linearalgebra_practice104105106.ipynb

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   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
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    "%%html\n",
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  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math, sympy, numpy\n",
    "sympy.init_printing(use_unicode=True)\n",
    "from sympy import *\n",
    "import pylab as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Computer Algebra I: Mathematica, SymPy, Sage, Maxima](http://hyperpolyglot.org/computer-algebra2)\n",
    "\n",
    "[Sympy. Matrices](https://docs.sympy.org/latest/tutorial/matrices.html)\n",
    "\n",
    "[LaTeX/Mathematics](https://en.wikibooks.org/wiki/LaTeX/Mathematics)\n",
    "\n",
    "[МЦНМО. Свободно распространяемые издания](https://www.mccme.ru/free-books/)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Типовой расчет 4 / Standard Calculation 4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Постановка  задачи / Formulation of the problem:\n",
    "Найти ранг данной матрицы методом Гаусса\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "-1 & -1 & 2 & -3 & -1 & 1 \\\\\n",
    "1 & -2 & -1 & -3 & 1 & -1 \\\\\n",
    "-2 & -1 & -2 & -1 & -3 & 1 \\\\\n",
    "-3 & -1 & -1 & 1 & -1 & 1 \\\\\n",
    "-3 & -3 & -3 & -3 & -1 & 1\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
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      "text/latex": [
       "$$\\left[\\begin{matrix}-1 & -1 & 2 & -3 & -1 & 1\\\\1 & -2 & -1 & -3 & 1 & -1\\\\-2 & -1 & -2 & -1 & -3 & 1\\\\-3 & -1 & -1 & 1 & -1 & 1\\\\-3 & -3 & -3 & -3 & -1 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-1  -1  2   -3  -1  1 ⎤\n",
       "⎢                      ⎥\n",
       "⎢1   -2  -1  -3  1   -1⎥\n",
       "⎢                      ⎥\n",
       "⎢-2  -1  -2  -1  -3  1 ⎥\n",
       "⎢                      ⎥\n",
       "⎢-3  -1  -1  1   -1  1 ⎥\n",
       "⎢                      ⎥\n",
       "⎣-3  -3  -3  -3  -1  1 ⎦"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sympy.Matrix([[-1,-1, 2,-3 ,-1,1],[1,-2,-1,-3,1,-1],[-2,-1,-2,-1,-3,1],[-3,-1,-1,1,-1,1],[-3 ,-3 ,-3,-3,-1,1]])\n",
    "A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение / Decision\n",
    "\n",
    "Приводим матрицу к ступенчатому виду.\n",
    "  \n",
    "Ко 2-ой, 3-ей, 4-ой и 5-ой строке прибавим 1-ую, умноженную на 1,-1, -1, -1 соответственно:\n",
    " \n",
    "$\\begin{pmatrix}\n",
    "-1 & -1 & 2 & -3 & -1 & 1 \\\\\n",
    "1 & -2 & -1 & -3 & 1 & -1 \\\\  \n",
    " -2 & -1 & -2 & -1 & -3 & 1 \\\\\n",
    " -3 & -1 & -1 & 1 & -1 & 1 \\\\\n",
    " -3 & -3 & -3 & -3 & -1 & 1\n",
    "\\end{pmatrix} = \n",
    "\\begin{pmatrix}\n",
    "-1 & -1 & 2 & -3 & -1 & 1 \\\\\n",
    "0 & -3 & 1 & -6 & 0 & 0 \\\\\n",
    "-1 & 0 & -4 & 2 & -2 & 0 \\\\ \n",
    "-2 & 0 & -3 & 4 & 0 & 0 \\\\\n",
    "-2 & -2 & -5 & 0 & 0 & 0\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}0 & -3 & 1 & -6 & 0 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}-1 & 0 & -4 & 2 & -2 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}-2 & 0 & -3 & 4 & 0 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}-2 & -2 & -5 & 0 & 0 & 0\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "([0  -3  1  -6  0  0], [-1  0  -4  2  -2  0], [-2  0  -3  4  0  0], [-2  -2  -\n",
       "5  0  0  0])"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(1, 6, [1,-2,-1,-3,1,-1])+sympy.Matrix(1, 6, [-1,-1, 2,-3 ,-1,1]), \\\n",
    "sympy.Matrix(1, 6, [-2,-1,-2,-1,-3,1])-sympy.Matrix(1, 6, [-1,-1, 2,-3 ,-1,1]), \\\n",
    "sympy.Matrix(1, 6, [-3,-1,-1,1,-1,1])-sympy.Matrix(1, 6, [-1,-1, 2,-3 ,-1,1]), \\\n",
    "sympy.Matrix(1, 6, [-3 ,-3 ,-3,-3,-1,1])-sympy.Matrix(1, 6, [-1,-1, 2,-3 ,-1,1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Поменяем 2-ую и 3-ю строки местами, \n",
    "\n",
    "первую строчку умножим на -2 и сложим со 2-ой,\n",
