmk5g7dy5Wrlxpqw9LOXiedAlmozythjS0jLUKWo5ms1nLiTf2PMx2FKt0DfhImbRRVu8rFArqhpIUEVE0RD/D/hH5spEFjaJrEYlLs0RM5Mpqu+w1JZsXS9LlMu613m8CsO+ccAv8s93d3S20gTn10L3qqqswZ84cYePy8DzpspEur93x0LNq8WOZBRuJinA6kI5JfUPNNFHXIl2+eo7127K6Lc1dK7JkiYpeQNbeZYeI6UNFu3/E43FHSUqLiNlqQvrg0PZJLHnyrgmCFhGXK3OuRmRoFmacE0BfD2Knz5HGFe3TveaaazB+/HgsWrRI2Jha8DzpEgKBQNnMPdm/ylm1zEoUWpFod3e35XOgFzaVSqlLfztuDF631fLbGtVtWaIiSxpv7zJLxPQRpA5lTroi9EASE23JxD4XrP7J/gHQL4rVIuJQKFSyYzJPxIB7RCUKWo6CQqGgyml6JcAinBP8B6SzsxODBg2yPB6LN954A08//TQmTZqEM888E5IkYfny5Zg9e7aQ8VnUDemWi/KM6qFGSZfXgtlI1GpCj7WpRaNRS7susJEW3w4SKO19QERjx2+rZ+8yQsQ0B/oIitg9wQyMED677NaqqOKJmNWIWU8xOx4RMVXUUbDgVL8Jt/RWukZGek6Ick6IjHTPOeccy3ZLs/A86eol0szqoVpj8GCTUHpJMrOkyy/9KaK0CkVR0NXVBQAqkbBRGPltnYosjRAxFbPQz7Kk5QZB0DW3QvhaRAygRCOmP3wiip4NtjEQXQMiaXYsL7eL1NPS7Tgn+A8I5R+8Bs+TLlDaaUxrSW2UWMoRJrthYznN1ky0rLW9udVEHhGJLMvq/Fjdlv6+nG7rFOgFpIRMOBxWl91WpQkrYKUEERV1LIg0jBAxgZUy+ChLq7pOr8xZj4jd8s8ajabtOifovOn3aEyvoS5IlyDLMo4dOya09wIlWKhSy05PA0DfOVFuDpXGI/ImEqHokdVtSV5x228L9F1DLcK3Ik2YJeJqaccsEdMqKZPJqL2F2WV3pQ5sgDYRy3LpbsBay3an77cdCcOscwIA3n77bWzZsgXhcBj5fN5y2Tth/vz5ePHFF9HS0oJt27bZGssI6oJ0c7kcUqkUFEXBgAEDbPdeoD9shy2jGzYajZb1kmRmImVetyVCp05lRGJuOAK0oFdNpgc7GrEeEbM2tGpox0DfRycQCCCRSPQLBrQ0YqNErFfmTNcJgCqHiag+cwta86St0dPpNP7yl79g27ZtGDhwIE4//XTccMMNuOqqqywdq6OjA9/61rcwb948EVOviLogXYpCk8mkrWUzPdhW5Qkao1xxg4homSVvXrdNJBKqbksPLj2sTnpsWdAHQa+azAysEDFppPl83hEpwSjKOSNYGDlHPSLmfcS0LKeudRTda2mo/FhWr4+bxRHBYBCf/exnMWXKFPzHf/wHnn/+eWzbtg1NTU2Wx5wxYwb27NkjcJblURekG4/HK3p0K4GiRKCXJO1sw842GqFkntFouVykyzsxyum2bESlR1I8CYtI0FitJjODSiTFNv6hD1I+n3ctCcXKGdQn2ewxzRCxlq5Lf0/PEx8RsxqqqDJgp6FVjRaPxzF9+vQqz8wc6oJ0WQcDkYlR8AUTkiSpXbaszoXI1mojdf4DotU0h/XbAighY177q/QCk8/SDhE7UU1mBkQQtP0NEb7IczQCJ+UMM0RMYKPrStJEuWRWJVeB25KFVxuYA3VCugTWdmMEWr1tu7q6LGd9yQ5EGzOK2C6dT7oB2n5bs7qtKCJ2u5pMC+wceCnB6Y+NkTk4CfY+0hxI1iEPsJUObOzf00pBz1XgpkuCyP3YsWNCS4DdRF2Qrp5XVw+8N5Zt4WjWPUCgaJkeSqsaEx2/nG4LQC3dFZmN1yJiNoPMbjNPmnChUEAoFKpK20egVM4wch2ciPr5RFk1ElXsHJqamjTLd810YCOCpf9HHxE2Wef2TsCsvCA60uW1cSdRF6RLqESYRnrbmiVdlsApystkMpbPgQiXtoovp9u6kY2nF4g9DhEdEVOxWEQqldJM1DkV7ZGEI0LOsErEkiSp+nG17HhmknVGG/+wUSz9Lv0sOx4RMZUy0/Pp1JZC7HspcqueK664AmvWrMHhw4cxcuRILFu2DB0dHULG1sJxQbrsMr2SxmqUdPUInDRFs6DxKGojC5iWbmvEfuUEyBOq5QhgX142kaUVLdqBW3JGOSKmCI89R7rvblaMUT8Rq6sdI0TMJuR4XZetrqMx2MIXQHs3ZztlzvSznZ2dGDx4sKnz1cMzzzwjZByjqAvS1ZMG2GU6LfkrRYaVSLcSgZuNlPnx4vE4MpkMcrmc+kDWgmZKFjC97Xr4iiy25JOIuFAolLzoZonYrJQgGmyyDugrs3Y7Wcf6n2k1JApmImICWfS0ImKgL5lXiYgrlTmz8kJ3dzdGjRol7LzdRF2QLoG9sZV2WTAyBgve5WC3uIGdI6vbUjNx6g0A9BJaJBKpSp15uWqyciCC4omYfXmz2ay6nC3nISYpgbXKuY1aSdbZtaJZAU/ElICme0v6LpEt3U+6/1pErNdvgt+IlCVilnRFdhhzG3VBumykWywW0d3drb6gZrPIWqRpdaNKPZTz2waDQUQiETVhRQRDLy8gfsmuBbPVZEZgJIriG6bTErZaUT7QF2EHg8GqJetYDbtalXXsh0frmeBXN/QHqNyBDdAmYrbMGeiV2J544gkcPnxY2LPw+9//HosXL4Ysy5g/fz5uvvlmIePqQaoQlbm345wNUDIhmUyqS65oNGrppmQyGRSLRTQ2NkKW5ZJSYCMbVSqKgqNHj2LQoEH9fpbvfBaJRFTSIbKvRHS83sYv2UOhkK1qM76azOzmnCLARnRAn4fUaOmvKDi5jAfKV52xJEwacrXuB1DqjojFYoY/9EaImB2L5yNWP6Z7cffdd+O1117Dvn37cOKJJ+LCCy/E448/bum8ZFnG2LFj8eqrr2LYsGFob2/Hc889J2LPNd2bVBeRrizL6OzsVKOLWCxmeSx6wam4wUqnMh68bqvlt2V123LLRi3tlCVi+mhYISg3qskqgSc6cgSw56m1cwV9bERopyLLmMuhUkRMLhagz4PuZmUdzYfcEVY+PHoyEx/5a0XEfJIW6JUK77vvPnz1q1/Fm2++iUOHDmHfvn2Wz2/Dhg049dRT1Z2E//3f/x3PP/+8I7sAE+qCdIPBIJqbm1U7lVWwxQ2SJNkqBSb9iZrSkA5MpE6w67c1smSvRFDVriajORvd/Vdr5wpR2in7DFXjw0PnSUREHx72g+NGsg7oi26NyipGoUXEgH4rTHqfNm7ciBNPPBHbtm3Du+++i3g8jtNOOw2nnXaa5bns27cPI0aMUP97+PDh2LBhg63zq4S6IF2gl3jJ0G0FfHGDnZ1AKcNNiSLS4Hi/LS2fnSgX1SJiLYKiB5pWCNWIbu0k60Rpp0b9rk5Dr9CC90o7layjse1Et1bBr+IKhQJSqZR67qtWrcIrr7yCgwcPor29Hbfddhu++93v2kqoafGF0/e9bkgXsFZNxie1AoGA6nqwAiL+VCqFWCym2Sehp6dHaILKCFiComUqvdz0QWA/Orw+7MQcnUrWmSVioLdtYLX2aaM50nNhhOicSNYBfUlj2qOvGh8elvQpIHnppZfwzjvv4Mknn0Rrayu2bNmCTZs2WdrWisXw4cPxwQcfqP+9d+9eDBs2zO4plEVdJNIAqPqXUSsJn9SixBu5H8zWdbO6LdC3NGWlhFwuV/WECC2fFUVRl64stJZ4gFjHBO/7tZr0tAOyOWlF/G5V1QF9khb11xB9LfgkFlvEwZ+nk0lDo2AljVgshq6uLnznO99BIBDAj370I+E2sWKxiNNOOw2vvvoqPv3pT2Pq1Kl49tlnMW7cOLtD13cijUAvDuvn40FfUb0OYFaKG/gCjFQqpRIKmeerueMtUL6ajEWlIgeSTEjCYEnKCFlYlRJEgu4Z79Bwq6qO4IYNTCsi1uunIUmSussFJWPd+hjykkYoFMKaNWtw11134bbbbsMll1ziyFyCwSAefvhhzJw5U7WMCSDcsqirSFeWZRw5ckTTrsWTo177RrJ8GSkx1PLvsllntiQ4FAohHA67mnkG+keVkUhESCmuVkRczjHhhJRgBSzpV9KwK9mdrBJxLdjygP5OEQAl56kVETvx7PJ2tJ6eHtx55504fPgwHn30UQwZMkTo8VxC/Ue6bIEEH+myLRyNVqeVi5b55uRUa06e2VAopFp9otGo2iGMnApay1gnol+nokqzjgmSbaqtmZpNlBmxO/GRfyWvdLXdEQTS9PnKNj03Ad9hTgQR88UWoVAI69evx6233opFixbhiiuuqMrHyGnUDekSWHmAHnAS5I1Up7FmbK1omZUmrPptyy1jzS7XtVCNqFKLiIn0ZVlWk3Xd3d2uRE8EVjMVQfp6RFzJK01FDtXcPgjoezZYV40eeKmJfl8EEVMCm1wauVwOS5cuxc6dO7Fq1SqcdNJJQs+7llA38gIlCTo7OxGLxdQkSSQSQSwWM/WAHz16tMSjy0sTNJ6e35aiW6OwslzXG6cWlq3lokotUzy9tPwHx+7cWc1UK2noJNh7yhc5mC1aETUftm+DyISdnr9WT4LhS4m3bt2KG2+8ER0dHViwYEFVVkIOQPfi1hXp5vN5dHV1QZZlhMNhxONxSzfw2LFjSCQSqkxA0gRldVnTNkU2em4Aq+ArsLRKRNmIgq0mi0ajVU1Q0YttVD8W7Zhgl6219PEhu57dj6tZVOPjo0fEQO9q4Y9//CNOO+00rFq1CuvXr8fjjz+O0aNHOz4vF1H/pJvNZtHZ2QlZlhGJRGz59zo7OxGNRtVyU1a3NdMnQTT07D80HyKYakQKrFZJBGMVemWipJeXIyerPQJEg+11W27lQ88UnSN9XMk/bbfIgT6C1f745HI59WNcLBYxb948bNmyBclkEtOnT8e0adOwfPnyetJw6z+RRi8Z9aG1CnrhU6kUotGorT4JosEXONB+WPRSyrKMZDIJQJw+XAlsNCdKqzSim/LdyEgzZbXKWtdMAXPVg2Z0Uza6rWbCjopuAKhVno888ggymQxee+01fOpTn8KmTZuwe/fueiLcsqibSFeW5ZL+tGab3rC6raIoiEajaGho0NRtg8FgTSzhtaIoUfqw3Xm4ASKnXC6n9stQFEVIlGhlLk5ppjQ+f0+1iJhK0GshuqXrQR/jf/7zn1i4cCE+97nP4eabb65aAYZLqH95gZYwVG1lRl7gdVva44l8tVSiKVq3NYtK1WR6MKsPOzUP0dCah1FyEknE7Dz0/N9OQE9uAnqtX7QqcrqqjgdF2XRfJEnCL37xCzz33HN45JFHcOaZZ7o2l507d+Kyyy5TP8i7d+/G3XffjYULFzp96OOHdNl+uJXA98sl3ZYiJ1b8p+IGarTsJoxWk5mB3gtbzj/shJRgde5mtjunc2V1UxElv6xbpNrXgy2ACYVCFZOSThCxlob84YcfYuHChZgyZQruuusuRCIRocc0A1mWMXz4cKxfv76ks5hDqH9Nly+OKAfWb0v9cunFBFDSSo8eYpIvqNigUkJHBPiXSXR7Pa3yUD3/MDkkqlngAFjb7tzsuRpxTJDPtJrlzDQPrWILPW+tU+XNvIYsSRKeffZZPPHEE3jwwQdx9tlnV12zXb16NcaMGeMG4ZZF3ZAugffPsmAliGAwqCbB2CQZPcTBYFDzZSqX0BGpI1ajR4FW3wXSBymjTu323Paait7FoVyPCbYtJ/+BDQQC6qqjmuXMZqLsSv007BIxBSO0pdLBgwexZMkSDB8+HH/6059sdwIThd/85je4/PLLqz2N+pEXAKiyQCqVQnNzc8nf8botRa9kueJ1KDM6ZTkdkX9hK6FWehToLeFF68NG5uFkgqrSsdkPLLv5IkXObhY4ENjoVlQPZP6jQ+dcrrER69SgebzwwgtYsWIF7r33Xnzuc5+renRLyOfzGDZsGLZv3+5WL4f6lxcIvLzA6rbUlIYeLnogqI+pVV3OzPJVT5bgq8mq2cu0XNksa3HS28FBVH8JvaWzW6BzJUcAfZCDwb6dHajHhJOJOoKTGrJZmx6tEDs7O9Hc3Izu7m7cdNNNiEajWL16db+gp9r43e9+h9bW1pponlNXpMv2Taik2wIo0UtFk1ylvczYB5jaPwYCgart9ApYbzVoVjOtFCGaTZQ5BT7KZp8R0b7aSmA/QG5p6loeYnpGCoUCQqEQ/vu//xvLly9HNBrFlClTMGfOHHz88cc1R7rPPvtsTUgLQJ3JC1TjfuzYMVVjpeUoq5brOr0AABRZSURBVPNS6W41S2YB9NNLiZityBJ24AbJGfUP05K1Wt5fgt3SWa2lOmBeM2XvTTXlJqB0V4lYLIZkMonbb78dqVQK1157LXbv3o2NGzdi9uzZuOSSS6oyRy309PRg5MiR2L17N5qamtw6bP1bxoDei9vd3Y1isaj2TtDTbau1ASNQvjdAuT4ETmiIbM8Gt8tmeX04n88D6NsPzM3iBnZOTjUNMttjghwS1S5pZq2C9AF64403cMcdd2DJkiWqD9ZHCY4PTZeST6lUStWc6GGoFX8pW8WltUw0I0vYISY2kqvWB4g0RGr4TveGJWI3+w87rSHruQgoGmYdExQkVLOfBlC6fU5TUxMymQy++93vYs+ePXj++efx6U9/2rW5dHZ2YsGCBfjrX/+KQCCAX/ziF5g2bZprxxeFuop0c7mcamkqFAolgj91vaq2n1JEFZcdt0StdOACjGfhnY7+a0VDBlDiBSe5xYky7krgo9twOIxNmzbhpptuwnXXXYerr77a9Q/B1VdfjfPOOw8dHR2q1EG9UWoQx4e8cM011+Cjjz7CWWedhUQigXfeeQf33HMP4vG4bhWS0w+OE9VkesepREz0ItXSctWKTimyv0StdCUrd03cLm3mr0mhUMB9992HzZs34/HHH8cpp5xi+xhm0d3djSlTpuC9995z/dgWcXyQrqIoePPNN/Gtb30Le/fuxbnnnot9+/bh1FNPRXt7O6ZPn44xY8YAgLqkY19UKvEV5S8VvTeZ2eOzy3S2DSTrMXVTLwWMtzw0C7P+YS2dspoJKlrCG70mRojYbN8FraTd9u3bccMNN+Cyyy7D9ddfX7WP0tatW3Hddddh/Pjx2Lp1K9ra2rBy5UrTja1cxPFBugDwyiuvYMeOHfjGN76h9u7csWMH1q5di3Xr1mH79u2IRCI466yz0N7ejqlTp2LgwIGaDy5LTGZgZvNDJ8EnhVi91G4Rh1lUo5G2Xn8JaoMZDAarfn/4Jbwd2GkGzyftZFnGj3/8Y6xevRqPPfYYTjvtNFtzs4tNmzZh+vTpWLt2Ldra2rB48WI0Nzdj2bJlVZ1XGRw/pFsJiqIgmUxi48aNWLt2LdavX48DBw5g5MiRaGtrw7Rp0zBhwgTVO2tGP6yVajKgdIlYzhbnhl5aC1sIAX2FMrRnG+vbdsodogcr0a0VGCFico7QM7tr1y4sXrwYs2bNwre//e2q+cZZHDhwAGeffTZ2794NAHj99ddx33334f/+7/+qPDNd+KRbDrIsY8+ePWo0vHXrViiKgsmTJ6OtrQ3Tp09HS0tLyQPMugcoA18LySm+R4HZZXMlvdSMLGGU+J2GVvcrVi91o/8wOxeR0a2V42vZ9F5//XU899xziMfj2Lp1K5544omacwacd955eOKJJzB27FgsW7YM6XQa9913X7WnpQefdM2AtK0tW7Zg3bp1WLduHfbs2YMTTjgB7e3tmDZtGqZMmYKGhgZ8+OGHGDx4cL8adbc1Qicjykr6IS/D2E2UiYSVPgVO9Zdg9Wyzm6WKBF9OHA6H8fbbb+OBBx7AoUOH0NPTg+3bt+Mb3/gGHnjggarMUQtbt27FggULkM/nMXr0aDz55JM1V/nGwCddu1AUBQcOHFBJ+M9//jPef/99hMNh3HTTTfjsZz+LUaNGlfgunUrS8aiGhqy3bCV/KRFLNd0AIm1gVvoPs79LpbPViG5ZsNvnEPE//fTT+OUvf4kf/ehHanRLew6eeOKJVZurx+GTrkhs2rQJs2bNwo033ogLLrgAmzZtwrp167Bz5040NjaitbUVU6dORVtbG5qamoQm6VjUmoZMJc1kT7MqS4iYixs2MCN6ON0jtzuk8WAlFvoIHThwADfccANGjx6N5cuX17ITwIvwSVckZFnGgQMH+lXjKIqCzs5ObNiwQU3SHTlyBKNGjVIta6eddppasEEvqdmG6Lwdrdovs15EaVaWEDGXasoalWx6bhU28ODblgYCAaxatQoPPfQQfvCDH+C8885z/fk55ZRT0NzcrFbobdiwwdXjuwCfdKsFWZbx3nvvqUm6d955B8FgEGeccYaqD59wwgklUVM57ZAiSkBcL1WrsBJRsvKLSLeEU/5fK6C5kD+bb37jZitIPoF49OhR3HjjjWhubsYPf/jDqlV0jR49Gps2bcKgQYOqcnwX4JNurUBRFKTTaVWS2LBhA/bt24ehQ4eqvuHJkyeX7HMFoKQJCu1U7FWHBAu+/4BZt4ToHSXsgG/qrWe1sqMPm5kL26YzEAjglVdewT333INly5bh85//fFWb1IwaNQobN27Epz71qarNwWH4pFvLUBQFe/fuVZN0mzdvRi6Xw8SJE3HWWWchlUohl8uho6NDlSaqoZW6tYuDEVmCbHq1IrHYvS4i/dLs9jmRSATd3d249dZbkc/n8dBDD2Hw4MGWzlMkRo8erbp+rrvuOlx77bXVnpJo+KTrNeRyOfz2t7/FHXfcgUKhgIkTJwIAWltbMW3aNLS2tiIWi7lWWWbFeiUSbDRM/wT6SInO223idarSzop/WGv7nL/85S+488478Z3vfAdz586tmRaM+/fvx9ChQ3Hw4EFceOGFePjhhzFjxoxqT0skfNL1Ir773e9i5MiRuOaaayBJEg4fPoz169dj7dq1eOutt9DV1aX2lZg2bRo+85nPAICtJB2PWurAxScQGxoahBRxWJ2LXsGFUyjnH6YthXK5HAYNGoRcLoe77roLH374IX7yk5+gpaXF0bnZwbJly9DU1IQlS5ZUeyoi4ZNuPcJoXwlZltVNFc00RKlmg3MeRiJtIiVWH9ZyS5hpAqMFXi+tZjKTfLeUgP3P//xPPPXUU6p1saOjAzNmzKiJvcEIVIqdSCSQSqUwc+ZMLF26FDNnzqz21ESivkj397//PRYvXgxZljF//nzcfPPN1Z5STUCvr8SIESNUEp44caJmXwmWlMh6VQvJKbvb1ZRzS/BEbGQst6PbcuC3z8nlcrjnnnuwY8cOXHLJJXj//fexYcMGzJ07F/Pnz6/aPHn885//xKWXXqpG51deeSVuueWWak9LNOqHdGVZxtixY/Hqq69i2LBhaG9vx3PPPYfTTz+92lOrSZTrK9Ha2orp06dj6NChJREi7ejQ0NDgaCVdJTjRFKaSW0KvepD3ulYzutXq37Bt2zYsWbIEV155Jb7xjW9UdVXiA0A9ke66deuwbNky/O53vwMA3HvvvZAkyY92DUKvr0RDQwMOHz6MyZMnY8WKFYhGo663f2Tn6GbZbCVZgiLcSCRSE9EtfYiowfiPfvQj/PnPf8Zjjz2GU0891fU5ybKMtrY2DB8+HC+88ILrx69R1M8eafv27cOIESPU/x4+fHg9VrM4BkmSEI1GcfbZZ+Pss88G0JvI+PGPf4zLL78c8XgcV111FdLpNE4//XQ1SUd9JWhLJCeqrCgCpcICdstzJ6G11TibtKOqMtpK3qwsIQJa0e2OHTuwePFifOELX8Af/vCHqkXfK1euxPjx49HV1VWV43sNniNdrci8VmwwXsVnP/tZfP3rXy/JcBcKBbz77rtYu3YtHnrooZK+Eu3t7Whvb0ckEoEsy8jlcrZ3LeCTU9Xs4cp34WpoaFD/P0XDZM1yo6kRW/mXSCSgKAoeffRRPP/88/jJT36i2gmrgb179+Lll1/G7bffjhUrVlRtHl6C50h3+PDh+OCDD9T/3rt3L4YNG1bFGXkfF154Yb//FwqFcMYZZ+CMM87A17/+9X59JX7+85+X9JWYNm0aTj/9dAQCATXZBFROWPEtKePxeFU/oqxLgt8RWJIklYCB/rKEiI8PC60k4p49e7Bw4ULMmDEDf/zjH6ua5ASAG264Affffz86OzurOg8vwXOk297ejl27dmHPnj349Kc/jeeeew7PPvus8OPMnz8fL774IlpaWrBt2zbh43sNkiRh4MCBmDlzpmrtYftKPP3005p9JYYMGaIZGRIZZTIZSJLkyJbnZqAV3VYiSj1Zgm0QbvTjw4PdPieRSAAA/uu//gu//vWvsXLlSrS3t9s8Y/t46aWX0NLSgilTpmDNmjWaq1Af/eG5RBrQaxlbtGiRahlzwm7y+uuvI5FIYN68eT7pGgTfV2L9+vX48MMPMXToULS1tWHq1Kk444wzIEkSPvjgA3WF4maSTgsU3VI/YpHHJ7cEW9BQTpbQim7379+PRYsWYdy4cbj77rsRjUaFzc8ObrvtNvz6179GKBRCT08Puru78aUvfQlPPfVUtadWC6gf94Kb2LNnDy6++GKfdG2A7yvxpz/9Cf/6179w6qmnYsGCBWhtbcXJJ59cskx3aqscrbnZ8QDbOa6WWyIQCKgEfeTIEZxyyin43//9Xzz66KP44Q9/iBkzZtRs/uK1117DAw884LsX+lA/7gUf3oIkSRgxYgRGjBiBYDCIZ599Fg8++CDGjh2LDRs24P7778d7772H5uZmNRpua2tTS3xF66QEfvnuZnTNyxJE/tlsFqFQCB999BFmz56NfD6PAQMGYN68eapM4cP78CPdMvAjXbFIJpPI5XL9ulwpiqLbV4J2aB47dmxJ9zHAWgeuakW3euC3zwkEAnjppZfwgx/8AEuWLFEbfO/evRv/8z//U7V5+jANX16wAp90qwcjfSUGDRrUr6qM14ZZQnVrGx8j0No+p6urSy3yWblyZT03+D4e4JOuFbz//vu4+OKL8c477zgy/t69ezFv3jzs378fwWAQ1157LRYuXOjIsbwORVHQ3d2NjRs3qkm6/fv3Y+TIkf36SpBeSo3B2f9HhQXVjm757XPWrFmDu+66C7feeqval8AtZLNZnHvuuWrhy9y5c7F06VLXjl+n8EnXLK644gqsWbMGhw8fRktLC5YtW4aOjg6hx9i/fz/279+PKVOmIJlMorW1Fc8//7zfR8Ig9PpKTJo0SZUljh49ikwmgwkTJkBRFNd2aNaCVsOcdDqNO++8E4cPH8ajjz5atW5g6XQa8XgcxWIR55xzDh566CFMnTq1KnOpE/iJNLN45plnHD/G0KFDMXToUABAIpHAuHHjsG/fPp90DSIQCGDUqFEYNWoUrrjiipK+Eq+99hrmzJmDjz/+GLNmzcKECRPQ3t6Os846C8FgUDNJJ3qjTBZsxV1jYyMCgQDWrVuHW2+9FYsWLcIVV1xR1eg7Ho8D6I16qczbhzPwI90awfvvv4/zzz8ff/3rX1UzvA/ruPrqqyHLMh588EHkcjlVkti4cWNJX4mpU6di9OjRQpJ0euC3z8lms/j+97+PnTt34rHHHsNJJ50k6rQtQ5ZltLa24r333sP111+Pe+65p9pT8jp8eaGWkUwmcf755+POO+/EnDlzqj2dukAmk9EtImD7Sqxbtw47d+5EPB5Ha2srpk6divb2dgwYMMBUkk4LWhtVvv3227jxxhvR0dGBBQsW1FwLxq6uLlxyySV4+OGHMX78+GpPx8vwSbdWUSgU8IUvfAGf//znsWjRIkeO4SdKyoPvK7F+/fqSvhJTp07FuHHj1ObvhUIBAPoVcLAEym7DHo1GUSgU8MMf/hDr1q3DY489hjFjxlTrdCvie9/7HhKJRL1tn+M2fNKtVcybNw8nnHCC4x2a/ESJOciyjF27dqkkvG3bNgSDQUyZMqWkr4RWJR1VmDU0NCAWi+Fvf/sbFi9ejC996UtYuHBhVXtMaOHQoUMIh8Nobm5GT08PZs2ahVtuuQUXXXRRtafmZfikW4t44403cO6552LSpElqhdXy5csxe/Zsx46ZTqdx7rnn4ic/+UlNNE3xCrT6Suzbtw9Dhw5VW10Wi0UcOHAAs2fPxrFjx9DW1oZTTz0Vhw4dwk033YS5c+fWZEe8d955B1/72tcgyzJkWcZll12G22+/vdrT8jp80j3e4SdKxIP6SqxZswYrVqzAe++9h3PPPRcnnXQSTj75ZKxevRrjx4/HkCFD8NZbb2HTpk3YvXs3YrFYtafuw3n4lrHjHYFAAFu2bFETJdu3b/cTJTZBfSV27dqFSZMm4Y9//CMaGxuxdetW/OpXv8INN9yAiy++WP152oHCx/ENP9I9DuEnSsSCtvCpNvwKx5qC7te1tvwqPhzBoUOH1M7+PT09WL16taMFGLIs46yzzsIXv/hFx45RS6gFwgV6d/tYsWIFtm/fjrVr1+KRRx7B3//+92pPywcHX144DvDRRx/1S5Q4mZn2NyqsDvwKR2/AJ93jAJMmTcLmzZtdOZa/UWFt4P3338fbb7+NadOmVXsqPjj48oIPoaCNCv2EUfWQTCYxd+5crFy50i8pr0H4pOtDGNiNCmlvMB/ugioOr7rqKr+kvEbhuxd8CIO/UWH14VaFo4+K8IsjfLgLNzYqPOWUU9Dc3IxAIKBua3M8oxoVjj504RdH+Kg/0I4L/rY2vTjnnHPU1pQ+ahd+pOvDsxg1ahQ2btyIT33qU9Weig8fPPziCB/1B0mSMGvWLLS3t+OJJ56o9nSEYf78+WhpacHkyZOrPRUfDsAnXR+exZtvvomNGzfi5ZdfxiOPPILXX3+92lMSgo6ODrzyyivVnoYPh+CTrg/PgqqvhgwZgksvvbRuEmkzZszwdeo6RiVN14ePmoQkSXEAAUVRkpIkNQL4A4BliqL8waHjNQP4GYCJAGQA1yiKst6JY31yvJMB/J+iKL7GUGfw3Qs+vIoWAKskSVLQ+xw/7RThfoKVAF5WFOUrkiSFAMQdPJaPOoYf6frwUQGSJDUBeFtRFNc2NvMj3fqFr+n68FEZowEckiTpSUmSNkuS9FNJkpze/kFCGduRD+/CJ10fPiojBOAsAI8oinIWgDSAW5w6mCRJzwB4E8BYSZI+kCSpw6lj+XAfvrzgw0cFSJLUAmCtoiijP/nvGQBuVhTl4vK/6cNHf/iRrg8fFaAoygEA/5Ikaewn/+vfAGyv4pR8eBj/PwLQHXBOi4cXAAAAAElFTkSuQmCC",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e33054240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "\n",