    "\n",
    "умножим получившуюся 3-ю строку на 2 и сложим с 4-ой, умноженной на 3:\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "1 & 2 & -8 & 8 & 0 & -2 \\\\\n",
    "-1 & 0 & -4 & 2 & -2 & 0 \\\\ \n",
    "0 & -3 & 1 & -6 & 0 & 0 \\\\\n",
    "-2 & 0 & -3 & 4 & 0 & 0 \\\\\n",
    "-2 & -2 & -5 & 0 & 0 & 0\n",
    "\\end{pmatrix} = \n",
    "\\begin{pmatrix}\n",
    "1 & 2 & -8 & 8 & 0 & -2 \\\\\n",
    "-1 & 0 & -4 & 2 & -2 & 0 \\\\ \n",
    "-6 & -6 & -7 & 0 & 0 & 0 \\\\\n",
    "-2 & 0 & -3 & 4 & 0 & 0 \\\\\n",
    "-2 & -2 & -5 & 0 & 0 & 0\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}1 & 2 & -8 & 8 & 0 & -2\\end{matrix}\\right], \\quad \\left[\\begin{matrix}-6 & -6 & -7 & 0 & 0 & 0\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "([1  2  -8  8  0  -2], [-6  -6  -7  0  0  0])"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(1, 6, [-1,0,-4,2,-2,0])+sympy.Matrix(1, 6, [-1,-1, 2,-3 ,-1,1]) * (-2), \\\n",
    "sympy.Matrix(1, 6, [0,-3,1,-6,0,0])*2+sympy.Matrix(1, 6, [-2,0,-3,4,0,0])*3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2-ую строку умножим на -2 и сложим с 4-ой, \n",
    "\n",
    "4-ую умножим на -2 и сложим с 1-ой,\n",
    "\n",
    "переставим 3-ю и 4-ую строки местами:\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "5 & 2 & -2 & 0 & 0 & -2 \\\\\n",
    "0 & 0 & 5 & 0 & 4 & 0 \\\\ \n",
    "-6 & -6 & -7 & 0 & 0 & 0 \\\\\n",
    "-2 & 0 & -3 & 4 & 0 & 0 \\\\\n",
    "-2 & -2 & -5 & 0 & 0 & 0\n",
    "\\end{pmatrix} = \n",
    "\\begin{pmatrix}\n",
    "5 & 2 & -2 & 0 & 0 & -2 \\\\\n",
    "0 & 0 & 5 & 0 & 4 & 0 \\\\ \n",
    "-2 & 0 & -3 & 4 & 0 & 0 \\\\\n",
    "-6 & -6 & -7 & 0 & 0 & 0 \\\\\n",
    "-2 & -2 & -5 & 0 & 0 & 0\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}0 & 0 & 5 & 0 & 4 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}5 & 2 & -2 & 0 & 0 & -2\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "([0  0  5  0  4  0], [5  2  -2  0  0  -2])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(1, 6, [-1,0,-4,2,-2,0])*(-2)+sympy.Matrix(1, 6, [-2,0,-3,4,0,0]), \\\n",
    "sympy.Matrix(1, 6, [1,2,-8,8,0,-2])+sympy.Matrix(1, 6, [-2,0,-3,4,0,0])*(-2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "К 4-ой строке прибавим 5-ую, умноженную на -1,\n",
    "\n",
    "разделим получившуюся 4-ую строку на -2:\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "5 & 2 & -2 & 0 & 0 & -2 \\\\\n",
    "0 & 0 & 5 & 0 & 4 & 0 \\\\ \n",
    "-2 & 0 & -3 & 4 & 0 & 0 \\\\\n",
    "-4 & -4 & -2 & 0 & 0 & 0 \\\\\n",
    "-2 & -2 & -5 & 0 & 0 & 0\n",
    "\\end{pmatrix} = \n",
    "\\begin{pmatrix}\n",
    "5 & 2 & -2 & 0 & 0 & -2 \\\\\n",
    "0 & 0 & 5 & 0 & 4 & 0 \\\\ \n",
    "-2 & 0 & -3 & 4 & 0 & 0 \\\\\n",
    "2 & 2 & 1 & 0 & 0 & 0 \\\\\n",
    "-2 & -2 & -5 & 0 & 0 & 0\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 2 & 1 & 0 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[2  2  1  0  0  0]"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sympy.Matrix(1, 6, [-6,-6,-7,0,0,0])-sympy.Matrix(1, 6, [-2,-2,-5,0,0,0]))/(-2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Умножим 4-ую строку на 2, -5, 3, 5 соответственно и сложим с 1-ой, 2-ой, 3-ей и 5-ой строками,\n",
    "\n",
    "разделим 2-ую и 3-ю строку на 2, 5-ую строку на 8\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "9 & 6 & 0 & 0 & 0 & -2 \\\\\n",
    "-10 & -10 & 0 & 0 & 4 & 0 \\\\ \n",
    "4 & 6 & 0 & 4 & 0 & 0 \\\\\n",
    "2 & 2 & 1 & 0 & 0 & 0 \\\\\n",
    "8 & 8 & 0 & 0 & 0 & 0\n",
    "\\end{pmatrix} = \n",
    "\\begin{pmatrix}\n",
    "9 & 6 & 0 & 0 & 0 & -2 \\\\\n",
    "-5 & -5 & 0 & 0 & 2 & 0 \\\\ \n",
    "2 & 3 & 0 & 2 & 0 & 0 \\\\\n",
    "2 & 2 & 1 & 0 & 0 & 0 \\\\\n",