    "subplot3d = plt.subplot(111, projection='3d')\n",
    "x_coords, y_coords, z_coords = zip(a,b)\n",
    "subplot3d.scatter(x_coords, y_coords, z_coords)\n",
    "subplot3d.set_zlim3d([0, 9])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hơi khó để minh họa chính xác 2 điểm này trong không gian, vì vậy ta sẽ thêm những đường thẳng đứng. Ta sẽ viết một hàm đơn giản để vẽ một danh sách các vector 3D kèm theo những đường thẳng đứng:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32f04e48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_vectors3d(ax, vectors3d, z0, **options):\n",
    "    for v in vectors3d:\n",
    "        x, y, z = v\n",
    "        ax.plot([x,x], [y,y], [z0, z], color=\"gray\", linestyle='dotted', marker=\".\")\n",
    "    x_coords, y_coords, z_coords = zip(*vectors3d)\n",
    "    ax.scatter(x_coords, y_coords, z_coords, **options)\n",
    "\n",
    "subplot3d = plt.subplot(111, projection='3d')\n",
    "subplot3d.set_zlim([0, 9])\n",
    "plot_vectors3d(subplot3d, [a,b], 0, color=(\"r\",\"b\"))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Chuẩn\n",
    "Chuẩn của một vector $\\textbf{u}$, được ký hiệu là $\\left \\Vert \\textbf{u} \\right \\|$, là đại lượng đo chiều dài (hay còn gọi là độ lớn) của $\\textbf{u}$. Có nhiều loại chuẩn, nhưng loại phổ biến nhất (và là loại duy nhất ta sẽ thảo luận ở đây) là chuẩn Euclid, được định nghĩa bởi:\n",
    "\n",
    "$\\left \\Vert \\textbf{u} \\right \\| = \\sqrt{\\sum_{i}{\\textbf{u}_i}^2}$\n",
    "\n",
    "Ta có thể lập trình hàm tính chuẩn một cách dễ dàng trong python, nhớ rằng $\\sqrt x = x^{\\frac{1}{2}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "|| [2 5] || =\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "5.3851648071345037"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def vector_norm(vector):\n",
    "    squares = [element**2 for element in vector]\n",
    "    return sum(squares)**0.5\n",
    "\n",
    "print(\"||\", u, \"|| =\")\n",
    "vector_norm(u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tuy nhiên, hàm `norm` của NumPy hiệu quả hơn nhiều, hàm này nằm trong module `linalg` (**Lin**ear **Alg**ebra):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.3851648071345037"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy.linalg as LA\n",
    "LA.norm(u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hãy vẽ một đồ thị nhỏ để xác nhận độ dài của vector $\\textbf{v}$ thật sự bằng $\\approx5.4$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32e93400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "radius = LA.norm(u)\n",
    "plt.gca().add_artist(plt.Circle((0,0), radius, color=\"#DDDDDD\"))\n",
    "plot_vector2d(u, color=\"red\")\n",
    "plt.axis([0, 8.7, 0, 6])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Trông có vẻ đúng!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Phép cộng\n",
    "Các vector có cùng kích thước có thể cộng được với nhau. Phép cộng được thực hiện theo từng phần tử (*elementwise*):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  [2 5]\n",
      "+ [3 1]\n",
      "----------\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([5, 6])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(\" \", u)\n",
    "print(\"+\", v)\n",
    "print(\"-\"*10)\n",
    "u + v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hãy xem biểu diễn hình học của phép cộng vector bằng đồ thị:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32e64c88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_vector2d(u, color=\"r\")\n",
    "plot_vector2d(v, color=\"b\")\n",
    "plot_vector2d(v, origin=u, color=\"b\", linestyle=\"dotted\")\n",
    "plot_vector2d(u, origin=v, color=\"r\", linestyle=\"dotted\")\n",
    "plot_vector2d(u+v, color=\"g\")\n",
    "plt.axis([0, 9, 0, 7])\n",
    "plt.text(0.7, 3, \"u\", color=\"r\", fontsize=18)\n",
    "plt.text(4, 3, \"u\", color=\"r\", fontsize=18)\n",
    "plt.text(1.8, 0.2, \"v\", color=\"b\", fontsize=18)\n",
    "plt.text(3.1, 5.6, \"v\", color=\"b\", fontsize=18)\n",
    "plt.text(2.4, 2.5, \"u+v\", color=\"g\", fontsize=18)\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Phép cộng vector có tính chất **giao hoán** (**commutative**), nghĩa là $\\textbf{u} + \\textbf{v} = \\textbf{v} + \\textbf{u}$. Ta có thể thấy điều này qua hình vẽ trên: đi theo $\\textbf{u}$ *kế đó* $\\textbf{v}$ sẽ đến cùng một điểm với việc đi theo $\\textbf{v}$ *kế đó* $\\textbf{u}$.\n",
    "\n",
    "Phép cộng vector cũng có tính chất **kết hợp** (**asociative**), nghĩa là $\\textbf{u} + (\\textbf{v} + \\textbf{w}) = (\\textbf{u} + \\textbf{v}) + \\textbf{w}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nếu ta có một hình được định nghĩa bởi một số điểm (vector), và ta cộng một vector $\\textbf{v}$ với tất cả các điểm này thì toàn bộ hình sẽ dịch chuyển bởi $\\textbf{v}$. Đây được gọi là [phép tịnh tiến hình học](https://en.wikipedia.org/wiki/Translation_(geometry)):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32e17a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t1 = np.array([2, 0.25])\n",
    "t2 = np.array([2.5, 3.5])\n",
    "t3 = np.array([1, 2])\n",
    "\n",
    "x_coords, y_coords = zip(t1, t2, t3, t1)\n",
    "plt.plot(x_coords, y_coords, \"c--\", x_coords, y_coords, \"co\")\n",
    "\n",
    "plot_vector2d(v, t1, color=\"r\", linestyle=\":\")\n",
    "plot_vector2d(v, t2, color=\"r\", linestyle=\":\")\n",
    "plot_vector2d(v, t3, color=\"r\", linestyle=\":\")\n",
    "\n",
    "t1b = t1 + v\n",
    "t2b = t2 + v\n",
    "t3b = t3 + v\n",
    "\n",
    "x_coords_b, y_coords_b = zip(t1b, t2b, t3b, t1b)\n",
    "plt.plot(x_coords_b, y_coords_b, \"b-\", x_coords_b, y_coords_b, \"bo\")\n",
    "\n",
    "plt.text(4, 4.2, \"v\", color=\"r\", fontsize=18)\n",
    "plt.text(3, 2.3, \"v\", color=\"r\", fontsize=18)\n",
    "plt.text(3.5, 0.4, \"v\", color=\"r\", fontsize=18)\n",
    "\n",
    "plt.axis([0, 6, 0, 5])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cuối cùng, phép trừ một vector giống như cộng một vector có chiều ngược lại."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Phép nhân với số vô hướng\n",
    "Các vector có thể được nhân với các đại lượng vô hướng. Khi đó, tất các các phần tử trong vector được nhân với số đó, ví dụ:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.5 * [2 5] =\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 3. ,  7.5])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(\"1.5 *\", u, \"=\")\n",
    "\n",
    "1.5 * u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Về mặt biểu diễn hình học, một phép nhân với số vô hướng dẫn đến thay đổi tỷ lệ của một hình, nên nó có tên là *vô hướng*. Khoảng cách từ gốc toạ độ (điểm có các toạ độ bằng 0) cũng được nhân với số vô hướng. Ví dụ, hãy thực hiện phép nhân với hệ số `k = 2.5`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32d3f0f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "k = 2.5\n",
    "t1c = k * t1\n",
    "t2c = k * t2\n",
    "t3c = k * t3\n",
    "\n",
    "plt.plot(x_coords, y_coords, \"c--\", x_coords, y_coords, \"co\")\n",
    "\n",
    "plot_vector2d(t1, color=\"r\")\n",
    "plot_vector2d(t2, color=\"r\")\n",
    "plot_vector2d(t3, color=\"r\")\n",
    "\n",
    "x_coords_c, y_coords_c = zip(t1c, t2c, t3c, t1c)\n",
    "plt.plot(x_coords_c, y_coords_c, \"b-\", x_coords_c, y_coords_c, \"bo\")\n",
    "\n",
    "plot_vector2d(k * t1, color=\"b\", linestyle=\":\")\n",
    "plot_vector2d(k * t2, color=\"b\", linestyle=\":\")\n",
    "plot_vector2d(k * t3, color=\"b\", linestyle=\":\")\n",
    "\n",
    "plt.axis([0, 9, 0, 9])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bạn có thể đoán được rằng, phép chia một vector cho một số vô hướng tương đương với phép nhân một vector với nghịch đảo (reciprocal):\n",
    "\n",
    "$\\dfrac{\\textbf{u}}{\\lambda} = \\dfrac{1}{\\lambda} \\times \\textbf{u}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Phép nhân với số vô hướng có tính chất **giao hoán**: $\\lambda \\times \\textbf{u} = \\textbf{u} \\times \\lambda$.\n",
    "\n",
    "Nó cũng có tính chất **kết hợp**: $\\lambda_1 \\times (\\lambda_2 \\times \\textbf{u}) = (\\lambda_1 \\times \\lambda_2) \\times \\textbf{u}$.\n",
    "\n",
    "Cuối cùng, nó cũng có tính chất **phân phối** (**distributive**) trên phép cộng các vector: $\\lambda \\times (\\textbf{u} + \\textbf{v}) = \\lambda \\times \\textbf{u} + \\lambda \\times \\textbf{v}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vector không, vector đơn vị, và vector chuẩn hóa\n",
    "* Một **vector không** (**zero-vector**) là một vector chứa toàn các phần tử có giá trị 0.\n",
    "* Một **vector đơn vị** (**unit vector**) là một vector có chuẩn bằng 1.\n",
    "* Một **vector chuẩn hoá** (**normalized vector**) của một vector không rỗng $\\textbf{u}$, được ký hiệu là $\\hat{\\textbf{u}}$, là một vector đơn vị  cùng hướng với $\\textbf{u}$. Nó bằng với: $\\hat{\\textbf{u}} = \\dfrac{\\textbf{u}}{\\left \\Vert \\textbf{u} \\right \\|}$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32f73e80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.gca().add_artist(plt.Circle((0,0),1,color='c'))\n",
    "plt.plot(0, 0, \"ko\")\n",
    "plot_vector2d(v / LA.norm(v), color=\"k\")\n",
    "plot_vector2d(v, color=\"b\", linestyle=\":\")\n",
    "plt.text(0.3, 0.3, \"$\\hat{u}$\", color=\"k\", fontsize=18)\n",
    "plt.text(1.5, 0.7, \"$u$\", color=\"b\", fontsize=18)\n",
    "plt.axis([-1.5, 5.5, -1.5, 3.5])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tích vô hướng\n",
    "### Định nghĩa\n",
    "Tích vô hướng (dot product) (hay còn được gọi là *scalar product* hoặc *inner product* trong không gian Euclid) của hai vector $\\textbf{u}$ và $\\textbf{v}$ là một phép toán hữu ích và xuất hiện khá thường xuyên trong đại số tuyến tính. Nó được ký hiệu là $\\textbf{u} \\cdot \\textbf{v}$, hoặc đôi khi $⟨\\textbf{u}|\\textbf{v}⟩$, hoặc $(\\textbf{u}|\\textbf{v})$, và nó được định nghĩa là:\n",
    "\n",
    "$\\textbf{u} \\cdot \\textbf{v} = \\left \\Vert \\textbf{u} \\right \\| \\times \\left \\Vert \\textbf{v} \\right \\| \\times cos(\\theta)$\n",
    "\n",
    "trong đó, $\\theta$ là góc giữa $\\textbf{u}$ và $\\textbf{v}$.\n",
    "\n",
    "Một cách khác để tính tích vô hướng là:\n",
    "\n",
    "$\\textbf{u} \\cdot \\textbf{v} = \\sum_i{\\textbf{u}_i \\times \\textbf{v}_i}$\n",
    "\n",
    "### Trong python\n",
    "Tích vô hướng được lập trình khá đơn giản:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def dot_product(v1, v2):\n",
    "    return sum(v1i * v2i for v1i, v2i in zip(v1, v2))\n",
    "\n",
    "dot_product(u, v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nhưng hàm `dot` của NumPy thì hiệu quả hơn *rất* nhiều:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(u,v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tương đương với đó, ta có thể sử dụng `dot` của `ndarray`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.dot(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Lưu ý**: toán tử `*` sẽ thực hiện phép nhân theo phần tử, *KHÔNG PHẢI* phép nhân vô hướng:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   [2 5]\n",
      "*  [3 1] (NOT a dot product)\n",
      "----------\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([6, 5])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(\"  \",u)\n",
    "print(\"* \",v, \"(NOT a dot product)\")\n",
    "print(\"-\"*10)\n",
    "\n",
    "u * v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Các tính chất quan trọng\n",
    "* Tích vô hướng có tính chất **giao hoán**: $\\textbf{u} \\cdot \\textbf{v} = \\textbf{v} \\cdot \\textbf{u}$.\n",
    "* Tích vô hướng chỉ được định nghĩa giữa hai vector, không phải giữa một số vô hướng với một vector. Điều này nghĩa là ta không thể tạo chuỗi tích vô hướng: ví dụ, biểu thức $\\textbf{u} \\cdot \\textbf{v} \\cdot \\textbf{w}$ không được định nghĩa vì $\\textbf{u} \\cdot \\textbf{v}$ là một số vô hướng và $\\textbf{w}$ là một vector.\n",
    "* Điều này cũng có nghĩa là tích vô hướng **KHÔNG CÓ tính chất kết hợp**: $(\\textbf{u} \\cdot \\textbf{v}) \\cdot \\textbf{w} ≠ \\textbf{u} \\cdot (\\textbf{v} \\cdot \\textbf{w})$ vì cả hai vế đều không được xác định.\n",
    "* Tuy nhiên, tích vô hướng **có tính chất kết hợp liên quan đến phép nhân với số vô hướng**: $\\lambda \\times (\\textbf{u} \\cdot \\textbf{v}) = (\\lambda \\times \\textbf{u}) \\cdot \\textbf{v} = \\textbf{u} \\cdot (\\lambda \\times \\textbf{v})$\n",
    "* Cuối cùng, tích vô hướng **có tính chất phân phối** trên phép cộng các vector: $\\textbf{u} \\cdot (\\textbf{v} + \\textbf{w}) = \\textbf{u} \\cdot \\textbf{v} + \\textbf{u} \\cdot \\textbf{w}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Tính toán góc giữa các vector\n",
    "Một trong nhiều ứng dụng của tích vô hướng là tính góc giữa hai vector khác không. Nhìn vào định nghĩa tích vô hướng, ta có thể ra công thức sau:\n",
    "\n",