    "1 & 1 & 0 & 0 & 0 & 0\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}9 & 6 & 0 & 0 & 0 & -2\\end{matrix}\\right], \\quad \\left[\\begin{matrix}-5 & -5 & 0 & 0 & 2 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}2 & 3 & 0 & 2 & 0 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1 & 1 & 0 & 0 & 0 & 0\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "([9  6  0  0  0  -2], [-5  -5  0  0  2  0], [2  3  0  2  0  0], [1  1  0  0  0\n",
       "  0])"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(1, 6, [5,2,-2,0,0,-2])+sympy.Matrix(1, 6, [2,2,1,0,0,0])*2, \\\n",
    "(sympy.Matrix(1, 6, [0,0,5,0,4,0])+sympy.Matrix(1, 6, [2,2,1,0,0,0])*(-5))/2, \\\n",
    "(sympy.Matrix(1, 6, [-2,0,-3,4,0,0])+sympy.Matrix(1, 6, [2,2,1,0,0,0])*3)/2, \\\n",
    "(sympy.Matrix(1, 6, [-2,-2,-5,0,0,0])+sympy.Matrix(1, 6, [2,2,1,0,0,0])*5)/8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Сложим 5-ую строку, умноженную соответственно на -6, 5, -3, -2 на 1-ую, 2-ую, 3-ю и 4-ую строки,\n",
    "\n",
    "разделим 1-ую, 2-ую, 3-ю строки на -2, 2 и 2 соответственно:\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "3 & 0 & 0 & 0 & 0 & -2 \\\\\n",
    "0 & 0 & 0 & 0 & 2 & 0 \\\\ \n",
    "-1 &  0 & 0 & 2 & 0 & 0 \\\\\n",
    "0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "1 & 1 & 0 & 0 & 0 & 0\n",
    "\\end{pmatrix} = \n",
    "\\begin{pmatrix}\n",
    "-3/2 & 0 & 0 & 0 & 0 & 1 \\\\\n",
    "0 & 0 & 0 & 0 & 1 & 0 \\\\ \n",
    "-1/2 & 0 & 0 & 1 & 0 & 0 \\\\\n",
    "0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "1 & 1 & 0 & 0 & 0 & 0\n",
    "\\end{pmatrix}$\n",
    "\n",
    "В матрице 5 линейно независимых строк. $\\implies$ Ранг матрицы равен 5."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}- \\frac{3}{2} & 0 & 0 & 0 & 0 & 1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & 0 & 0 & 1 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}- \\frac{1}{2} & 0 & 0 & 1 & 0 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & 1 & 0 & 0 & 0\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "([-3/2  0  0  0  0  1], [0  0  0  0  1  0], [-1/2  0  0  1  0  0], [0  0  1  0\n",
       "  0  0])"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sympy.Matrix(1, 6, [9,6,0,0,0,-2])+sympy.Matrix(1, 6, [1,1,0,0,0,0])*(-6))/(-2), \\\n",
    "(sympy.Matrix(1, 6, [-5,-5,0,0,2,0])+sympy.Matrix(1, 6, [1,1,0,0,0,0])*5)/2, \\\n",
    "(sympy.Matrix(1, 6, [2,3,0,2,0,0])+sympy.Matrix(1, 6, [1,1,0,0,0,0])*(-3))/2, \\\n",
    "sympy.Matrix(1, 6, [2,2,1,0,0,0])+sympy.Matrix(1, 6, [1,1,0,0,0,0])*(-2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Приведем матрицу к ступенчатому виду другим способом.\n",
    "  \n",
    "Ко 2-ой, 3-ей, 4-ой и 5-ой строке прибавим 1-ую, умноженную на 1,-2, -3, -3 соответственно:\n",
    " \n",
    "$\\begin{pmatrix}\n",
    "-1 & -1 & 2 & -3 & -1 & 1 \\\\\n",
    "1 & -2 & -1 & -3 & 1 & -1 \\\\  \n",
    " -2 & -1 & -2 & -1 & -3 & 1 \\\\\n",
    " -3 & -1 & -1 & 1 & -1 & 1 \\\\\n",
    " -3 & -3 & -3 & -3 & -1 & 1\n",
    "\\end{pmatrix} = \n",
    "\\begin{pmatrix}\n",
    "-1 & -1 & 2 & -3 & -1 & 1 \\\\\n",
    "0 & -3 & 1 & -6 & 0 & 0 \\\\\n",
    "0 & 1 & -6 & 5 & -1 & -1 \\\\ \n",
    "0 & 2 & -7 & 10 & 2 & -2 \\\\\n",
    "0 & 0 & -9 & 6 & 2 & -2\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}0 & -3 & 1 & -6 & 0 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 1 & -6 & 5 & -1 & -1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 2 & -7 & 10 & 2 & -2\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & -9 & 6 & 2 & -2\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "([0  -3  1  -6  0  0], [0  1  -6  5  -1  -1], [0  2  -7  10  2  -2], [0  0  -9\n",
       "  6  2  -2])"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(1, 6, [1,-2,-1,-3,1,-1])+sympy.Matrix(1, 6, [-1,-1, 2,-3 ,-1,1]), \\\n",
    "sympy.Matrix(1, 6, [-2,-1,-2,-1,-3,1])-2*sympy.Matrix(1, 6, [-1,-1, 2,-3 ,-1,1]), \\\n",
    "sympy.Matrix(1, 6, [-3,-1,-1,1,-1,1])-3*sympy.Matrix(1, 6, [-1,-1, 2,-3 ,-1,1]), \\\n",
    "sympy.Matrix(1, 6, [-3 ,-3 ,-3,-3,-1,1])-3*sympy.Matrix(1, 6, [-1,-1, 2,-3 ,-1,1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "К 1-ой строке прибавим 3-ю, из 4-ой вычтем 5-ую,\n",
    "\n",
    "разделим 4-ую строку на 2:\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "-1 & 0 & -4 & 2 & -2 & 0 \\\\\n",
    "0 & -3 & 1 & -6 & 0 & 0 \\\\\n",
    "0 & 1 & -6 & 5 & -1 & -1 \\\\ \n",
    "0 & 2 & 2 & 4 & 0 & 0 \\\\\n",
    "0 & 0 & -9 & 6 & 2 & -2\n",
    "\\end{pmatrix} = \n",
    "\\begin{pmatrix}\n",
    "-1 & 0 & -4 & 2 & -2 & 0 \\\\\n",
    "0 & -3 & 1 & -6 & 0 & 0 \\\\\n",
    "0 & 1 & -6 & 5 & -1 & -1 \\\\ \n",
    "0 & 1 & 1 & 2 & 0 & 0 \\\\\n",
    "0 & 0 & -9 & 6 & 2 & -2\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAc4AAAAZBAMAAABKhNOgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVESZEM27ZiLviTJ23asWSsTfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAD4ElEQVRYCeWYzWsTURTFT5rUTJM2LS7d2P+ghVBwJbNxoajVjTu1K0UECYigCzUgiBu1WxF0FuJCBLNy4caALhSsVHShSDCCuvGjWusHfhDvuy/z5r43b9LVdONAOufeN3PO/WUmEw2A9cDHThNrty10JnxhuQzBWZGKK0wDu3y5+fXKDZ93PkOorNsqbgu98onwwXBvrTnLbYo9SC/JWX3Gs8R/Kjc3xdLeV6dlHdT3tGQd6zMzh2PJ+xv3mkAmpxvmzAKn9poncezGWYQUzNOC4Nw5s5IcSWo3djStRlyMdWOl9hdQWZZ1Xwdd3JoU/eGJ0vMBnE6YO4tT+82TNHZjzsdAMaIFwYlRi7O8iMpccqpQC11R4A1wSdZ9XQsxot7IeDsJXM/mTIXZs8CZzW8eR0G7MedsiPFJWsjmXNdFwQKPbarHurFU+6shlpqyofXICmo/RfsIsDHMvG9TYYM5/eYmTbsx53ADd0NayOYcn8bQF3OqEJXhrqjwEVhqyYbWtUXUfov2V2BqIpMzFTaY029u0rQbcxbnsKD62ZyzxPnDnCrEA5uTVl6JVSHHPidFSXE2MjlTYYM5yddjbtK0G3OOzuOt6mdzXoxQ/WNOTUTQcDm9h9EJd9virF9cc3bS7atdSIWtyukxN77ajbOGVnBA9Qdz0nCprQKXc2w6dRA3nop2oDijzOupJrPCVuX0mJs47cac1d/Yp/rEuaWjtpfuM202wpC8nvFhx1OcdRNAorSX7Vr0PJf4JaKYamdyumHO89WdzW9uxtBuzBksg98R4jSb/R7Sh7ngeQ4FbZezLHmMGXBUaOArPW8HPofsMHuWNKfP3OTp0ZmztGyup1m2vdfNoez5Xhn69GnpSmTOIbEBQVPWfV2IcE606VNyesD3ihtmz5Li9JqbND26vp7JfWuWbe/yPIryapvDMNJNNFCNMBrKRl/fAe6L9glgBpn3bSrMniXF6TU3adpNfz5/4p3qSxLH+wm2tsypUozPyep8vU5foakteFrfvCi6YxOlyzFnKPpK0hBumDOL83n1myeu7Mac9Lx92I+Il2uH/n6Itdpve/RelkYX939vmwLY3+t9E2Ush3u9Hs1vttLNx60+Z8k9no5zwtxZnNpvbrK0G3PSpZ0KaUGOkhyXm+JsdFSy2PIZgrOKXYxMUlQ+EYLBlpqzENrdfIbgLPr3bTGitHwibAxRac6K6CiZzxCcNRuiOp9bhMORlJpze9JglSPnKQp4Ta98IhyOpNScjaTBKp8hOIv+b//f/D70H/zed5bvF/r9dm+nyXJt/iy8cG9Zzs1lCM66BvwD3mNLVbPDWXEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}-1 & 0 & -4 & 2 & -2 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 1 & 1 & 2 & 0 & 0\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "([-1  0  -4  2  -2  0], [0  1  1  2  0  0])"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(1, 6, [-1,-1,2,-3,-1,1])+sympy.Matrix(1, 6, [0,1,-6,5,-1,-1]), \\\n",
    "(sympy.Matrix(1, 6, [0,2,-7,10,2,-2])-sympy.Matrix(1, 6, [0,0,-9,6,2,-2])) / 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4-ую строку умножим на 3 и на -1 соответственно и сложим со 2-ой и 3-ей строками,\n",
    "\n",
    "поделим 2-ую строку на 4, переместим 4-ую строку на 2-ую: \n",
    "\n",
    "$\\begin{pmatrix}\n",
    "-1 & 0 & -4 & 2 & -2 & 0 \\\\\n",
    "0 & 0 & 4 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & -7 & 3 & -1 & -1 \\\\ \n",
    "0 & 1 & 1 & 2 & 0 & 0 \\\\\n",
    "0 & 0 & -9 & 6 & 2 & -2\n",
    "\\end{pmatrix} = \n",
    "\\begin{pmatrix}\n",
    "-1 & 0 & -4 & 2 & -2 & 0 \\\\\n",
    "0 & 1 & 1 & 2 & 0 & 0 \\\\\n",
    "0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & -7 & 3 & -1 & -1 \\\\ \n",
    "0 & 0 & -9 & 6 & 2 & -2\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}0 & 0 & 1 & 0 & 0 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & -7 & 3 & -1 & -1\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "([0  0  1  0  0  0], [0  0  -7  3  -1  -1])"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sympy.Matrix(1, 6, [0,-3,1,-6,0,0])+sympy.Matrix(1, 6, [0,1,1,2,0,0])*3)/4, \\\n",
    "(sympy.Matrix(1, 6, [0,1,-6,5,-1,-1])-sympy.Matrix(1, 6, [0,1,1,2,0,0])) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3-ю строку умножим на 4, -1, 7, 9 соответственно и сложим с 1-ой, 2-ой, 4-ой и 5-ой,\n",
    "\n",