    "$\\theta = \\arccos{\\left ( \\dfrac{\\textbf{u} \\cdot \\textbf{v}}{\\left \\Vert \\textbf{u} \\right \\| \\times \\left \\Vert \\textbf{v} \\right \\|} \\right ) }$\n",
    "\n",
    "Lưu ý rằng nếu $\\textbf{u} \\cdot \\textbf{v} = 0$, theo sau là $\\theta = \\dfrac{π}{2}$. Nói cách khác, nếu tích vô hướng của hai vector khác rỗng là không thì chúng trực giao.\n",
    "\n",
    "Hãy sử dụng công thức này để tính gốc giữa $\\textbf{u}$ và $\\textbf{v}$ (theo radian):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Angle = 0.868539395286 radians\n",
      "      = 49.7636416907 degrees\n"
     ]
    }
   ],
   "source": [
    "def vector_angle(u, v):\n",
    "    cos_theta = u.dot(v) / LA.norm(u) / LA.norm(v)\n",
    "    return np.arccos(np.clip(cos_theta, -1, 1))\n",
    "\n",
    "theta = vector_angle(u, v)\n",
    "print(\"Angle =\", theta, \"radians\")\n",
    "print(\"      =\", theta * 180 / np.pi, \"degrees\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lưu ý: do lỗi dấu chấm động nhỏ, `cos_theta` có thể nằm ngoài khoảng $[-1, 1]$  chút, điều này sẽ làm cho `acros` bị lỗi. Đây là lý do vì sao ta cắt bớt giá trị trong khoảng, bằng cách sử dụng hàm `clip` của NumPy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Chiếu một điểm lên một trục\n",
    "Tích vô hướng cũng rất hữu dụng để chiếu các điểm lên một trục. Phép chiếu vector $\\textbf{v}$ lên trục của vector $\\textbf{u}$ được biểu diễn bởi công thức sau:\n",
    "\n",
    "$\\textbf{proj}_{\\textbf{u}}{\\textbf{v}} = \\dfrac{\\textbf{u} \\cdot \\textbf{v}}{\\left \\Vert \\textbf{u} \\right \\| ^2} \\times \\textbf{u}$\n",
    "\n",
    "Điều này tương đương với:\n",
    "\n",
    "$\\textbf{proj}_{\\textbf{u}}{\\textbf{v}} = (\\textbf{v} \\cdot \\hat{\\textbf{u}}) \\times \\hat{\\textbf{u}}$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32c344e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "u_normalized = u / LA.norm(u)\n",
    "proj = v.dot(u_normalized) * u_normalized\n",
    "\n",
    "plot_vector2d(u, color=\"r\")\n",
    "plot_vector2d(v, color=\"b\")\n",
    "\n",
    "plot_vector2d(proj, color=\"k\", linestyle=\":\")\n",
    "plt.plot(proj[0], proj[1], \"ko\")\n",
    "\n",
    "plt.plot([proj[0], v[0]], [proj[1], v[1]], \"b:\")\n",
    "\n",
    "plt.text(1, 2, \"$proj_u v$\", color=\"k\", fontsize=18)\n",
    "plt.text(1.8, 0.2, \"$v$\", color=\"b\", fontsize=18)\n",
    "plt.text(0.8, 3, \"$u$\", color=\"r\", fontsize=18)\n",
    "\n",
    "plt.axis([0, 8, 0, 5.5])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ma trận\n",
    "Ma trận là một mảng chữ nhật chứa các số vô hướng (cụ thể, với bất kỳ dạng số nào: nguyên, thực, phức) được sắp xếp thành hàng và cột, ví dụ:\n",
    "\n",
    "\\begin{bmatrix} 10 & 20 & 30 \\\\ 40 & 50 & 60 \\end{bmatrix}\n",
    "\n",
    "Ta cũng có thể xem ma trận như một danh sách (_list_) các vector: ví dụ, ma trận trên chứa 2 vector hàng 3D, chứa 3 vector cột 2D.\n",
    "\n",
    "Ma trận thuận tiện và rất hiệu quả để thực hiện các phép toán trên nhiều vector cùng một lúc. Ta cũng sẽ thấy rằng ma trận rất hữu dụng để biểu diễn và thực hiện các phép biến đổi tuyến tính, ví dụ như: phép xoay, tịnh tiến, và co giãn."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ma trận trong python\n",
    "Trong python, ma trận có thể được biểu diễn bằng nhiều cách. Cách đơn giản nhất là một danh sách (_list_) các danh sách python:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[10, 20, 30], [40, 50, 60]]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[\n",
    "    [10, 20, 30],\n",
    "    [40, 50, 60]\n",
    "]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Một cách hiệu quả hơn rất nhiều là sử dụng thư viện NumPy. Thư viện này cung cấp các hàm được lập trình tối ưu hóa cho nhiều phép toán ma trận."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 20, 30],\n",
       "       [40, 50, 60]])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([\n",
    "    [10,20,30],\n",
    "    [40,50,60]\n",
    "])\n",
    "A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Theo quy ước, ma trận thường có tên in hoa, ví dụ như $A$.\n",
    "\n",
    "Phần còn lại của notebook này, chúng tôi sẽ sử dụng các mảng NumPy (loại `ndarray`) để biểu diễn ma trận."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Kích thước\n",
    "Kích thước của ma trân được định nghĩa bởi số dòng và số cột. Nó được ký hiệu $dòng \\times cột$. Ví dụ, ma trận $A$ ở trên là một ví dụ cho một ma trận $2 \\times 3$: 2 dòng, 3 cột. Thận trọng: một ma trận $3 \\times 2$ sẽ có 3 dòng và 2 cột.\n",
    "\n",
    "Để có được kích thước của ma trận trong NumPy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 3)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Lưu ý**: thuộc tính `size` đại diện cho số lượng phần tử trong `ndarray`, không phải là kích thước của ma trận:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.size"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Lập chỉ mục phần tử\n",
    "Phần tử nằm ở hàng thứ $i$ và cột thứ $j$ của một ma trận $X$ đôi khi được ký hiệu là $X_{i,j}$ hoặc $X_{ij}$. Nhưng chưa có một quy ước chuẩn nào, nên mọi người thường chọn cách đặt tên rõ ràng cho phần tử, ví dụ như: \"*đặt $X = (x_{i,j})_{1 ≤ i ≤ m, 1 ≤ j ≤ n}$*\". Điều này có nghĩa là $X$ bằng với:\n",
    "\n",
    "$X = \\begin{bmatrix}\n",
    "  x_{1,1} & x_{1,2} & x_{1,3} & \\cdots & x_{1,n}\\\\\n",
    "  x_{2,1} & x_{2,2} & x_{2,3} & \\cdots & x_{2,n}\\\\\n",
    "  x_{3,1} & x_{3,2} & x_{3,3} & \\cdots & x_{3,n}\\\\\n",
    "  \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "  x_{m,1} & x_{m,2} & x_{m,3} & \\cdots & x_{m,n}\\\\\n",
    "\\end{bmatrix}$\n",
    "\n",
    "Tuy nhiên, trong notebook này, chúng ta sẽ sử dụng ký hiệu $X_{i,j}$, bởi vì nó khá trùng khớp với ký hiệu của NumPy. Chú ý rằng chỉ mục trong toán thường bắt đầu với 1, nhưng trong các ngôn ngữ lập trình, chúng thường bắt đầu với 0. Vì vậy, để truy cập $A_{2,3}$, trong ngôn ngữ lập trình, ta viết như sau:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "60"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[1,2]  # 2nd row, 3rd column"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vector hàng thứ $i$ đôi khi được ký hiệu là $M_i$ or $M_{i,*}$. Nhưng vẫn chưa có ký hiệu chuẩn, vì vậy mọi người thường định nghĩa rõ ràng tên của chúng, ví dụ: \"*đặt **x**$_{i}$ là vector hàng thứ $i$ của ma trận $X$*\". Chúng ta sẽ sử dụng $M_{i,*}$ vì lý do tương tự được nói đến ở trên. Ví dụ, để truy cập $A_{2,*}$ (nghĩa là  hàng thứ 2 của $A$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([40, 50, 60])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[1, :]  # 2nd row vector (as a 1D array)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tương tự, vector cột thứ $j$ đôi khi được ký hiệu là $M^j$ or $M_{*,j}$, nhưng cũng chưa có ký hiệu chuẩn. Chúng ta sẽ sử dụng $M_{*,j}$. Ví dụ, để truy cập $A_{*,3}$ (nghĩa là vector cột thứ 3 của $A$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([30, 60])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[:, 2]  # 3rd column vector (as a 1D array)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lưu ý rằng kết quả là một mảng NumPy một chiều: không có cái gọi là mảng một chiều *dọc* hoặc *ngang*. Nếu bạn cần trình bày một vector hàng như môt ma trận môth hàng (nghĩa là mảng NumPy 2D), hoặc một vector cột như một ma trận một cột, thì bạn cần sử dụng  pháp cắt chọn thay vì một số nguyên khi truy cập hàng hoặc cột, ví dụ:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[40, 50, 60]])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[1:2, :]  # rows 2 to 3 (excluded): this returns row 2 as a one-row matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[30],\n",
       "       [60]])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[:, 2:3]  # columns 3 to 4 (excluded): this returns column 3 as a one-column matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ma trận vuông, ma trận tam giác, ma trận đường chéo và ma trận đơn vị\n",
    "Một **ma trận vuông** (**square matrix**) là một ma trận có số hàng và số cột bằng nhau, ví dụ một ma trận $3 \\times 3$:\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  4 & 9 & 2 \\\\\n",
    "  3 & 5 & 7 \\\\\n",
    "  8 & 1 & 6\n",
    "\\end{bmatrix}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Một **ma trận tam giác trên** (**upper triangular matrix**) là một loại đặc biệt của ma trận vuông khi tất cả các thành phần *dưới* đường chéo chính (từ góc trên-trái đến góc dưới-phải) bằng không, ví dụ:\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  4 & 9 & 2 \\\\\n",
    "  0 & 5 & 7 \\\\\n",
    "  0 & 0 & 6\n",
    "\\end{bmatrix}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tương tự, một **ma trận tam giác dưới** (**lower triangular matrix**) là một ma trận vuông trong đó tất cả các phần tử *trên* đường chéo chính bằng không, ví dụ:\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  4 & 0 & 0 \\\\\n",
    "  3 & 5 & 0 \\\\\n",
    "  8 & 1 & 6\n",
    "\\end{bmatrix}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Một **ma trận tam giác** (**triangular matrix**) hoặc là một ma trận tam giác trên hoặc là một ma trận tam giác dưới."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Một ma trận cả tam giác trên và dưới được gọi là **ma trận đường chéo** (**diagonal**), ví dụ:\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  4 & 0 & 0 \\\\\n",
    "  0 & 5 & 0 \\\\\n",
    "  0 & 0 & 6\n",
    "\\end{bmatrix}\n",
    "\n",
    "Bạn có thể xây dựng một ma trận đường chéo sử dụng hàm `diag` của NumPy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4, 0, 0],\n",
       "       [0, 5, 0],\n",
       "       [0, 0, 6]])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.diag([4, 5, 6])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nếu bạn truyền vào hàm `diag` một ma trân, nó sẽ trích xuất ra các giá trị đường chéo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 5, 9])"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = np.array([\n",
    "        [1, 2, 3],\n",
    "        [4, 5, 6],\n",
    "        [7, 8, 9],\n",
    "    ])\n",
    "np.diag(D)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cuối cùng, **ma trận đơn vị** (**identity matrix**) với kích thước $n$, được ký hiệu là $I_n$, là một ma trận đường chéo có kích thước $n \\times n$ với $1$ nằm trên đường chéo chính, ví dụ $I_3$:\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  1 & 0 & 0 \\\\\n",
    "  0 & 1 & 0 \\\\\n",
    "  0 & 0 & 1\n",
    "\\end{bmatrix}\n",
    "\n",
    "Hàm `eye` của NumPy trả về ma trận đơn vị với kích thước mong muốn:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.,  0.,  0.],\n",
       "       [ 0.,  1.,  0.],\n",
       "       [ 0.,  0.,  1.]])"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.eye(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ma trận đơn vị thường được ký hiệu đơn giản là $I$ (thay vì $I_n$) khi kích thước được cho rõ ràng trong ngữ cảnh. Nó được gọi là ma trân *đơn vị* vì khi nhân với một ma trận khác ta thu được một ma trận không đổi như bạn sẽ thấy ở bên dưới."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Phép cộng ma trận\n",
    "Nếu hai ma trận $Q$ và $R$ có kích thước giống nhau $m \\times n$, chúng có thể cộng được với nhau. Phép cộng được thực hiện *theo phần tử*: kết quả cũng là một ma trận $S$ $m \\times n$ khi mỗi phần tử của ma trận tổng là tổng của các phần tử ở vị trí tương ứng: $S_{i,j} = Q_{i,j} + R_{i,j}$\n",
    "\n",
    "$S =\n",
    "\\begin{bmatrix}\n",
    "  Q_{11} + R_{11} & Q_{12} + R_{12} & Q_{13} + R_{13} & \\cdots & Q_{1n} + R_{1n} \\\\\n",
    "  Q_{21} + R_{21} & Q_{22} + R_{22} & Q_{23} + R_{23} & \\cdots & Q_{2n} + R_{2n}  \\\\\n",
    "  Q_{31} + R_{31} & Q_{32} + R_{32} & Q_{33} + R_{33} & \\cdots & Q_{3n} + R_{3n}  \\\\\n",
    "  \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "  Q_{m1} + R_{m1} & Q_{m2} + R_{m2} & Q_{m3} + R_{m3} & \\cdots & Q_{mn} + R_{mn}  \\\\\n",
    "\\end{bmatrix}$\n",
    "\n",
    "Ví dụ, hãy tạo một ma trận $B$ $2 \\times 3$ và tính $A + B$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2, 3],\n",
       "       [4, 5, 6]])"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = np.array([[1,2,3], [4, 5, 6]])\n",
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 20, 30],\n",
       "       [40, 50, 60]])"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[11, 22, 33],\n",
       "       [44, 55, 66]])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A + B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Phép cộng có tính chất *giao hoán***, nghĩa là $A + B = B + A$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[11, 22, 33],\n",
       "       [44, 55, 66]])"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B + A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Nó cũng có *tính chất kết hợp***, nghĩa là $A + (B + C) = (A + B) + C$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[111, 222, 333],\n",
       "       [444, 555, 666]])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = np.array([[100,200,300], [400, 500, 600]])\n",
    "\n",
    "A + (B + C)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[111, 222, 333],\n",
       "       [444, 555, 666]])"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A + B) + C"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Phép nhân với số vô hướng\n",
    "Ma trận $M$ có thể được nhân với một số vô hướng $\\lambda$. Kết quả được ký hiệu là $\\lambda M$, và nó là một ma trận có cùng kích thước với $M$ với tất cả các phần tử được nhân với $\\lambda$.\n",
    "\n",
    "$\\lambda M =\n",
    "\\begin{bmatrix}\n",
    "  \\lambda \\times M_{11} & \\lambda \\times M_{12} & \\lambda \\times M_{13} & \\cdots & \\lambda \\times M_{1n} \\\\\n",
    "  \\lambda \\times M_{21} & \\lambda \\times M_{22} & \\lambda \\times M_{23} & \\cdots & \\lambda \\times M_{2n} \\\\\n",
    "  \\lambda \\times M_{31} & \\lambda \\times M_{32} & \\lambda \\times M_{33} & \\cdots & \\lambda \\times M_{3n} \\\\\n",
    "  \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "  \\lambda \\times M_{m1} & \\lambda \\times M_{m2} & \\lambda \\times M_{m3} & \\cdots & \\lambda \\times M_{mn} \\\\\n",
    "\\end{bmatrix}$\n",
    "\n",
    "Ta có thể viết ngắn gọn hơn như sau:\n",
    "\n",
    "$(\\lambda M)_{i,j} = \\lambda (M)_{i,j}$\n",
    "\n",
    "Trong NumPy, đơn giản sử dụng toán tử `*` để nhân một ma trận với một số vô hướng. Ví dụ:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 20,  40,  60],\n",
       "       [ 80, 100, 120]])"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Phép nhân với số vô hướng cũng được định nghĩa ở vế bên phải và cho kết quả giống nhau: $M \\lambda = \\lambda M$. Ví dụ:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 20,  40,  60],\n",
       "       [ 80, 100, 120]])"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Điều này khiến phép nhân số vô hướng có **tính chất giao hoán**.\n",
    "\n",
    "Nó cũng có **tính chất kết hợp**, nghĩa là $\\alpha (\\beta M) = (\\alpha \\times \\beta) M$, trong đó $\\alpha$ and $\\beta$ là số vô hướng. Ví dụ:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 60, 120, 180],\n",
       "       [240, 300, 360]])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * (3 * A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 60, 120, 180],\n",
       "       [240, 300, 360]])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(2 * 3) * A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cuối cùng, nó cũng có **tính phân phối trên phép cộng** ma trận, nghĩa là $\\lambda (Q + R) = \\lambda Q + \\lambda R$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 22,  44,  66],\n",
       "       [ 88, 110, 132]])"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * (A + B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 22,  44,  66],\n",
       "       [ 88, 110, 132]])"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * A + 2 * B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Phép nhân ma trận\n",
    "Đến giờ, các phép toán ma trận vẫn khá trực quan. Nhưng phép nhân hai ma trận phức tạp hơn một chút.\n",
    "\n",
    "Một ma trận $Q$ kích thước $m \\times n$ có thể được nhân với một ma trận $R$ kích thước $n \\times q$. Nó được ký hiệu đơn giản là $QR$ mà không cần có ký hiệu nhân hay chấm. Kết quả $P$ là một ma trận $m \\times q$ trong đó mỗi phần tử được tính toán như là tổng của các tích:\n",
    "\n",
    "$P_{i,j} = \\sum_{k=1}^n{Q_{i,k} \\times R_{k,j}}$\n",
    "\n",
    "Phần tử ở vị trí $i,j$ trong ma trận kết quả là tổng của các tích của các phần tử ở hàng $i$ trong ma trận $Q$ với các phần tử ở cột $j$ trong ma trận $R$\n",
    "\n",
    "$P =\n",
    "\\begin{bmatrix}\n",
    "Q_{11} R_{11} + Q_{12} R_{21} + \\cdots + Q_{1n} R_{n1} &\n",
    "  Q_{11} R_{12} + Q_{12} R_{22} + \\cdots + Q_{1n} R_{n2} &\n",
    "    \\cdots &\n",
    "      Q_{11} R_{1q} + Q_{12} R_{2q} + \\cdots + Q_{1n} R_{nq} \\\\\n",
    "Q_{21} R_{11} + Q_{22} R_{21} + \\cdots + Q_{2n} R_{n1} &\n",
    "  Q_{21} R_{12} + Q_{22} R_{22} + \\cdots + Q_{2n} R_{n2} &\n",
    "    \\cdots &\n",
    "      Q_{21} R_{1q} + Q_{22} R_{2q} + \\cdots + Q_{2n} R_{nq} \\\\\n",
    "  \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "Q_{m1} R_{11} + Q_{m2} R_{21} + \\cdots + Q_{mn} R_{n1} &\n",
    "  Q_{m1} R_{12} + Q_{m2} R_{22} + \\cdots + Q_{mn} R_{n2} &\n",
    "    \\cdots &\n",
    "      Q_{m1} R_{1q} + Q_{m2} R_{2q} + \\cdots + Q_{mn} R_{nq}\n",
    "\\end{bmatrix}$\n",
    "\n",
    "Ta có thể thấy, mỗi phần tử $P_{i,j}$ là một tích vô hướng của vector hàng $Q_{i,*}$ và vector cột $R_{*,j}$:\n",
    "\n",
    "$P_{i,j} = Q_{i,*} \\cdot R_{*,j}$\n",
    "\n",
    "Vì vậy chúng ta có thể viết lại $P$ một cách chính xác hơn:\n",
    "\n",
    "$P =\n",
    "\\begin{bmatrix}\n",
    "Q_{1,*} \\cdot R_{*,1} & Q_{1,*} \\cdot R_{*,2} & \\cdots & Q_{1,*} \\cdot R_{*,q} \\\\\n",
    "Q_{2,*} \\cdot R_{*,1} & Q_{2,*} \\cdot R_{*,2} & \\cdots & Q_{2,*} \\cdot R_{*,q} \\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "Q_{m,*} \\cdot R_{*,1} & Q_{m,*} \\cdot R_{*,2} & \\cdots & Q_{m,*} \\cdot R_{*,q}\n",
    "\\end{bmatrix}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ta hãy nhân hai ma trận trong NumPy, sử dụng phương thức `dot` của `ndarray`:\n",
    "\n",
    "$E = AD = \\begin{bmatrix}\n",
    "  10 & 20 & 30 \\\\\n",
    "  40 & 50 & 60\n",
    "\\end{bmatrix} \n",
    "\\begin{bmatrix}\n",
    "  2 & 3 & 5 & 7 \\\\\n",
    "  11 & 13 & 17 & 19 \\\\\n",
    "  23 & 29 & 31 & 37\n",
    "\\end{bmatrix} = \n",
    "\\begin{bmatrix}\n",
    "  930 & 1160 & 1320 & 1560 \\\\\n",
    "  2010 & 2510 & 2910 & 3450\n",
    "\\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 930, 1160, 1320, 1560],\n",
       "       [2010, 2510, 2910, 3450]])"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = np.array([\n",
    "        [ 2,  3,  5,  7],\n",
    "        [11, 13, 17, 19],\n",
    "        [23, 29, 31, 37]\n",
    "    ])\n",
    "E = A.dot(D)\n",
    "E"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hãy kiểm tra kết quả này bằng cách kiểm tra một phần tử để đảm bảo rằng phép tính được thực hiện chính xác: ví dụ xét $E_{2,3}$, chúng ta cần nhân các phần tử trong hàng thứ 2 của $A$ với cột thứ 3 của $D$, và tính tổng các tích thu được:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2910"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "40*5 + 50*17 + 60*31"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2910"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E[1,2]  # row 2, column 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Trông có vẻ ổn! Bạn có thể kiểm tra với các phần tử còn lại đến khi bạn đã làm quen với thuật toán.\n",
    "\n",
    "Chúng ta đã nhân ma trận $2 \\times 3$ với ma trận $3 \\times 4$, vì vậy kết quả là ma trận $2 \\times 4$. Số cột của ma trận đầu tiên phải bằng với số hàng trong ma trận thứ hai. Nếu chúng ta nhân $D$ với $A$, chúng ta sẽ gặp lỗi vì D có 4 cột trong khi A có 2 hàng:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ValueError: shapes (3,4) and (2,3) not aligned: 4 (dim 1) != 2 (dim 0)\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    D.dot(A)\n",
    "except ValueError as e:\n",
    "    print(\"ValueError:\", e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Điều này minh họa rằng **phép nhân ma trận *KHÔNG* có tính chất giao hoán**: thông thường $QR ≠ RQ$\n",
    "\n",
    "Thật sự, $QR$ và $RQ$ được *cùng* tồn tại nếu $Q$ có kích thước $m \\times n$ và $R$ có kích thước $n \\times m$. Hãy xem một ví dụ khi *cả hai* đều tồn tại và thấy rằng chúng (thường) *KHÔNG* bằng nhau:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[400, 130],\n",
       "       [940, 310]])"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F = np.array([\n",
    "        [5,2],\n",
    "        [4,1],\n",
    "        [9,3]\n",
    "    ])\n",
    "A.dot(F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[130, 200, 270],\n",
       "       [ 80, 130, 180],\n",
       "       [210, 330, 450]])"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F.dot(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mặt khác, **nhân ma trận *có* tính chất kết hợp**, nghĩa là $Q(RS) = (QR)S$. Hãy tạo một ma trận $G$ $4 \\times 5$ để minh hoạ điều này:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[21640, 28390, 27320, 31140, 13570],\n",
       "       [47290, 62080, 60020, 68580, 29500]])"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G = np.array([\n",
    "        [8,  7,  4,  2,  5],\n",
    "        [2,  5,  1,  0,  5],\n",
    "        [9, 11, 17, 21,  0],\n",
    "        [0,  1,  0,  1,  2]])\n",
    "A.dot(D).dot(G)     # (AB)G"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[21640, 28390, 27320, 31140, 13570],\n",
       "       [47290, 62080, 60020, 68580, 29500]])"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.dot(D.dot(G))     # A(BG)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nó cũng **có *tính chất phân phối* trên phép cộng** ma trận, nghĩa là $(Q + R)S = QS + RS$. Ví dụ:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1023, 1276, 1452, 1716],\n",
       "       [2211, 2761, 3201, 3795]])"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A + B).dot(D)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1023, 1276, 1452, 1716],\n",
       "       [2211, 2761, 3201, 3795]])"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.dot(D) + B.dot(D)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nhận ma trận $M$ với ma trận đơn vị (có cùng kích thước) thu được ma trận không đổi $M$. Cụ thể hơn, nếu $M$ là ma trận $m \\times n$ thì:\n",
    "\n",
    "$M I_n = I_m M = M$\n",
    "\n",
    "Biểu thức này thường được viết ngắn gọn hơn như sau (bởi vì kích thước của các ma trận đơn vị thường rõ ràng trong ngữ cảnh):\n",
    "\n",
    "$MI = IM = M$\n",
    "\n",
    "Ví dụ:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 10.,  20.,  30.],\n",
       "       [ 40.,  50.,  60.]])"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.dot(np.eye(3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 10.,  20.,  30.],\n",
       "       [ 40.,  50.,  60.]])"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.eye(2).dot(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Lưu ý**: Toán tử `*` của NumPy thực hiện nhân *theo phần tử*, *KHÔNG* phải là một phép nhân ma trận:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 10,  40,  90],\n",
       "       [160, 250, 360]])"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * B   # NOT a matrix multiplication"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Toán tử @ infix**\n",
    "\n",
    "Python 3.5 [đã giới thiệu](https://docs.python.org/3/whatsnew/3.5.html#pep-465-a-dedicated-infix-operator-for-matrix-multiplication) toán tử `@` infix cho phép nhân ma trận, và NumPy 1.10 đã thêm hỗ trợ cho nó. Nếu bạn đang sử dụng Python 3.5+ và NumPy 1.10+, bạn có thể viết đơn giản `A @ D` thay vì `A.dot(D)`, giúp cho mã nguồn của bạn dễ đọc hơn (nhưng khó di động hơn). Toán tử này cũng hoạt động với tích vô hướng cho vector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python version: 3.5.3\n",
      "Numpy version: 1.12.1\n"
     ]
    }
   ],
   "source": [
    "import sys\n",
    "print(\"Python version: {}.{}.{}\".format(*sys.version_info))\n",
    "print(\"Numpy version:\", np.version.version)\n",
    "\n",
    "# Uncomment the following line if your Python version is ≥3.5\n",
    "# and your NumPy version is ≥1.10:\n",
    "\n",
    "#A @ D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lưu ý: `Q @ R` thực sự tương đương với `Q.__matmul__(R)`, được lập trình bởi NumPy là `np.matmul(Q, R)`, không giống như `Q.dot(R)`. Khác biệt chính là `matmul` không hỗ trợ phép nhân số vô hướng, trong khi `dot` hỗ trợ, vì vậy bạn có thể viết `Q.dot(3)`, tương đương với `Q * 3`, nhưng bạn không thể viết `Q @ 3` ([thông tin chi tiết](http://stackoverflow.com/a/34142617/38626))."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ma trận chuyển vị\n",
    "Ma trận chuyển vị của ma trận $M$ là một ma trận được kí hiệu $M^T$ trong đó hàng thứ $i$ của $M^T$ bằng với cột thứ $i$ của $M$:\n",
    "\n",
    "$ A^T =\n",
    "\\begin{bmatrix}\n",
    "  10 & 20 & 30 \\\\\n",
    "  40 & 50 & 60\n",
    "\\end{bmatrix}^T =\n",
    "\\begin{bmatrix}\n",
    "  10 & 40 \\\\\n",
    "  20 & 50 \\\\\n",
    "  30 & 60\n",
    "\\end{bmatrix}$\n",
    "\n",
    "Nói cách khác, ($A^T)_{i,j}$ = $A_{j,i}$\n",
    "\n",
    "Rõ ràng, nếu $M$ là một ma trận $m \\times n$, thì $M^T$ là một ma trận $n \\times m$.\n",
    "\n",
    "Lưu ý: có nhiều ký hiệu khác, ví dụ như: $M^t$, $M′$, hay ${^t}M$.\n",
    "\n",
    "Trong NumPy, chuyển vị của một ma trận có thể thu được bằng cách sử dụng thuộc tính `T`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 20, 30],\n",
       "       [40, 50, 60]])"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 40],\n",
       "       [20, 50],\n",
       "       [30, 60]])"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Như bạn có thể đã biết, chuyển vị một ma trận hai lần sẽ thu được ma trận ban đầu:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 20, 30],\n",