    "разделим 1-ую строку на -1, 5-ую на 2:\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "-1 & 0 & 0 & 2 & -2 & 0 \\\\\n",
    "0 & 1 & 0 & 2 & 0 & 0 \\\\\n",
    "0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 3 & -1 & -1 \\\\ \n",
    "0 & 0 & 0 & 6 & 2 & -2\n",
    "\\end{pmatrix} = \n",
    "\\begin{pmatrix}\n",
    "1 & 0 & 0 & -2 & 2 & 0 \\\\\n",
    "0 & 1 & 0 & 2 & 0 & 0 \\\\\n",
    "0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 3 & -1 & -1 \\\\ \n",
    "0 & 0 & 0 & 3 & 1 & -1\n",
    "\\end{pmatrix}$\n",
    "\n",
    "4-ую строку умножим  -1 и сложим 5-ой строкой,\n",
    "\n",
    "1-ую, 2-ую строки умножим на 3 и -3 соответственно и сложим с 4-ой, умноженной на 2,\n",
    "\n",
    "поделим 5-ую строку на 2:\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "3 & 0 & 0 & 0 & 4 & -2 \\\\\n",
    "0 & -3 & 0 & 0 & 0 & -2 \\\\\n",
    "0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 3 & -1 & -1 \\\\ \n",
    "0 & 0 & 0 & 0 & 2 & 0\n",
    "\\end{pmatrix} = \n",
    "\\begin{pmatrix}\n",
    "3 & 0 & 0 & 0 & 4 & -2 \\\\\n",
    "0 & -3 & 0 & 0 & 0 & -2 \\\\\n",
    "0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 3 & -1 & -1 \\\\ \n",
    "0 & 0 & 0 & 0 & 1 & 0\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Умножим 5-ую строку на -4 и 1 соответственно и сложим с 1-ой и 4-ой строкой,\n",
    "\n",
    "разделим на 3, -3 и 3 соответственно 1-ую, 2-ую и 4-ую строки:\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "3 & 0 & 0 & 0 & 0 & -2 \\\\\n",
    "0 & -3 & 0 & 0 & 0 & -2 \\\\\n",
    "0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 3 & 0 & -1 \\\\ \n",
    "0 & 0 & 0 & 0 & 1 & 0\n",
    "\\end{pmatrix} = \n",
    "\\begin{pmatrix}\n",
    "1 & 0 & 0 & 0 & 0 & -2/3 \\\\\n",
    "0 & 1 & 0 & 0 & 0 & 2/3 \\\\\n",
    "0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 1 & 0 & -1/3 \\\\ \n",
    "0 & 0 & 0 & 0 & 1 & 0\n",
    "\\end{pmatrix}$\n",
    "\n",
    "В матрице 5 линейно независимых строк. $\\implies$ Ранг матрицы равен 5."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}1 & 0 & 0 & 0 & 0 & - \\frac{2}{3}\\\\0 & 1 & 0 & 0 & 0 & \\frac{2}{3}\\\\0 & 0 & 1 & 0 & 0 & 0\\\\0 & 0 & 0 & 1 & 0 & - \\frac{1}{3}\\\\0 & 0 & 0 & 0 & 1 & 0\\end{matrix}\\right], \\quad \\left ( 0, \\quad 1, \\quad 2, \\quad 3, \\quad 4\\right )\\right )$$"
      ],
      "text/plain": [
       "⎛⎡1  0  0  0  0  -2/3⎤                 ⎞\n",
       "⎜⎢                   ⎥                 ⎟\n",
       "⎜⎢0  1  0  0  0  2/3 ⎥                 ⎟\n",
       "⎜⎢                   ⎥                 ⎟\n",
       "⎜⎢0  0  1  0  0   0  ⎥, (0, 1, 2, 3, 4)⎟\n",
       "⎜⎢                   ⎥                 ⎟\n",
       "⎜⎢0  0  0  1  0  -1/3⎥                 ⎟\n",
       "⎜⎢                   ⎥                 ⎟\n",
       "⎝⎣0  0  0  0  1   0  ⎦                 ⎠"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.rref()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAPBAMAAADE5vg+AAAALVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAMhAi783diatmdpm7RL9tftMAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABPSURBVAgdY2BkYFBmYGB5GpbEwMCWl67AwMDBAAIo5KzVAgwMPArMb8FSx8DkPgOGowwM+wQY3BgY+hgYChgYfBgYrBm4nBgYmHd3KzAAAOU9DBlFth6pAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$5$$"
      ],
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.rank()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-1 -1  2 -3 -1  1]\n",
      " [ 1 -2 -1 -3  1 -1]\n",
      " [-2 -1 -2 -1 -3  1]\n",
      " [-3 -1 -1  1 -1  1]\n",
      " [-3 -3 -3 -3 -1  1]]\n"
     ]
    }
   ],
   "source": [
    "a = numpy.matrix([[-1,-1, 2,-3 ,-1,1],[1,-2,-1,-3,1,-1],[-2,-1,-2,-1,-3,1],[-3,-1,-1,1,-1,1],[-3 ,-3 ,-3,-3,-1,1]])\n",
    "print (a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "numpy.linalg.matrix_rank(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Типовой расчет 5 / Standard Calculation 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Решить систему линейных алгебраических уравнений методом Гаусса\n",
    "\n",
    "$\\begin{cases}\n",
    "2x_1 + 3x_2 + 2x_3 - 3x_4 - 9x_5 - 9x_6 = 0 \\\\\n",
    "2x_1 - 2x_2 - 3x_3 - 3x_4 + x_5 + 6x_6 = 0 \\\\\n",
    "-x_1 + 2x_2 + x_3 + 3x_4 - x_5 - 3x_6 = 0 \\\\\n",
    "6x_1 - 3x_2 - 3x_3 - 12x_4 - 6x_5 + 3x_6 = 0 \\\\\n",
    "\\end{cases}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение / Decision\n",
    "\n",
    "Число уравнений *меньше* числа неизвестных переменных, \n",
    "\n",
    "значит система *совместна* и имеет *бесконечное множество решений*.\n",
    "\n",
    "Матричный вид записи: $Ax = b$, где\n",
    "    \n",
    "$ A = \\begin{bmatrix}\n",
    "2 & 3 & 2 & -3 & -9 & -9  \\\\\n",
    "2 & -2 & -3 & -3 & 1 & 6 \\\\\n",
    "-1 & 2 & 1 & 3 & -1 & -3 \\\\\n",
    "6 & -3 & -3 & -12 & -6 & 3 \\end{bmatrix}, \\ \n",
    "b = \\begin{bmatrix}\n",
    "0 \\\\ 0 \\\\ 0 \\\\ 0\n",
    " \\end{bmatrix}$.\n",
    "\n",
    "Все коэффициенты $b_1 = b_2 = b_3 = b_4 = 0$, значит система является *однородной*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "В матрице $A$ выполним элементарные преобразования.\n",
    "\n",
    "Поделим 4-ую строку на 3.\n",
    "\n",
    "$ \\begin{bmatrix}\n",
    "2 & 3 & 2 & -3 & -9 & -9  \\\\\n",
    "2 & -2 & -3 & -3 & 1 & 6 \\\\\n",
    "-1 & 2 & 1 & 3 & -1 & -3 \\\\\n",
    "2 & -1 & -1 & -4 & -2 & 1\n",
    " \\end{bmatrix}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & -1 & -1 & -4 & -2 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[2  -1  -1  -4  -2  1]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(1, 6, [6,-3,-3,-12,-6,3])/3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Выберем в качестве разрешающего элемента 1-ый в 1-ой строке.\n",
    "\n",
    "К 2-ой, 3-ой, 4-ей строкам, умноженным соответственно на -1, 2, -1, прибавим 1-ую:\n",
    "    \n",
    "$ \\begin{bmatrix}\n",
    "2 & 3 & 2 & -3 & -9 & -9  \\\\\n",
    "0 & 5 & 5 & 0 & -10 & -15 \\\\\n",
    "0 & 7 & 4 & 3 & -11 & -15 \\\\\n",
    "0 & 4 & 3 & 1 & -7 & -10\n",
    " \\end{bmatrix}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAwsAAAAZBAMAAACMfq+EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVESZEM27ZiLviTJ23asWSsTfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGPklEQVRoBe1a22scVRj/9tLdzV6SxX8gCz75oFlYUwSxDkhARd34IH1KuxQvCF7yoKhQ6ULAogiJIvogNgNaH2ofFnzwQaWBvlhomhTFK1tXgxW8pK221ao1ft85M+d855yZ3bUlWWE9sDPf73z385szM7sJAFwD8HO7CVs5ltvlqHRDV0axfRIgL5YiUQWYjlqUzZzLzkZFH8IyngIoNGgtpvAT0X8L0k3UhKMAcHsoX9VZBuqXhtwTV5Us3vnflREfp6tmv9dVTUqkAfaS8BB+iIZDR5t4VGPxvranAMDIhZlPGIT84Zs4VHKBrKxQhlIGiqfB9E1sbGxcUv4kFKocvvDmHg6VLMoQBzWlBZrvUYbb3rvaHyB3aHuFY/jQM2AAxrH6hSgFmcsUREMJUY7skIZt5fTnKKnRvvyWklEoff2lx/G9cHeT40C+Z/K8Gyq0E0oZKJYGq4z8rloNb59slDoMpKuwr8xwKIpM4hDOsHM/ZbjtnWIR4DbInmM4vbbbY1CJB9dqt0SUJ81lCqKh2AFI+SggDc8CvI2SGh0lCWHUhNlVyDfMKYmKSIMdStmRUgaKpcHyTeF1UlX+JCx3GCyegzrW7g7KBOLg6sR89zLc9ooGDSvl9AUj8IpnwABg5XNR84DmQQqiIY3FjlVQwFYeAxj3UAxHJxTkWZat5jIdSFCnzqDW7VDKqA8aLN9EyFsYo/BkJxTxnHwZxjaDBre993axtHBdBfqhoQzAq9URkIYgBdEAOwGOeHjGVn4FmEA3NTpKEoJFw1gVknxbKmNaaTuUoZSBYneD6/uOcichv61jYBifNbFEYiOIQ4S299XgtucbNAAkfjMCR+8GLLdlmIUAzYMUgobtuMlJNQ1p6p939P3xm0MnOo/uPcoj1pEGs5DAGFt0Qqk4on8RKI4G1zfXUO4kfGTRkPzMUIdAMCAO4Qw79y7Daa/YtGi4nq8V0F0mcjwXOUvmQQpBwzGA78hwGnJ/4MZYYk73w3yFwYyX+4XBeR8KfzGsRGzRCcWVIAPF0eD6jlaUOwq5WZOGwokfuVrJPWnoXobT3n4wafhgnV+U8TR8qyoyBKQhSCFomPDgQdIHNPiGbWbVgPAogxQEiXNHSIPvqtQzEwN1pcHwnTcC5cGkAR9oZcMgAD1pQLsuZTjt+RYNkDBvySteVBWQOxs5TbshSCFoONKS4fGmhIs6sWQ4pX43INza1LjuQ5Lvhqk2ja/ESjuhuJJCYKA4GhxfOKiTovS0Q8MIf1MwMwkutLup7FqG3V62adMAi56ObO8GlSk/zY3SM2KVWsI8SCFowPvOx2SI1vhs4BdWZhpSf7IYD2DZ6B+Osap9PQQaat0KFfrI3SADxdHg+n6q3fGetGTSUGhBxrwsWRldX1i7l2G39yKYNLwP8HiZFxazG0oNbqRlNA9SCBrqlXA30M1pn6cNR1Zh9KyG8AbAGQYzDcjyy1CpiAYrlNKJdZGBYmmwfdN/a3d8QV1fP3PA1zP181dKQ/cy7PZuWF+/9INOm97w+qOh3tFOXEIaghRyN6ibEjwDMMks8/h9v8GwD/Aqg9kFSOEWcgfRYIXSRqT0RSBJg6dVQsKItm8St5YxRjoMIjBuSkpFmbruBr9rGRHtHVCxUUCw6PGJmN0wUeVGWkbzIIV8NjThNCmx/1I5/bq2o++uxzyG56D4CoNwEu5ocRzK1L8VKlTJdZGBBA3WN9GoMiBhvwiMNXQ8esRc4feGHmW47V1kafEHuf6+N4z73EvLxJpMIWiY8ADfWUX/6cMnjHWd23GjdsO7cm0NXfW483jkm+Low5d/AjtU6CWUMpDcDcaPh2iFV4Ptm3sp9Jbn1O6LS2zm2toehpQoMomDmtJCP2U47U1uPKIjQLK2vcIgnF48tcRxKD9fDiXjLMxlCkED3gKQCdG/YbfpQNKQoNxsIA1bOwZfhqBhBz4BKtj5gPrHB5AxhrAMQcNO/QursR6bDuRleJeVZ0A0DLIMooF+YS0soDCg/mf/GzQMsgyiIdvBwzf4GRANmNkYQ1gG0VBq4mEKP0PYP3aN16G9EcTsVq4G0SD+Fv3/f2aItWeHLaYh+Ckf/09ppt1kZWy+uPxF5GU4dGUU26/hn4Y8gH8AKz4iBTQlLvIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}0 & 5 & 5 & 0 & -10 & -15\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 7 & 4 & 3 & -11 & -15\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 4 & 3 & 1 & -7 & -10\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "([0  5  5  0  -10  -15], [0  7  4  3  -11  -15], [0  4  3  1  -7  -10])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(1, 6, [2,-2,-3,-3,1,6])*(-1)+sympy.Matrix(1, 6, [2,3,2,-3,-9,-9]), \\\n",
    "sympy.Matrix(1, 6, [-1,2,1,3,-1,-3])*2+sympy.Matrix(1, 6, [2,3,2,-3,-9,-9]), \\\n",
    "sympy.Matrix(1, 6, [2,-1,-1,-4,-2,1])*(-1)+sympy.Matrix(1, 6, [2,3,2,-3,-9,-9])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Поделим 2-ую строку на 5.\n",
    "\n",
    "$ \\begin{bmatrix}\n",
    "2 & 3 & 2 & -3 & -9 & -9  \\\\\n",
    "0 & 1 & 1 & 0 & -2 & -3 \\\\\n",
    "0 & 7 & 4 & 3 & -11 & -15 \\\\\n",
    "0 & 4 & 3 & 1 & -7 & -10\n",
    " \\end{bmatrix}$.\n",
    "    \n",
    "Выберем в качестве разрешающего элемента 2-ой во 2-ой строке.\n",
    "\n",
    "К 1-ой, 3-ой, 4-ей строке прибавим 2-ую, умноженную на -3, -7, -4 соответственно:\n",
    "\n",
    "$ \\begin{bmatrix}\n",
    "2 & 0 & -1& -3 & -3 & 0  \\\\\n",
    "0 & 1 & 1 & 0 & -2 & -3 \\\\\n",
    "0 & 0 & -3 & 3 & 3 & 6 \\\\\n",
    "0 & 0 & -1 & 1 & 1 & 2\n",
    " \\end{bmatrix}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}2 & 0 & -1 & -3 & -3 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & -3 & 3 & 3 & 6\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & -1 & 1 & 1 & 2\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "([2  0  -1  -3  -3  0], [0  0  -3  3  3  6], [0  0  -1  1  1  2])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(1, 6, [0,1,1,0,-2,-3])*(-3)+sympy.Matrix(1, 6, [2,3,2,-3,-9,-9]), \\\n",
    "sympy.Matrix(1, 6, [0,1,1,0,-2,-3])*(-7)+sympy.Matrix(1, 6, [0,7,4,3,-11,-15]), \\\n",
    "sympy.Matrix(1, 6, [0,1,1,0,-2,-3])*(-4)+sympy.Matrix(1, 6, [0,4,3,1,-7,-10])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Поделим 3-ю строку на -3.\n",
    "\n",
    "$ \\begin{bmatrix}\n",
    "2 & 0 & -1& -3 & -3 & 0  \\\\\n",
    "0 & 1 & 1 & 0 & -2 & -3 \\\\\n",
    "0 & 0 & 1 & -1 & -1 & -2 \\\\\n",
    "0 & 0 & -1 & 1 & 1 & 2\n",
    " \\end{bmatrix}$.\n",
    " \n",
    "Выберем в качестве разрешающего элемента 3-ий во 3-ой строке.\n",
    "\n",
    "К 1-ой, 2-ой, 4-ей строке прибавим 3-ую, умноженную на 1, -1, 1 соответственно:\n",
    "\n",
    "$ \\begin{bmatrix}\n",
    "2 & 0 & 0 & -4 & -4 & -2  \\\\\n",
    "0 & 1 & 0 & 1 & -1 & -1 \\\\\n",
    "0 & 0 & 1 & -1 & -1 & -2 \\\\\n",
    "0 & 0 & 0 & 0 & 0 & 0\n",
    " \\end{bmatrix}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}2 & 0 & 0 & -4 & -4 & -2\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 1 & 0 & 1 & -1 & -1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "([2  0  0  -4  -4  -2], [0  1  0  1  -1  -1], [0  0  0  0  0  0])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(1, 6, [0,0,1,-1,-1,-2])+sympy.Matrix(1, 6, [2,0,-1,-3,-3,0]), \\\n",
    "sympy.Matrix(1, 6, [0,0,1,-1,-1,-2])*(-1)+sympy.Matrix(1, 6, [0,1,1,0,-2,-3]), \\\n",
    "sympy.Matrix(1, 6, [0,0,1,-1,-1,-2])+sympy.Matrix(1, 6, [0,0,-1,1,1,2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Поделим 1-ю строку на 2 и удалим строку с нулевыми коэффициентами.\n",
    "\n",
    "$ \\begin{bmatrix}\n",
    "1 & 0 & 0 & -2 & -2 & -1  \\\\\n",
    "0 & 1 & 0 & 1 & -1 & -1 \\\\\n",