       "       [40, 50, 60]])"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.T.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Phép chuyển vị có tính chất phân phối trên phép cộng ma trận, nghĩa là $(Q + R)^T = Q^T + R^T$. Ví dụ:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[11, 44],\n",
       "       [22, 55],\n",
       "       [33, 66]])"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A + B).T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[11, 44],\n",
       "       [22, 55],\n",
       "       [33, 66]])"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.T + B.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ngoài ra, $(Q \\cdot R)^T = R^T \\cdot Q^T$. Lưu ý rằng thứ tự được bảo toàn. Ví dụ:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 930, 2010],\n",
       "       [1160, 2510],\n",
       "       [1320, 2910],\n",
       "       [1560, 3450]])"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A.dot(D)).T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 930, 2010],\n",
       "       [1160, 2510],\n",
       "       [1320, 2910],\n",
       "       [1560, 3450]])"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D.T.dot(A.T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Một **ma trận đối xứng** (**symmetric matrix**) $M$ được định nghĩa là một ma trận bằng với chuyển vị của nó: $M^T = M$. Định nghĩa này hàm ý rằng, nó là một ma trận vuông có các phần tử đối xứng so với đường chéo chính, ví dụ:\n",
    "\n",
    "\\begin{bmatrix}\n",
    "  17 & 22 & 27 & 49 \\\\\n",
    "  22 & 29 & 36 & 0 \\\\\n",
    "  27 & 36 & 45 & 2 \\\\\n",
    "  49 & 0 & 2 & 99\n",
    "\\end{bmatrix}\n",
    "\n",
    "Tích của ma trận với chuyển vị của nó luôn là một ma trận đối xứng, ví dụ:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  87,  279,  547],\n",
       "       [ 279,  940, 1860],\n",
       "       [ 547, 1860, 3700]])"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D.dot(D.T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Chuyển đổi mảng 1D thành mảng 2D trong NumPy\n",
    "Như đã nhắc đến trước đó, trong NumPy (ngược lại với Matlab, chẳng hạn), 1D thực sự có nghĩa là 1D: không có cái gọi là mảng 1D dọc và mảng 1D ngang. Ví vậy bạn không nên ngạc nhiên khi thấy chuyển vị mảng 1D sẽ không làm bất cứ điều gì cả:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 5])"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 5])"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ta muốn chuyển đổi $\\textbf{u}$ thành một vector hàng trước khi chuyển vị nó. Có một vài cách để thực hiện:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 5]])"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u_row = np.array([u])\n",
    "u_row"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lưu ý các dấu ngoặc vuông được thêm vào: đây là một mảng 2D có chỉ một hàng (tức là ma trận 1x2). Nói cách khác, nó thực sự là một **vector hàng**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 5]])"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u[np.newaxis, :]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Điều này khá rõ ràng: chúng ta đang cần một trục tung mới, giữ dữ liệu hiện có làm trục hoành."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 5]])"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u[np.newaxis]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cách này tương đương nhưng ít rõ ràng hơn."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 5]])"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u[None]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Đây là phiên bản ngắn nhất, nhưng bạn có thể muốn tránh sử dụng nó bởi vì nó không được rõ ràng. Nguyên nhân nó hoạt động là do `np.newaxis` bằng với `None`, vì vậy nó tương đương với phiên bản trước.\n",
    "\n",
    "Được rồi, bây giờ hãy chuyển vị vector hàng của chúng ta:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2],\n",
       "       [5]])"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u_row.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tuyệt vời! Bây giờ ta đã có một **vector cột**.\n",
    "\n",
    "Thay vì tạo một vector hàng sau đó chuyển vị nó, ta có thể chuyển đổi một mảng 1 chuyển trực tiếp thành một vector cột:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2],\n",
       "       [5]])"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u[:, np.newaxis]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Biểu diễn một ma trận\n",
    "Ta đã thấy rằng các vector có thể được biểu diễn bởi các điểm hoặc mũi tên trong không gian N chiều. Liệu có một cách biểu diễn nào cho ma trận không? Bạn có thể đơn giản xem một ma trận là danh sách các vector, vì vậy biểu diễn một ma trận sẽ cho ta nhiều điểm hoặc mũi tên. Ví dụ, hãy tạo một ma trận `P` $2 \\times 4$  và vẽ nó dưới dạng các điểm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32d8af60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P = np.array([\n",
    "        [3.0, 4.0, 1.0, 4.6],\n",
    "        [0.2, 3.5, 2.0, 0.5]\n",
    "    ])\n",
    "x_coords_P, y_coords_P = P\n",
    "plt.scatter(x_coords_P, y_coords_P)\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dĩ nhiên chúng ta cũng có thể lưu trữ 4 vector như các vector hàng thay vì là vector cột, khi đó ta có ma trận $4 \\times 2$ (chuyển vị của $P$). Đó hoàn toàn là một lựa chọn ngẫu nhiên.\n",
    "\n",
    "Vì các vector có thứ tự, bạn có thể thấy ma trận như là một con đường và nó được biểu diễn bằng các chấm liên kết với nhau:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32d74d30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x_coords_P, y_coords_P, \"bo\")\n",
    "plt.plot(x_coords_P, y_coords_P, \"b--\")\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hoặc bạn có thể biểu diễn nó như một đa giác: lớp `Polygon` của matplotlib cần một mảng NumPy $n \\times 2$, không phải là một mảng $2 \\times n$, vì vậy chúng ta chỉ cần truyền vào $P^T$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e330592b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from matplotlib.patches import Polygon\n",
    "plt.gca().add_artist(Polygon(P.T))\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ứng dụng hình học của các phép toán ma trận\n",
    "Chúng ta đã biết phép cộng vector tạo ra chuyển vị hình học, phép nhân vector với số vô hướng tạo ra phép tái co giãn (phóng đại hoặc thu nhỏ, căn giữa theo gốc tọa độ), và tích vô hướng tạo ra phép chiếu vector lên một vector khác, tái co giãn và tính tọa độ thu được.\n",
    "\n",
    "Tương tự, các phép toán ma trận có những ứng dụng hình học rất hữu ích."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Phép cộng = Phép dịch chuyển đa hình học\n",
    "Đầu tiên, cộng hai ma trận với nhau tương đương với cộng tất cả vector của chúng với nhau. Ví dụ, hãy tạo một ma trận $H$ $2 \\times 4$ và cộng nó với $P$ rồi xem kết quả:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e3302d4a8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "H = np.array([\n",
    "        [ 0.5, -0.2, 0.2, -0.1],\n",
    "        [ 0.4,  0.4, 1.5, 0.6]\n",
    "    ])\n",
    "P_moved = P + H\n",
    "\n",
    "plt.gca().add_artist(Polygon(P.T, alpha=0.2))\n",
    "plt.gca().add_artist(Polygon(P_moved.T, alpha=0.3, color=\"r\"))\n",
    "for vector, origin in zip(H.T, P.T):\n",
    "    plot_vector2d(vector, origin=origin)\n",
    "\n",
    "plt.text(2.2, 1.8, \"$P$\", color=\"b\", fontsize=18)\n",
    "plt.text(2.0, 3.2, \"$P+H$\", color=\"r\", fontsize=18)\n",
    "plt.text(2.5, 0.5, \"$H_{*,1}$\", color=\"k\", fontsize=18)\n",
    "plt.text(4.1, 3.5, \"$H_{*,2}$\", color=\"k\", fontsize=18)\n",
    "plt.text(0.4, 2.6, \"$H_{*,3}$\", color=\"k\", fontsize=18)\n",
    "plt.text(4.4, 0.2, \"$H_{*,4}$\", color=\"k\", fontsize=18)\n",
    "\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nếu cộng một ma trận gồm toàn các vector giống nhau, chúng ta sẽ thu được phép dịch chuyển hình học đơn giản:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32faaba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "H2 = np.array([\n",
    "        [-0.5, -0.5, -0.5, -0.5],\n",
    "        [ 0.4,  0.4,  0.4,  0.4]\n",
    "    ])\n",
    "P_translated = P + H2\n",
    "\n",
    "plt.gca().add_artist(Polygon(P.T, alpha=0.2))\n",
    "plt.gca().add_artist(Polygon(P_translated.T, alpha=0.3, color=\"r\"))\n",
    "for vector, origin in zip(H2.T, P.T):\n",
    "    plot_vector2d(vector, origin=origin)\n",
    "\n",
    "plt.axis([0, 5, 0, 4])\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mặc dù các ma trận chỉ có thể được cộng với nhau nếu chúng có cùng kích thước, NumPy cho phép cộng một vector hàng hoặc một vector cột với một ma trận: phép toán này được gọi là *lan truyền* (*broadcasting*) và được giải thích rõ hơn trong [Hướng dẫn sử dụng NumPy](tools_numpy.ipynb). Ta có thể thu được kết quả giống như trên với:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.5,  3.5,  0.5,  4.1],\n",
       "       [ 0.6,  3.9,  2.4,  0.9]])"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P + [[-0.5], [0.4]]  # same as P + H2, thanks to NumPy broadcasting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Phép nhân với số vô hướng\n",
    "Nhân ma trận với số vô hướng thu được tất cả vector của nó nhân với số vô hướng đó, không có gì ngạc nhiên khi kết quả hình học là một phép tái co giãn của toàn bộ hình. Ví dụ, hãy tái co giãn đa giác của chúng ta theo hệ số 60% (co lại, căn giữa theo gốc tọa độ):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32cdd5c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_transformation(P_before, P_after, text_before, text_after, axis = [0, 5, 0, 4], arrows=False):\n",
    "    if arrows:\n",
    "        for vector_before, vector_after in zip(P_before.T, P_after.T):\n",
    "            plot_vector2d(vector_before, color=\"blue\", linestyle=\"--\")\n",
    "            plot_vector2d(vector_after, color=\"red\", linestyle=\"-\")\n",
    "    plt.gca().add_artist(Polygon(P_before.T, alpha=0.2))\n",
    "    plt.gca().add_artist(Polygon(P_after.T, alpha=0.3, color=\"r\"))\n",
    "    plt.text(P_before[0].mean(), P_before[1].mean(), text_before, fontsize=18, color=\"blue\")\n",
    "    plt.text(P_after[0].mean(), P_after[1].mean(), text_after, fontsize=18, color=\"red\")\n",
    "    plt.axis(axis)\n",
    "    plt.grid()\n",
    "\n",
    "P_rescaled = 0.60 * P\n",
    "plot_transformation(P, P_rescaled, \"$P$\", \"$0.6 P$\", arrows=True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Phép nhân ma trận - Phép chiếu lên một trục\n",
    "Phép nhân ma trận thì phức tạp hơn để minh hoạ, nhưng nó cũng là một công cụ mạnh nhất.\n",
    "\n",
    "Hãy bắt đầu bằng cách định nghĩa một ma trận $1 \\times 2$ $U = \\begin{bmatrix} 1 & 0 \\end{bmatrix}$. Vector hàng này chỉ là một vector đơn vị nằm ngang."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "U = np.array([[1, 0]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bây giờ hãy nhìn vào tích vô hướng $U \\cdot P$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 3. ,  4. ,  1. ,  4.6]])"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.dot(P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Đây là các tọa độ ngang của các vector trong $P$. Nói cách khác, chúng ta chỉ chiếu $P$ lên trục hoành:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32cc1be0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_projection(U, P):\n",
    "    U_P = U.dot(P)\n",
    "    \n",
    "    axis_end = 100 * U\n",
    "    plot_vector2d(axis_end[0], color=\"black\")\n",
    "\n",
    "    plt.gca().add_artist(Polygon(P.T, alpha=0.2))\n",
    "    for vector, proj_coordinate in zip(P.T, U_P.T):\n",
    "        proj_point = proj_coordinate * U\n",
    "        plt.plot(proj_point[0][0], proj_point[0][1], \"ro\")\n",
    "        plt.plot([vector[0], proj_point[0][0]], [vector[1], proj_point[0][1]], \"r--\")\n",
    "\n",
    "    plt.axis([0, 5, 0, 4])\n",
    "    plt.grid()\n",
    "    plt.show()\n",
    "\n",
    "plot_projection(U, P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Chúng ta có thể chiếu lên bất cứ trục nào, đơn giản bằng cách thay $U$ với bất cứ một vector đơn vị khác. Ví dụ, hãy chiếu lên trục hợp với trục hoành một góc 30°:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32d68828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "angle30 = 30 * np.pi / 180  # angle in radians\n",
    "U_30 = np.array([[np.cos(angle30), np.sin(angle30)]])\n",
    "\n",
    "plot_projection(U_30, P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tốt lắm! Hãy nhớ rằng tích vô hướng của một vector đơn vị và một ma trận cơ bản thực hiện một phép chiếu lên trục và cho ta toạ độ của các điểm kết quả trên trục đó."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Phép nhân ma trận - Phép xoay\n",
    "Bây giờ hãy tạo một ma trận $2 \\times 2$ $V$ chứa hai vector đơn vị hợp với trục hoành một góc 30° và 120°:\n",
    "\n",
    "$V = \\begin{bmatrix} \\cos(30°) & \\sin(30°) \\\\ \\cos(120°) & \\sin(120°) \\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.8660254,  0.5      ],\n",
       "       [-0.5      ,  0.8660254]])"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "angle120 = 120 * np.pi / 180\n",
    "V = np.array([\n",
    "        [np.cos(angle30), np.sin(angle30)],\n",
    "        [np.cos(angle120), np.sin(angle120)]\n",
    "    ])\n",
    "V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hãy nhìn vào tích $VP$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.69807621,  5.21410162,  1.8660254 ,  4.23371686],\n",
       "       [-1.32679492,  1.03108891,  1.23205081, -1.8669873 ]])"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V.dot(P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hàng đầu tiên bằng với $V_{1,*} P$  toạ độ của hình chiếu của $P$ lên trục 30°, như chúng ta đã thấy ở trên. Hàng thứ hai là $V_{2,*} P$, là toạ độ của hình chiếu $P$ lên trục 120°. Vì vậy, về cơ bản, chúng ta thu được toạ độ của $P$ sau khi xoay các trục hoành và trung 30° (hoặc tương đương sau khi xoay đa giác -30° xung quanh gốc toạ độ)! Hay vẽ $VP$ để thấy điều này:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32be3da0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P_rotated = V.dot(P)\n",
    "plot_transformation(P, P_rotated, \"$P$\", \"$VP$\", [-2, 6, -2, 4], arrows=True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ma trận $V$ được gọi là **ma trận xoay** (**rotation matrix**)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Phép nhân ma trận - Các phép biến đổi tuyến tính khác\n",
    "Khái quát hơn, bất kỳ phép biến đổi tuyển tính $f$ nào ánh xạ vector n chiều sang vector m chiều có thể được biểu diễn dưới dạng ma trận $m \\times n$. Ví dụ, giả sử $\\textbf{u}$ là một vector 3 chiều:\n",
    "\n",
    "$\\textbf{u} = \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}$\n",
    "\n",
    "và $f$ được định nghĩa như sau:\n",
    "\n",
    "$f(\\textbf{u}) = \\begin{pmatrix}\n",
    "ax + by + cz \\\\\n",
    "dx + ey + fz\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Phép biến đổi $f$ này ánh xạ vector 3 chiều thành 2 chiều một cách tuyến tính (nghĩa là toạ độ thu được chỉ liên quan đến tổng của các toạ độ gốc). Ta có thể biểu diễn biến đổi này như một ma trận $F$:\n",
    "\n",
    "$F = \\begin{bmatrix}\n",
    "a & b & c \\\\\n",
    "d & e & f\n",
    "\\end{bmatrix}$\n",
    "\n",
    "Bây giờ, để tính $f(\\\\textbf{u})$, ta có thể đơn giản thực hiện phép nhân ma trận:\n",
    "\n",
    "$f(\\textbf{u}) = F \\textbf{u}$\n",
    "\n",
    "Nếu ta có ma trận $G = \\begin{bmatrix}\\textbf{u}_1 & \\textbf{u}_2 & \\cdots & \\textbf{u}_q \\end{bmatrix}$, trong đó mỗi $\\textbf{u}_i$ là một vector cột 3 chiều, thì $FG$ sẽ dẫn đến biến đổi tuyến tính của tất cả vector $\\textbf{u}_i$ như được định nghĩa bởi ma trận $F$:\n",
    "\n",
    "$FG = \\begin{bmatrix}f(\\textbf{u}_1) & f(\\textbf{u}_2) & \\cdots & f(\\textbf{u}_q) \\end{bmatrix}$\n",
    "\n",
    "Tóm tắt lại, ma trận vế bên trái của tích vô hướng xác định phép chuyển đổi tuyến tính nào sẽ sử dụng cho các vector vế bên phải. Chúng ta đã chỉ ra rằng phép biến đổi này có thể được sử dụng để thực hiện các phép chiếu và quay, nhưng bất kỳ phép biến đổi tuyến tính nào khác đều có thể được thực hiện. Ví dụ, ở đây là phép biến đổi được gọi là *ánh xạ cắt* (*shear mapping*):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32bdf358>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_shear = np.array([\n",
    "        [1, 1.5],\n",
    "        [0, 1]\n",
    "    ])\n",
    "plot_transformation(P, F_shear.dot(P), \"$P$\", \"$F_{shear} P$\",\n",
    "                    axis=[0, 10, 0, 7])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hãy xem phép biến đổi này ảnh hưởng đến **hình vuông đơn vị** như thế nào:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32d62da0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Square = np.array([\n",
    "        [0, 0, 1, 1],\n",
    "        [0, 1, 1, 0]\n",
    "    ])\n",
    "plot_transformation(Square, F_shear.dot(Square), \"$Square$\", \"$F_{shear} Square$\",\n",
    "                    axis=[0, 2.6, 0, 1.8])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bây giờ hãy xem **ánh xạ bóp** (**squeeze mapping**):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32b74518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_squeeze = np.array([\n",
    "        [1.4, 0],\n",
    "        [0, 1/1.4]\n",
    "    ])\n",
    "plot_transformation(P, F_squeeze.dot(P), \"$P$\", \"$F_{squeeze} P$\",\n",
    "                    axis=[0, 7, 0, 5])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ảnh hưởng lên hình vuông đơn vị:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32b40b38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_transformation(Square, F_squeeze.dot(Square), \"$Square$\", \"$F_{squeeze} Square$\",\n",
    "                    axis=[0, 1.8, 0, 1.2])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hãy xem cái cuối cùng: phép đối xứng qua trục hoành:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32f119e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_reflect = np.array([\n",
    "        [1, 0],\n",
    "        [0, -1]\n",
    "    ])\n",
    "plot_transformation(P, F_reflect.dot(P), \"$P$\", \"$F_{reflect} P$\",\n",
    "                    axis=[-2, 9, -4.5, 4.5])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Nghịch đảo ma trận\n",
    "Bây giờ chúng ta đã hiểu một ma trận có thể biểu diễn bất kỳ phép biến đổi tuyến tính nào, một câu hỏi tự nhiên: chúng ta có thể tìm một ma trận biến đổi mà có thể đảo ngược phép biến đổi của ma trận biến đổi $F$ đã cho hay không? Câu trả lời là có ... đôi khi! Khi nó tồn tại, ma trận như thế được gọi là **nghịch đảo** (**inverse**) của $F$ và được ký hiệu $F^{-1}$.\n",
    "\n",
    "Ví dụ, phép quay, ánh xạ cắt và ánh xạ bóp ở trên đều có biến đổi nghịch đảo. Hãy chứng minh điều này trên ánh xạ cắt:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32fc3f28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_inv_shear = np.array([\n",
    "    [1, -1.5],\n",
    "    [0, 1]\n",
    "])\n",
    "P_sheared = F_shear.dot(P)\n",
    "P_unsheared = F_inv_shear.dot(P_sheared)\n",
    "plot_transformation(P_sheared, P_unsheared, \"$P_{sheared}$\", \"$P_{unsheared}$\",\n",
    "                    axis=[0, 10, 0, 7])\n",
    "plt.plot(P[0], P[1], \"b--\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Chúng ta đã áp dụng ánh xạ cắt trên $P$, như chúng ta đã làm ở trên, nhưng bây giờ chúng ta áp dụng một phép biến đổi thứ hai lên kết quả, và điều này có tác dụng quay trở lại $P$ ban đầu (ta đã vẽ biểu đồ phác thảo của $P$ ban đầu để kiểm tra kỹ). Phép biến đổi thứ hai là nghịch đảo của phép biến đổi thứ nhất.\n",
    "\n",
    "Chúng ta đã định nghĩa ma trận nghịch đảo $F_{shear}^{-1}$ theo cách thủ công, nhưng NumPy hỗ trợ hàm `inv` để tính nghịch đảo của ma trận, vì vậy chúng ta có thể sử dụng nó thay thế:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1. , -1.5],\n",
       "       [ 0. ,  1. ]])"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F_inv_shear = LA.inv(F_shear)\n",
    "F_inv_shear"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Chỉ các ma trận vuông mới có thể nghịch đảo. Điều này có lý khi bạn nghĩ về nó nếu bạn có một phép biến đổi có thể giảm số chiều, kế đó một vài thông tin bị mất và không có cách nào bạn có thể lấy lại nó. Ví dụ, giả sử bạn sử dụng ma trận $2 \\times 3$ để chiếu một vật thể 3D lên một mặt phẳng. Kết quả có thể trông như thế này:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32e643c8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot([0, 0, 1, 1, 0, 0.1, 0.1, 0, 0.1, 1.1, 1.0, 1.1, 1.1, 1.0, 1.1, 0.1],\n",
    "         [0, 1, 1, 0, 0, 0.1, 1.1, 1.0, 1.1, 1.1, 1.0, 1.1, 0.1, 0, 0.1, 0.1],\n",
    "         \"r-\")\n",
    "plt.axis([-0.5, 2.1, -0.5, 1.5])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nhìn vào hình này, không thể nói liệu đây là một phép chiếu của khối lập phương hay hình chiếu của một vật thể chữ nhật hẹp. Một vài thông tin đã bị mất trong phép chiếu.\n",
    "\n",
    "Thậm chí các ma trận biến đổi vuông có thể gây mất thông tin. Ví dụ, xét ma trận biến đổi này:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32f36b38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_project = np.array([\n",
    "        [1, 0],\n",
    "        [0, 0]\n",
    "    ])\n",
    "plot_transformation(P, F_project.dot(P), \"$P$\", \"$F_{project} \\cdot P$\",\n",
    "                    axis=[0, 6, -1, 4])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ma trận biến đổi này thực hiện phép chiếu lên trục hoành. Đa giác của chúng ta bị làm phẳng hoàn toàn nên một số thông tin hoàn toàn bị mất và không thể quay về đa giác ban đầu bằng cách sử dụng biến đổi tuyến tính. Nói cách khác, $F_{project}$ không có nghịch đảo. Một ma trận vuông không thể đảo ngược như vậy gọi là một **ma trận suy biến** (**singular matrix** hay **degenerate matrix**). Nếu chúng ta yêu cầu NumPy tính nghịch đảo của nó, nó sẽ trả về một ngoại lệ:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "LinAlgError: Singular matrix\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    LA.inv(F_project)\n",
    "except LA.LinAlgError as e:\n",
    "    print(\"LinAlgError:\", e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Đây là một ví dụ khác về ma trận suy biến. Phép biến đổi này thực hiện một phép chiếu lên trục 30° trên trục hoành:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32fcf2b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "angle30 = 30 * np.pi / 180\n",
    "F_project_30 = np.array([\n",
    "               [np.cos(angle30)**2, np.sin(2*angle30)/2],\n",
    "               [np.sin(2*angle30)/2, np.sin(angle30)**2]\n",
    "         ])\n",
    "plot_transformation(P, F_project_30.dot(P), \"$P$\", \"$F_{project\\_30} \\cdot P$\",\n",
    "                    axis=[0, 6, -1, 4])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nhưng lần này, do lỗi làm tròn dấu chấm động, NumPy đã cố gắng tính toán thu được nghịch đảo như sau (dù vậy, hãy lưu ý độ lớn của các phần tử này):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  1.20095990e+16,  -2.08012357e+16],\n",
       "       [ -2.08012357e+16,   3.60287970e+16]])"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.inv(F_project_30)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Như bạn có thể biết, tích vô hướng của một ma trận với nghịch đảo của nó sẽ thu được một ma trận đơn vị:\n",
    "\n",
    "$M \\cdot M^{-1} = M^{-1} \\cdot M = I$\n",
    "\n",
    "Điều này có lý vì thực hiện biến đổi tuyến tính theo sau là phép biến đổi nghịch đảo dẫn đến không có sự thay đổi nào cả."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.,  0.],\n",
       "       [ 0.,  1.]])"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F_shear.dot(LA.inv(F_shear))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Một cách khác để diễn đạt điều này là nghịch đảo của ma trận $M$ là chính nó $M$:\n",
    "\n",
    "$((M)^{-1})^{-1} = M$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1. ,  1.5],\n",
       "       [ 0. ,  1. ]])"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.inv(LA.inv(F_shear))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ngoài ra, nghịch đảo của phép co giãn theo tỷ lệ $\\lambda$ tất nhiên là có giãn theo tỷ lệ $\\frac{1}{\\lambda}$:\n",
    "\n",
    "$ (\\lambda \\times M)^{-1} = \\frac{1}{\\lambda} \\times M^{-1}$\n",
    "\n",
    "Một khi bạn hiểu được ý nghĩa hình học của ma trận như những phép biến đổi tuyến tính, hầu hết những tính chất này có vẻ khá trực quan.\n",
    "\n",
    "Một ma trận là nghịch đảo của chính nó được gọi là một **ma trận tự nghịch đảo** (**involution**). Các ví dụ đơn giản nhất là ma trận đối xứng, hoặc quay 180°, nhưng cũng có những phép phức tạp hơn, ví dụ như tưởng tượng một phép biến hình ép theo chiều ngang, sau đó lấy đối xứng qua trục tung và cuối cùng quay 90° theo chiều kim đồng hồ. Nhấc một chiếc khăn ăn và thử làm điều đó hai lần: bạn sẽ kết thúc ở vị trí ban đầu. Đây là ma trận tự nghịch đảo tương ứng:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32c29470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_involution  = np.array([\n",
    "        [0, -2],\n",
    "        [-1/2, 0]\n",
    "    ])\n",
    "plot_transformation(P, F_involution.dot(P), \"$P$\", \"$F_{involution} \\cdot P$\",\n",
    "                    axis=[-8, 5, -4, 4])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cuối cùng, một ma trận vuông $H$ có nghịch đảo là chuyển vị của chính nó được gọi là một **ma trận trực giao**:\n",
    "\n",
    "$H^{-1} = H^T$\n",
    "\n",
    "Vì vậy:\n",
    "\n",
    "$H \\cdot H^T = H^T \\cdot H = I$\n",
    "\n",
    "Nó tương ứng với phép biến đổi giữ nguyên khoảng cách, như phép xoay hay phép đối xứng, và tổ hợp của chúng, nhưng không phải là phép co giãn, cắt hoặc bóp. Hãy kiểm tra xem liệu $F_{reflect}$ có thực sự trục giao:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0],\n",
       "       [0, 1]])"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F_reflect.dot(F_reflect.T)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Định thức\n",
    "Định thức của ma trận vuông $M$, được ký $\\det (M)$ hoặc $\\det M$ hoặc $|M|$, là một giá trị có thể được tính từ các phần tử của nó $(M_{i,j})$ bằng cách sử dụng các phương pháp tương đương khác nhau. Một trong những phương pháp đơn giản nhất là phương pháp đệ quy này:\n",
    "\n",
    "$|M| = M_{1,1}\\\\times|M^{(1,1)}| - M_{1,2}\\\\times|M^{(1,2)}| + M_{1,3}\\\\times|M^{(1,3)}| - M_{1,4}\\\\times|M^{(1,4)}| + \\\\cdots ± M_{1,n}\\\\times|M^{(1,n)}|$\n",
    "\n",
    "* Trong đó $M^{(i,j)}$ là ma trận $M$ không có hàng $i$ và cột $j$.\n",
    "\n",
    "Ví dụ, hãy tính định thức của ma trận $3 \\times 3$ sau:\n",
    "\n",
    "$M = \\begin{bmatrix}\n",
    "  1 & 2 & 3 \\\\\n",
    "  4 & 5 & 6 \\\\\n",
    "  7 & 8 & 0\n",
    "\\end{bmatrix}$\n",
    "\n",
    "Sử dụng phương pháp trên, ta thu được:\n",
    "\n",
    "$|M| = 1 \\times \\left | \\begin{bmatrix} 5 & 6 \\\\ 8 & 0 \\end{bmatrix} \\right |\n",
    "     - 2 \\times \\left | \\begin{bmatrix} 4 & 6 \\\\ 7 & 0 \\end{bmatrix} \\right |\n",
    "     + 3 \\times \\left | \\begin{bmatrix} 4 & 5 \\\\ 7 & 8 \\end{bmatrix} \\right |$\n",
    "\n",
    "Bây giờ ta cần tính định thức của từ ma trận trong các ma trận $2 \\times 2$ (các định thức này được gọi là định thức con):\n",
    "\n",
    "$\\left | \\begin{bmatrix} 5 & 6 \\\\ 8 & 0 \\end{bmatrix} \\right | = 5 \\times 0 - 6 \\times 8 = -48$\n",
    "\n",
    "$\\left | \\begin{bmatrix} 4 & 6 \\\\ 7 & 0 \\end{bmatrix} \\right | = 4 \\times 0 - 6 \\times 7 = -42$\n",
    "\n",
    "$\\left | \\begin{bmatrix} 4 & 5 \\\\ 7 & 8 \\end{bmatrix} \\right | = 4 \\times 8 - 5 \\times 7 = -3$\n",
    "\n",
    "Bây giờ ta có thể tính toán kết quả cuối cùng:\n",
    "\n",
    "$|M| = 1 \\times (-48) - 2 \\times (-42) + 3 \\times (-3) = 27$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Để thu được định thức của một ma trận, bạn có thể gọi hàm `det` của NumPy trong mô-đun `numpy.linalg`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "27.0"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = np.array([\n",
    "        [1, 2, 3],\n",
    "        [4, 5, 6],\n",
    "        [7, 8, 0]\n",
    "    ])\n",
    "LA.det(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Một trong những ứng dụng chính của định thức là để *xác định* liệu một ma trận vuông có khả nghịch hay không: nếu định thức bằng 0, ma trận *không thể* nghịch đảo (nó là một ma trận suy biến), và nếu định thức khác 0 thì ma trận đó *có thể* nghịch đảo.\n",
    "\n",
    "Ví dụ, hãy tính định thức cho các ma trận $F_{project}$, $F_{project\\_30}$ and $F_{shear}$ mà ta đã định nghĩa ở trên:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.det(F_project)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Đúng vậy, $F_{project}$ là một ma trận suy biến, như chúng ta đã thấy ở trên."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0816681711721642e-17"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.det(F_project_30)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Định thức này gần với 0 một cách đáng ngờ: nó thực sự phải là 0, nhưng nó không bằng 0 do lỗi dấu chấm động nhỏ. Ma trận này thật sự là một ma trận suy biến."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.det(F_shear)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hoàn hảo! Ma trận này *có thể* được nghịch đảo như chúng ta đã thấy trước đó. Wow, toán học thực sự hiệu quả!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Định thức cũng có thể được sử dụng để đo lường mức độ ảnh hưởng của một phép biến đổi tuyến tính đến diện tích bề mặt: ví dụ, ma trận phép chiếu $F_{project}$ và $F_ {project\\_30}$ làm phẳng hoàn toàn đa giác $ P $, cho đến khi diện tích của nó bằng không. Đây là lý do tại sao định thức của các ma trận này là 0. Ánh xạ cắt đã sửa đổi hình dạng của đa giác, nhưng nó không ảnh hưởng đến diện tích bề mặt của nó, đó là lý do tại sao định thức là 1. Bạn có thể thử tính toán định thức của ma trận xoay, và bạn sẽ thu được kết quả bằng 1. Còn ma trận co giãn thì sao? Hãy xem ví dụ dưới đây:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32cd89b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_scale = np.array([\n",
    "        [0.5, 0],\n",
    "        [0, 0.5]\n",
    "    ])\n",
    "plot_transformation(P, F_scale.dot(P), \"$P$\", \"$F_{scale} \\cdot P$\",\n",
    "                    axis=[0, 6, -1, 4])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Chúng ta đã co giãn đa giác theo hệ số 1/2 trên cả trục tung và trục hoành để diện tích bề mặt của đa giác thu được bằng 1/4 của đa giác ban đầu. Hãy tính định thức và kiểm tra:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.25"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.det(F_scale)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Chính xác!\n",
    "\n",
    "Định thức thực sự có thể là số âm, khi phép biến đổi dẫn đến một phiên bản \"lật ngược\" của đa giác ban đầu (ví dụ: găng tay trái trở thành găng tay phải). Ví dụ, định thức của ma trận `F_reflect` là -1 vì diện tích bề mặt được giữ nguyên nhưng đa giác bị lật ngược:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.0"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.det(F_reflect)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Kết hợp các phép biến đổi tuyến tính\n",
    "Một số phép biến đổi tuyến tính có thể được xâu chuỗi đơn giản bằng cách thực hiện nhiều tích vô hướng liên tiếp. Ví dụ, để thực hiện ánh xạ bóp, theo sau là ánh xạ cắt, chỉ cần viết:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [],
   "source": [
    "P_squeezed_then_sheared = F_shear.dot(F_squeeze.dot(P))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vì tích vô hướng có tính chất kết hợp, đoạn mã sau tương đương:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [],
   "source": [
    "P_squeezed_then_sheared = (F_shear.dot(F_squeeze)).dot(P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lưu ý rằng thứ tự của các phép biến đổi là ngược lại với thứ tự tích vô hướng.\n",
    "\n",
    "Nếu chúng ta định thực hiện kết hợp các phép biến đổi tuyến tính này nhiều lần, chúng ta cũng có thể lưu ma trận thành phần như sau:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "F_squeeze_then_shear = F_shear.dot(F_squeeze)\n",
    "P_squeezed_then_sheared = F_squeeze_then_shear.dot(P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Từ giờ trở đi, ta có thể thực hiện cả hai phép biến đổi chỉ trong một tích vô hướng, điều này có thể dẫn đến việc tăng hiệu suất rất đáng kể."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nếu bạn muốn thực hiện nghịch đảo của phép biến đổi kép này thì sao? Vâng, nếu bạn bóp và sau đó bạn cắt, và bạn muốn hoàn tác những gì bạn đã làm, điều hiển nhiên là bạn nên hoàn tác việc bóp trước rồi mới hoàn tác việc cắt. Theo ngôn ngữ toán học, cho hai ma trận khả nghịch (hay còn gọi là không suy biến) $Q$ và $R$:\n",
    "\n",
    "$(Q \\cdot R)^{-1} = R^{-1} \\cdot Q^{-1}$\n",
    "\n",
    "Và trong NumPy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ True,  True],\n",
       "       [ True,  True]], dtype=bool)"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.inv(F_shear.dot(F_squeeze)) == LA.inv(F_squeeze).dot(LA.inv(F_shear))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Phân tích giá trị suy biến\n",
    "Hóa ra rằng bất kỳ ma trận $M$ $m \\times n$ nào cũng có thể được phân tích thành tích vô hướng của ba ma trận đơn giản:\n",
    "* một ma trận xoay $U$ (một ma trận trực giao $m \\times m$)\n",
    "* một ma trận co giãn và chiếu $\\Sigma$ (một ma trận đường chéo $m \\times n$)\n",
    "* một ma trận xoay khác $V^T$ (một ma trận trực giao $n \\times n$)\n",
    "\n",
    "$M = U \\cdot \\Sigma \\cdot V^{T}$\n",
    "\n",
    "Ví dụ, hãy phân tích phép biến đổi cắt:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.89442719, -0.4472136 ],\n",
       "       [ 0.4472136 ,  0.89442719]])"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U, S_diag, V_T = LA.svd(F_shear) # note: in python 3 you can rename S_diag to Σ_diag\n",
    "U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2. ,  0.5])"
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S_diag"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lưu ý rằng đây chỉ là một mảng 1D chứa các giá trị đường chéo của Σ. Để có được ma trận Σ thực, chúng ta có thể sử dụng hàm `diag` của NumPy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2. ,  0. ],\n",
       "       [ 0. ,  0.5]])"
      ]
     },
     "execution_count": 126,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S = np.diag(S_diag)\n",
    "S"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bây giờ, hãy kiểm tra xem $U \\cdot \\Sigma \\cdot V^T$ có thực sự bằng với `F_shear` không:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1. ,  1.5],\n",
       "       [ 0. ,  1. ]])"
      ]
     },
     "execution_count": 127,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.dot(np.diag(S_diag)).dot(V_T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1. ,  1.5],\n",
       "       [ 0. ,  1. ]])"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F_shear"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nó hoạt động rất mượt mà! Hãy áp dụng lần lượt các phép biến đổi này (theo thứ tự ngược lại) trên hình vuông đơn vị để hiểu điều gì đang xảy ra. Đầu tiên, hãy áp dụng phép xoay đầu tiên $V^T$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32faa860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_transformation(Square, V_T.dot(Square), \"$Square$\", \"$V^T \\cdot Square$\",\n",
    "                    axis=[-0.5, 3.5 , -1.5, 1.5])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bây giờ, hãy tái co giãn theo trục tung và trục hoành bằng cách sử dụng $\\Sigma$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32ca8dd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_transformation(V_T.dot(Square), S.dot(V_T).dot(Square), \"$V^T \\cdot Square$\", \"$\\Sigma \\cdot V^T \\cdot Square$\",\n",
    "                    axis=[-0.5, 3.5 , -1.5, 1.5])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cuối cùng, ta hãy áp dụng phép xoay thứ hai $U$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Qz6AN7oUXxh9vPJ5/Xu9FeP/95tecay4Kf//7+tq0aboPqqt1uqxM8/SRN6z17KnLhAYE7PHBB5rDnzhRi8UDBmjePy+v5XoeeEBrOnV1up/CFi7UlFBkIxo2dqz2kHrrrebXwndY79qV+H7pQhIeO0hEpgGznXOfCk3fDDjn3N0Ry1wBnOScu6YD67Oxg/wuaGP8L1miB4y+ffUA1revHkwTbbRmz4YPPjh8PeFhIyor9YB30UXRUxbQsTH+d+/WG9NWrEj9A2AuvVQbsttu04Y0L0+7b55yCnznOy2vfh55BP7zH31EZe/emor57W9b1kkeekh780yapGnDggI9yI8fr72JcnI0ZTR/vt5zkZ+vDeKNN+rVwQ036AkHaO+oDRvg/vujx75pE9x0k647O1v/L1/+cuz7HQIsrUNJi8iFwCedc18PTX8JmOKc+++IZa4A7gQ+AtYD1zvntsRYnzUCfmVj/HurK4/xv3KlnqGvWwfHHJPuaLq8dA8gF23DrY/iTwBznXP1IvIN4CHg7FgrLCkpYWSotc7Pz6eoqOhQhT6cn0vndFlZGdeFioB+iCfWdGQuM6H1NTRQXFAAy5ZRunEj5OdTHLrjt7RVz5l4psvee4/rZs3ybH3Jmo6sCSS0voMHKS4shGHDKC0o0OlQA9Bl3p/r18PAgZRu2QJbtiT3/ZnkaV/sz1bT4Z/Ly8tJlFfpoNucczND04elg1otnwXscs5F7ZcXhCuB0oAMKJVwnCka49/3D2sJSTjOhgZN/fToobnykSOT8sAcX7w/S0o0JfbPf8ZcxBdxdkAQ4kx3OigbWIee2W8DlgGXOufeiVhmkHNue+jnzwLfc86dGmN9vm8EMoKN8e8d57Q+UFurPWMmTtQaQFf05pvaBfSZZ7RnzoUX6t26JqnS/nhJEZkJ3Iv2NvqTc26OiNwOLHfOPSkidwIXAPXALuBbzrn1MdZljUA6tR7jP9WFyK6mulpz/6NH65g2ybij2GS8RBoBT+4TcM4945wb65w71jk3J/TabOfck6Gfb3HOfcw5N8k5d3asBiAoIvNyftapOJ3Tm6f+8Q9tAAYNSlkDkNT7BDzUqTjr67UXVWOj3t07c2bKGoAu+f5Mo6DEGS+7Y9jomeqSJXrQ6t9f0z8mPs7p3b6NjdqNcsIE/3ehNRnNnjGcyVqP8d/OnaimHfv2aZ/+sWNh6tSWI2Mak0Rprwl4yRqBFGg9xn9hYerH+O9KwmP8FxRor5/Bg9Mdkckwaa8JZJqg5AijxtnBMf5TKbA1gaamlmP8X3yxLxqAQL8/fSgoccbLagKZwk9j/HcFe/bo13HHabdP60JrAsrSQV1dY6Peuv/KK3rmWliY/Ae7d2W1tXr2P3Cgnv13cmx6Y5Ih3cNGGL/y6xj/QRQe4z87W8f4HzPGGlPTJdi7OA6+zxGGxvgvveuuuMf4TyXf1wRC4+CX1tbqE6/GjfN1A+D792eIxekPdiXQlTQ06EM2XntND1IDBvhzjP+gOHBAH1wyeLA+dOWttxJ7/oAxPmQ1ga4iaGP8+1lHxvg3xkfsPoFMZmP8eytyjP9Jk3ydRjMmzO4TSDFf5Ajr6/Vu37lz9Spg2LDDGgDf59pDfBFnTY0Om33kkXDJJc2PS4zgi/97B1ic3gpKnPGymkDQpGiM/4wROcb/eefBiBFJGePfGL+ydFCQ2Bj/3smkMf5Nl2c1ga7Oxvj3lo3xb7oYqwmkWMpyhAmO8e+LXHsHpCzO8Bj/TU0wa1anx/hP9P9eX19PTU0NVVVVVFRU8MEHW1m79n1WrFjH4sWrWbZsDZWVlTQ1NSW0naDksC1Of7CagF/ZGP/eScEY//X19dTX11NXV0ddXR21tfXs319HTU1d6Hs9zmUDuaGvbojk0q1bX3JyutGtWy4HDtSyfHkFPXt+yOjR/Rk0aCA5OfYRNcll6SC/sTH+veXBGP/xH+BzDx3gc3K6kdXBew1qaw+yZ88OsrN3c/TR/RgyZCA97CY10warCXQFNsa/tzo4xn+qD/Cd0dDQQFXVRzQ1fcTQoT0YMaKQPnYHuInCGoEUKy0tpbi42LsVVlTA4sV60BowAPLyPFlt6erVFE+c6Mm6ksnTOJuadKgH56g/+WTqR4+mrrHRkwP866+XctJJxd7E2QnOOfbs2UVd3Q4KCmD06IEUFBTEbHg8f38micXpHRtFNKhsjP+4NTQ00NjYSH19PQ0NDdTXN1JfuZuGXbvZPXgkH40YS8N2ge1biJWDLyhIzhm810SE/PwjgSPZv38fy5fvsLqB8YxdCaSDjfHfpmgH+NraBmpr6zl4sIHa2kacy0LPYXKQ+kbyqqpo7D+I6klnQOHQpKVo/MLqBiaSpYOCJMPH+O/sAR5yEMkhO1u/cnJyyMrOJkuyoKmRnF0VkJVN9QmnUTsi88b4t7qBAWsEUi6uHGF1NSxdCmvXQn5+SoZ4TnVNIN4D/IrN65l2zPEtD/DtyN67m6yafdSMKeLAuBNxeck/C05XTaAjIusG77+/jM9//vw26wZ+EIRcOwQjTqsJ+FnrMf6HDg3k2Wr8Z/A9D53B9+od/QDfI68neR08iEvtAXJ2f0T9gMHsmX4ujfl29zS0rBusX1/O8uVVVjcwHWJXAskUkDH+PU3RJEtjIzm7duByc6kuOp26IV1vjP+6Orj4YpgxQ29knj8fhg+HoiL48EPYuhV+97uOr8/qBpnD0kF+46Mx/gNxgG9HdtVOsmprqBl3IgfGTsLlds06yosv6s3hEybo9Kc/DXPmNE8/+CCUlHR+vVY36PosHZRiMXOE9fWwejUsW6Zn/cOGJXVY4vYO8C+vXcfkkWOIJ0WTSks3rGbqMYfXLuRgDTlVldQNGs7+Ey6gsW96755Odk2gqkqvAkD7D1RV6eOMwwYN6th6WseZk5ND//5H4dwgKip2sWXLlg7db5BsQci1Q3DijJcnjYCIzAR+iQ5I9yfn3N2t5ucCfwEmA5XAF5xzm73Yti8kYYx/L3LwPXruoW/fUV78hanV2EC3nTto6t6DvdPPo+6o5I3x/+lP6wF36lS9T+9QCI064sT69XovX0mJLtNZP/6x3gReWQnHHgvTp8O3v908v6EBrr1WzxuOOkpj+cY3YOVKHd068qbxmTPj/jMBPVtcvPhIli07kp49azl48CCDB+/gwguzWby4P3fc0bXSa6ZjEk4HiUgWsB44G/gQWA5c4pxbG7HMt4CJzrlvi8gXgM865y6Jsb5gpYPiGOO/K6RoksI5cqoqkfpaaiZM4cAxE3HdkjvG//LlcNVV+iTJ3/8++jK/+pU2BHfcEd825s3TtM7f/gZjxhw+v6ICfvITXSZcv73zTm2Uvva1+LYZzezZen4S2QgtWlTHD36Qwy23lHPZZdlWNwiodKeDpgDvOuc2hYJ5GJgFrI1YZhYwO/TzY8B9Hmw3vVqP8T98OJDcXjRdWdaBanKqdlI7bDTVx59KU6/UjPF/8snwxS/qAfqBB+ArXzl8mauvhltuiX8bgwfrxeKWLdEbgQcegNtvb24AQMcPvPHG+LfZ2tNP61XN7be3fP3MM3MZOBBOP304Gzd+xLvvvmt1gwzjRSMwBPggYnoL2jBEXcY51ygiVSJS4Jzb5cH2U+6ZBx/klD17cNu3U9OngLqGPRzc8CF1dU1oYxw+wGcfOoPPysomN6c7ednZZGVFabCbGvSr3rs4X3v/HaaNGu/dCpPBwdJ1K5ly3ElUzZhFfeHQlD/e8aqr9IrgD3/QlE+4EBuWlaWpmnhrAkOH6vetWw+f99RTmiKKvICsqtIG42Mf6/SmgOhxLlyof0M0J54IffrkAKmtGwQl1x6UOOPlRSMQ7RPbOp/TehmJsswhJSUljBw5EoD8/HyKiooO/RPCD3hI5/Qrryyl74gTOJgzmvWbPiQ7O4dTxk0mNzuHZRtWATB1TBEAS9eXQWM9U8cU0QS8ur7s8PlJmq6vqmRxVWXKthfv9Ns5ORz78c9DTjdef133d/gglqrpO+4o5vLL4bvfreEHP3id0047o8X8a64pZsWK+Nbf2ChkZc1g69aW8ysr4cUXP+DiizcCzcuvXNmfsWM/Rl5efNtbt67ssPk9exbz7LPw05++y0knVXDmmacdmn/WWVmA/r0vv7yE+fNHMmrUcdTU1FJQsJDS0uN4/PFcVqwYyNNPb+btt/vwzDNH0q+ffh4WLRrAihUTmD+/+fMyeXIxs2dDZWU5DQ1ZnHvucO65B37xC51fXFzMY48N5Re/2EplZXeuuaY/S5ZAaekeJkzYyyOP6Dhazz33Io88Mpy+fUcxYAAsXbqVr33tPc4773QgNZ/3srIyXx1/wkpLSykvLydRXtQEpgG3OedmhqZvBlxkcVhEng4ts1REsoFtzrmBMdYXiJrA/v372bx5B5s378O5I8nPH0i3JOevTXLNmwd33aXPm7/tNm/XfcEF+gz7X/+6+bU774Rrrml+xME77+hgsosXw4EDWgg+//yO9wpqS3k5fOtbOlqJc5q9nDkTvvzl5pFLamrgm9+ET3wCvvQlfe3rX4e9e5uYPftDFi8Wvve9Bs4/fxhz5mTxmc/oMhddpI9nnjtXp/fvhzPPhEsugeuv19emT9dHZbz+uk6vWaM/n3EGHH00/Nd/6ZXYJz6ho6n87W/axyK8nhtu0N+bN0+L5bNmJb5PupJ01wSWA8eIyAhgG3AJcGmrZeYDVwBLgYuBhR5sN62OOOIIxo8/mqOPrmXbtgo2blxDXV1fevceSI8e9gD4IPrc5+CFF6BfP+/XPWSIpnjCnn4aTjut5TNuxo/Xr69/3fvtjxwJjz+ufRiWL9cb2O+/X4vSt96qy/zmN3qgDjcAoAfcE0/MYuPGoVx4YQMLFuxmwwahd+/32Lu3P3369GHxYm3Qwm65RRuUcAMAOjr6Kac0T+/apfv7+ec1FXbvvfr6c881L3P99XrLTbgB2L5db6CbM8fTXZPxEk70OecagauB54C3gYedc++IyO0icn5osT8B/UXkXeA64OZEt5tOkZdk3bt3Z+TIYRQXT2TSpJ7Ae1RUrGPfvirSfUUTTgX4nV/ifPttLeJee+3h87Zvjx7nsmVw7rnNZ7ixDBmi63BOu4uuXt18T0Aiom2/dZwHDuj37t31zPrGG/WMesYMrRWAnnU//njLbqjh214mTdIrkiOPzGHx4gGcfDJAX15+eQsPP7yBykpHcbE+F3nPHj2jv+yylut59VXddtj06RrnokXaGPbs2fLv2rsX/vpXGDUKfvpTbWT++U+45x7t4ZRK9ozhDnDOPQOMbfXa7Iifa4HPe7Etv8rOzmbQoEIKCwdSVVXFe+9tp6JiCzk5A8nP7+/rgbwMbNoE//hH7DTQ3/8e/aBdV6dnvQcPtr3+IUP03oNt27Q30DXXJBxyh7a/bZuebV9++eHzpkzREc1BeyPV1xM6wKvVq3X9kyY1v7ZwIXz1q83jFC1YcIABAxrYsuUd8vIGsHLlQGprsznrrObfWbpU13366YfHsHBhyyuPsPXrtQPezTdDF67J+oLdMRyHtnoKiAj9+vVj8uR+EXWDbWmpG/h1xMvW0h1nZaWmQn74w5bdNMNqajSVEy3O6dOhIyeKQ4boVcAf/6hnvn096gEbbfuRcS5dqo1PNK++qrUK0IM0aH4+bPlyvTLqHxqjb+9eTR9F9p5as6YHU6dCt25juPXWA4wYsRkYydFHHwT0foNFi7QHUu/ecPfdcNNN+rsTJhSzZk3LK4Sw3r21k1i0esjq1XojXap05Z5B4EE6yMQWrhsUF49j3DjH/v1rqKh4nwMH9qc7NBNSXa0Hpptvjv5Uz61bNWee6PBPQ4bo9wMH4OyzE1tXZyxdCk8+2bJ7qnOasqmthSuv1NcmTtSb3MOpo3Xr4OGHW14F5OVpATjcg3fTJq0tHHMM7NyZx9Ch/Zg6dRjdujkWLHif1avfZcmSah54AI4/Xq8qwusHvQro1QtOOunwuMeO1W2/9Vbzaw0N2lDvCmTHcv+yAeSboDXzAAAOP0lEQVTiEG+/4cbGRj76qJINGyrYty+XHj0K6dWrL5KkfvF+Hv8+UjrjvPpqvdk7P7/l601NOmzEjh160CspgSlT4o9z71649FJNK7Xeltci9+ctt8B3vqNF4OpqrQvs368H5csuazksxXPP6RAXw4drsfa++/SsPXy1ANqgLF6sB+mCAn0sxhNPaO7+qqv0Sur552HxYseAAQfo0WMPp5xSwwMPDOOEE7rx3e/KofsmvvGNjTg3mvvvj/53bNqk2x8/XuNsatLeTKHe4ykThPsEbBTRFEv0TeGcC9UNdlBR0ZC0uoE1At7KpDjXrtUD7r/+lfijr6ur91JTU0HPnjUcc8wACgsHkJOTE4iDK1gjkHJBaAS8ZPcbGD+aN0/HUnr2We/Wac83SB5rBLqA2trw/QY77X4Dk3a33ab5+7vvbnfRTrPnG3gvkUbACsNxSEa/4WTcb+CX/vftsTi9lUic69frTVrPPqv3Tfz8597FFRZ+vsHmzTupqCjg5Ze3sHTpGiorK2lqavJ+gwmy+wRMStn9BiadxoxJzoE/uub7Daqr97J8eQU9e37Yom5gks/SQQFgdQOTKaxuEB+rCWQIqxuYTGF1g86xmkCKpStH2Nm6QSbksFPJ4vRWW3GG6wYDBkxMe93AagLGd6xuYDJFU1MTeXk9yckZyo4dO9m8eRNHHbWVKVPGk5trKVEvWDqoi4isG0B/+vYdYHUD42v6GNY6GhrqaWioo76+Dn20Xl3oq56cHEfPnrmhr24ccUQu3bvnJu1pZ0FlNQFzSOu6QZ8+heTl9Wz/F43xUCIH+G7dupGbm0tubi7ZkeNamJisEUixINxG3tjYyL///W8GDjyWvXtz6dkzueMUJSKThmNIhWTH6dUBfvHixb7/HEEwPu/pfrKY8aHs7GwKCgo47bSPWd3AdFhnDvB9+0Ye4I+gW7d8O4MPILsSyCBWN8hslqLpuiwdZDrF6gZdjx3gM5s1AikWhBwhtB9n5PMN0lk3sFx72zp7gF+9+g3OOusM3x/gu8rnyA+sJmDiYvcbpF8ycvBVVVsZMmRwmv4iEzR2JWBasLqBdyxFY1LF0kHGc1Y3aJsd4I2fWCOQYkHIEYI3caaibuC3mkCsA/yqVYs5/vhJ+P0An0nvz1QIQpxWEzBJ09XqBonk4Bsa8pkx42g7gzddil0JmE7za93AUjQmU1k6yKRFKusGdoA3JjZrBFIsCDlCSF2cidYNli17geOPP9X3B3j7v3vL4vSO1QRMWrVVN+jTpx8NDQ1tHuBrat5DZJCNRWNMGiR0JSAi/YBHgBFAOfB559yeKMs1Am8CAmxyzn2mjXX6/krAtC9cN9i2bS95ed0sRWNMEqUtHSQidwM7nXM/EZGbgH7OuZujLLfXOdehB4RaI2CMMZ2TzmcMzwIeCv38EBDrDN9/g9gnICjPHLU4vWVxesvi9IdEG4GBzrkdAM657cCAGMt1F5FlIvKKiMxKcJvGGGM80m5hWEQWAIWRLwEO+EEntjPcObddREYBC0VklXPu/VgLl5SUMHLkSADy8/MpKio6VJ0Pt8rpng7zSzzRpouLi30VT1vTYX6Jx/Zn8qdtfyYWT2lpKeXl5SQq0ZrAO0Cxc26HiAwCFjnnxrfzO38G5jvn5sWYbzUBY4zphHTWBJ4ASkI/XwH8u/UCIpIvIrmhn/sDpwJrEtxuWrU+O/Ari9NbFqe3LE5/SLQRuBv4uIisA84B5gCIyGQRuT+0zHjgdRFZCbwA3OWcW5vgdo0xxnjA7hg2xpiAS2c6yBhjTIBZIxCHoOQILU5vWZzesjj9wRoBY4zJYFYTMMaYgLOagDHGmLhYIxCHoOQILU5vWZzesjj9wRoBY4zJYFYTMMaYgLOagDHGmLhYIxCHoOQILU5vWZzesjj9wRoBY4zJYFYTMMaYgLOagDHGmLhYIxCHoOQILU5vWZzesjj9wRoBY4zJYFYTMMaYgLOagDHGmLhYIxCHoOQILU5vWZzesjj9wRoBY4zJYFYTMMaYgLOagDHGmLhYIxCHoOQILU5vWZzesjj9wRoBY4zJYFYTMMaYgLOagDHGmLhYIxCHoOQILU5vWZzesjj9wRoBY4zJYFYTMMaYgEtbTUBELhKRt0SkUURObGO5mSKyVkTWi8hNiWzTGGOMdxJNB60GPgu8GGsBEckC7gM+CUwALhWRcQluN62CkiO0OL1lcXrL4vSHnER+2Tm3DkBE2roMmQK865zbFFr2YWAWsDaRbRtjjEmcJzUBEVkE3OCcWxFl3oXAJ51zXw9NfwmY4pz77xjrspqAMcZ0QiI1gXavBERkAVAY+RLggFudc/M7El+U1+wob4wxPtBuI+Cc+3iC29gCDI+YHgp82NYvlJSUMHLkSADy8/MpKiqiuLgYaM7PpXO6rKyM6667zjfxxJqOzGX6IZ5Y07Y/bX/6IZ5Y037cn+Gfy8vLSZhzLuEvYBEwOca8bGADMALIBcqA8W2sy/ndokWL0h1Ch1ic3rI4vWVxeid03Izr+J1QTUBEPgP8GugPVAFlzrlPichRwB+cc+eHlpsJ3Iv2RvqTc25OG+t0icRkjDGZJpGagN0sZowxAWcDyKVYZF7OzyxOb1mc3rI4/cEaAWOMyWCWDjLGmICzdJAxxpi4WCMQh6DkCC1Ob1mc3rI4/cEaAWOMyWBWEzDGmICzmoAxxpi4WCMQh6DkCC1Ob1mc3rI4/cEaAWOMyWBWEzDGmICzmoAxxpi4WCMQh6DkCC1Ob1mc3rI4/cEaAWOMyWBWEzDGmICzmoAxxpi4WCMQh6DkCC1Ob1mc3rI4/cEaAWOMyWBWEzDGmICzmoAxxpi4WCMQh6DkCC1Ob1mc3rI4/cEaAWOMyWBWEzDGmICzmoAxxpi4WCMQh6DkCC1Ob1mc3rI4/cEaAWOMyWBWEzDGmICzmoAxxpi4JNQIiMhFIvKWiDSKyIltLFcuIm+KyEoRWZbINv0gKDlCi9NbFqe3LE5/SPRKYDXwWeDFdpZrAoqdc5Occ1MS3GbalZWVpTuEDrE4vWVxesvi9IecRH7ZObcOQETay0UJXSj1VFVVle4QOsTi9JbF6S2L0x9SdWB2wLMislxEvpaibRpjjGlHu1cCIrIAKIx8CT2o3+qcm9/B7ZzqnNsuIgOABSLyjnNuSefD9Yfy8vJ0h9AhFqe3LE5vWZz+4EkXURFZBNzgnFvRgWVnA/uccz+PMd/6hxpjTCfF20U0oZpAK1EDEJGeQJZzrlpEjgA+AdweayXx/iHGGGM6L9Euop8RkQ+AacCTIvJ06PWjROTJ0GKFwBIRWQm8Bsx3zj2XyHaNMcZ4w3d3DBtjjEmdtHfbFJF+IvKciKwTkWdFpG+M5RpFZEXohrPHUxTbTBFZKyLrReSmKPNzReRhEXlXRF4VkeGpiCuOOK8QkYrQ/lshIl9JQ4x/EpEdIrKqjWV+FdqXZSJSlMr4ImJoM04RmSEiVRH78gepjjEUx1ARWSgia0RktYj8d4zl0rZPOxKjH/aniHQXkaWhY8vqUN2y9TJp/6x3MM7Of9adc2n9Au4Gbgz9fBMwJ8Zye1McVxawARgBdAPKgHGtlvkW8NvQz18AHk7D/utInFcAv0rz/3k6UASsijH/U8B/Qj9PBV7zaZwzgCfSuS9DcQwCikI/9wLWRfm/p3WfdjBGv+zPnqHv2Wjaekqr+Wn/rHcwzk5/1tN+JQDMAh4K/fwQ8JkYy6W6YDwFeNc5t8k5Vw88jMYaKTL2x4CzUxhfWEfihNTvvxacdgne3cYis4C/hJZdCvQVkcI2lk+KDsQJad6XAM657c65stDP1cA7wJBWi6V1n3YwRvDH/qwJ/dgd7TDTOk/uh896R+KETu5PPzQCA51zO0DfNMCAGMt1F5FlIvKKiEQ7yHltCPBBxPQWDn8DH1rGOdcIVIlIQQpiixpDSLQ4AT4XSgk8KiJDUxNap7T+O7YS/e/wg2mhS/L/iMhx6Q5GREaiVy9LW83yzT5tI0bwwf4UkaxQ55XtwALn3PJWi/jhs96ROKGTn/WUNAIiskBEVkV8rQ59v6ATqxnudNyhLwK/FJFRSQo3LFpr2rrVbb2MRFkm2ToS5xPASOdcEfACzWc0ftKRv8MP3gBGOOcmAfcBKalPxSIivdAz02tDZ9stZkf5lZTv03Zi9MX+dM41hWIYCkyN0hj54bPekTg7/VlPSSPgnPu4c+74iK+Joe9PADvCl6giMgioiLGO7aHv7wOlwKQkh70FiCz+DAU+bLXMB8AwABHJBvo459pLJXit3Tidc7tDqSKAPwCTUxRbZ2whtC9Dou3vtHPOVYcvyZ1zTwPd0nFGCCAiOejB9a/OuX9HWSTt+7S9GP20P0Mx7EWPLzNbzfLDZ/2QWHHG81n3QzroCaAk9PMVwGFvFBHJF5Hc0M/9gVOBNUmOazlwjIiMCG37klCskeajMQNcDCxMckzRtBtnqHENm0Xy910sQux85RPAlwFEZBpQFU4TpkHMOCNz6iIyBe1mvStVgbXyALDGOXdvjPl+2KdtxuiH/Ski/SXUK1FEegDnAGtbLZb2z3pH4ozrs56OCneranYB8Dzac2ABkB96fTJwf+jnU4BVwErgTaAkRbHNDMX1LnBz6LXbgfNDP3cHHg3Nfw29DEvHPmwvzjuBt0L77wVgTBpinIuehdYCm4ErgW8AX49Y5j60p9ObwIlp2pdtxglcFbEvXwGmpinO04BGtDfYSmBF6H3gm33akRj9sD+BiaHYykLHmVtDr/vqs97BODv9WbebxYwxJoP5IR1kjDEmTawRMMaYDGaNgDHGZDBrBIwxJoNZI2CMMRnMGgFjjMlg1ggYY0wGs0bAGGMy2P8HF74GI87mD5gAAAAASUVORK5CYII=",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6e32d9c7f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_transformation(S.dot(V_T).dot(Square), U.dot(S).dot(V_T).dot(Square),\"$\\Sigma \\cdot V^T \\cdot Square$\", \"$U \\cdot \\Sigma \\cdot V^T \\cdot Square$\",\n",
    "                    axis=[-0.5, 3.5 , -1.5, 1.5])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Và chúng ta có thể thấy rằng kết quả thực sự là một ánh xạ cắt của hình vuông đơn vị ban đầu."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vector riêng và trị riêng\n",
    "Một **vector riêng** (**eigenvector**) của một ma trận vuông $M$ (còn được gọi là **characteristic vector**) là một vector khác không nằm trên cùng một dòng sau khi biến đổi bởi một phép biến đổi tuyến tính có liên quan đến $M$. Một định nghĩa chính thức hơn là bất kỳ vector nào $v$ sao cho:\n",
    "\n",
    "$M \\cdot v = \\lambda \\times v$\n",
    "\n",
    "Trong đó $\\lambda$ là một giá trị vô hướng được gọi là **trị riêng** (**eigenvalue**) được liên kết với vector $v$.\n",
    "\n",
    "Ví dụ, bất kỳ vector ngang nào vẫn nằm ngang sau khi áp dụng ánh xạ cắt (như bạn có thể thấy trên hình trên), vì vậy nó là một vector riêng của $M$. Một vector thẳng đứng kết thúc nghiêng về bên phải, vì vậy vector thẳng đứng *KHÔNG* là các vector riêng của $M$.\n",
    "\n",
    "Nếu chúng ta nhìn vào ánh xạ bóp, chúng ta thấy rằng bất kỳ vector ngang hoặc dọc nào đều giữ nguyên hướng của nó (mặc dù độ dài của nó thay đổi), vì vậy tất cả các vector ngang và dọc đều là ký hiệu riêng của $F_{shear}$.\n",
    "\n",
    "Tuy nhiên, ma trận xoay hoàn toàn không có vector riêng (ngoại trừ nếu góc quay là 0° hoặc 180°, trong trường hợp đó tất cả các vector khác 0 đều là vector riêng).\n",
    "\n",
    "Hàm `eig` của NumPy trả về danh sách các vector riêng đơn vị và các giá trị riêng tương ứng của chúng cho bất kỳ ma trận vuông nào. Hãy cùng xem xét các vector riêng và giá trị riêng của ma trận ánh xạ nén $F_{squeeze}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.4       ,  0.71428571])"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigenvalues, eigenvectors = LA.eig(F_squeeze)\n",
    "eigenvalues # [λ0, λ1, …]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.,  0.],\n",
       "       [ 0.,  1.]])"
      ]
     },
     "execution_count": 133,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigenvectors # [v0, v1, …]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thật vậy, các vector ngang được giãn theo hệ số 1.4, và các vector dọc được co theo hệ số 1 / 1.4 = 0.714…, cho đến nay vẫn rất tốt. Hãy xem xét ma trận ánh xạ cắt $F_{shear}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.,  1.])"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigenvalues2, eigenvectors2 = LA.eig(F_shear)\n",
    "eigenvalues2 # [λ0, λ1, …]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  1.00000000e+00,  -1.00000000e+00],\n",
       "       [  0.00000000e+00,   1.48029737e-16]])"
      ]
     },
     "execution_count": 135,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigenvectors2 # [v0, v1, …]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Chờ một chút. Chúng ta chỉ mong đợi một vector riêng đơn vị, không phải hai. Vector thứ hai gần như bằng với $\\begin{pmatrix}-1 \\\\ 0 \\end{pmatrix}$, nằm trên cùng hàng với vector đầu tiên $\\begin{pmatrix}1 \\\\ 0 \\end{pmatrix}$. Điều này do lỗi của dấu chấm động. Chúng ta có thể bỏ qua một cách an toàn các vector (gần như) đồng tuyến (tức là trên cùng một dòng)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vết\n",
    "Vết (_trace_) của một ma trận vuông $M$, ký hiệu $tr(M)$ là tổng của các giá trị trên đường chéo chính của nó. Ví dụ:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "123"
      ]
     },
     "execution_count": 136,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = np.array([\n",
    "        [100, 200, 300],\n",
    "        [ 10,  20,  30],\n",
    "        [  1,   2,   3],\n",
    "    ])\n",
    "np.trace(D)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vết không có một cách giải thích hình học nào đơn giản (nói chung), nhưng nó có một số tính chất có ích trong nhiều lĩnh vực:\n",
    "* $tr(A + B) = tr(A) + tr(B)$\n",
    "* $tr(A \\cdot B) = tr(B \\cdot A)$\n",
    "* $tr(A \\cdot B \\cdot \\cdots \\cdot Y \\cdot Z) = tr(Z \\cdot A \\cdot B \\cdot \\cdots \\cdot Y)$\n",
    "* $tr(A^T \\cdot B) = tr(A \\cdot B^T) = tr(B^T \\cdot A) = tr(B \\cdot A^T) = \\sum_{i,j}X_{i,j} \\times Y_{i,j}$\n",
    "* …\n",
    "\n",
    "Tuy nhiên, nó có một giải thích hình học hữu ích trong trường hợp ma trận chiếu (chẳng hạn như $F_{project}$ mà chúng ta đã thảo luận trước đó): nó tương ứng với số chiều sau khi chiếu."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 137,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.trace(F_project)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Tiếp theo là gì?\n",
    "Notebook này kết thúc phần giới thiệu về Đại số tuyến tính. Mặc dù những điều cơ bản này bao gồm hầu hết những gì bạn cần biết về Học máy, nhưng nếu bạn muốn đi sâu hơn vào chủ đề này, có rất nhiều lựa chọn có sẵn: [Sách] Đại số tuyến tính (http://linear.axler.net/), [Khan Academy](https://www.khanacademy.org/math/linear-algebra), hoặc chỉ các trang [Wikipedia](https://en.wikipedia.org/wiki/Linear_algebra)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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