    "0 & 0 & 1 & -1 & -1 & -2 \\\\\n",
    " \\end{bmatrix}$.\n",
    " \n",
    " Система уравнений примет вид:\n",
    "\n",
    "$\\begin{cases}\n",
    "x_1 - 2x_4 - 2x_5 - x_6 = 0 \\\\\n",
    "x_2 + x_4 - x_5 - x_6 = 0 \\\\\n",
    "x_3 - x_4 - x_5 - 2x_6 = 0 \\\\\n",
    "\\end{cases} \\iff\n",
    "\\begin{cases}\n",
    "x_1 = 2x_4 + 2x_5 + x_6 \\\\\n",
    "x_2 = -x_4 + x_5 + x_6 \\\\\n",
    "x_3 = x_4 + x_5 + 2x_6 \\\\\n",
    "\\end{cases}$.\n",
    "\n",
    "Решение системы уравнений:\n",
    "\n",
    "$\\big(2x_4 + 2x_5 + x_6, \\ -x_4 + x_5 + x_6, \\ x_4 + x_5 + 2x_6, \\ x_4, \\ x_5, \\ x_6\\big)$, где \n",
    "\n",
    "$x_4, x_5, x_6$ - произвольные числа."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ x_{1} : 2 x_{4} + 2 x_{5} + x_{6}, \\quad x_{2} : - x_{4} + x_{5} + x_{6}, \\quad x_{3} : x_{4} + x_{5} + 2 x_{6}\\right \\}$$"
      ],
      "text/plain": [
       "{x₁: 2⋅x₄ + 2⋅x₅ + x₆, x₂: -x₄ + x₅ + x₆, x₃: x₄ + x₅ + 2⋅x₆}"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x1,x2,x3,x4,x5,x6 = sympy.symbols('x1,x2,x3,x4,x5,x6')\n",
    "system = sympy.Matrix([ [2,3,2,-3,-9,-9,0], [2,-2,-3,-3,1,6,0], [-1,2,1,3,-1,-3,0], [6,-3,-3,-12,-6,3,0] ])\n",
    "sympy.solve_linear_system(system, x1,x2,x3,x4,x5,x6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Типовой расчет 6 / Standard Calculation 6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Постановка  задачи / Formulation of the problem:\n",
    "Найти общее решение системы линейных алгебраических уравнений \n",
    "\n",
    "$\\begin{cases}\n",
    "x_1 + 3x_2 + x_3 - 3x_5 - 6x_6 = 47 \\\\\n",
    "x_1  + x_2 + 3x_3 - 4x_4 - 3x_5 - 8x_6 = 55 \\\\\n",
    "-x_1 + x_2 - 3x_3 + 6x_4 + x_5 + 6x_6 = -33 \\\\\n",
    "x_1 - x_2 + 3x_3 - 6x_4 - x_5 - 6x_6 = 33 \\\\\n",
    "\\end{cases}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение / Decision\n",
    "\n",
    "Число уравнений *меньше* числа неизвестных переменных, \n",
    "\n",
    "значит система *совместна* и имеет *бесконечное множество решений*. \n",
    "\n",
    "Матричный вид записи: $Ax = b$, где\n",
    "    \n",
    "$ A = \n",
    "\\begin{bmatrix}\n",
    "1 & 3 & 1 & 0 & -3 & -6  \\\\\n",
    "1 & 1 & 3 & -4 & -3 & -8 \\\\\n",
    "-1 & 1 & -3 & 6 & 1 & 6 \\\\\n",
    "1 & -1 & 3 & -6 & -1 & -6\n",
    " \\end{bmatrix}, \\ \n",
    "b = \n",
    "\\begin{bmatrix}\n",
    "47 \\\\ 55 \\\\ -33 \\\\ 33\n",
    " \\end{bmatrix}\n",
    "$.\n",
    "\n",
    "Система является *неоднородной*.\n",
    "\n",
    "Заметим, что 3-я и 4-ая строки *линейно зависимы*  \n",
    "\n",
    "(при умножении на -1 из 3-его уравнения получается 4-ое) \n",
    "\n",
    "$\\implies$ 4-ую строку можно исключить."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Применим метод Гаусса и преобразуем расширенную матрицу:\n",
    "    \n",
    "$ \\left(\\begin{array}{cccccc|c}\n",
    "1 & 3 & 1 & 0 & -3 & -6 & 47 \\\\\n",
    "1 & 1 & 3 & -4 & -3 & -8 & 55 \\\\\n",
    "-1 & 1 & -3 & 6 & 1 & 6 & -33 \\\\\n",
    " \\end{array}\\right)$\n",
    " \n",
    "Выберем в качестве разрешающего элемента 1-ый в 1-ой строке.\n",
    " \n",
    "К 2-ой и 3-ей строке прибавим 1-ю, умноженную на -1, 1 соответственно:\n",
    " \n",
    "$ \\left(\\begin{array}{cccccc|c}\n",
    "1 & 3 & 1 & 0 & -3 & -6 & 47 \\\\\n",
    "0 & -2 & 2 & -4 & 0 & -2 & 8\\\\\n",
    "0 & 4 & -2 & 6 & -2 & 0 & 14 \\\\\n",
    "\\end{array}\\right)$\n",
    " \n",
    "Разделим 2-ую и 4-ую строки на -2 и 2 соответственно.\n",
    " \n",
    "$ \\left(\\begin{array}{cccccc|c}\n",
    "1 & 3 & 1 & 0 & -3 & -6 & 47 \\\\\n",
    "0 & 1 & -1 & 2 & 0 & 1& -4 \\\\\n",
    "0 & 2 & -1 & 3& -1 & 0 & 7 \\\\\n",
    "\\end{array}\\right)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}0 & -2 & 2 & -4 & 0 & -2 & 8\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 4 & -2 & 6 & -2 & 0 & 14\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "([0  -2  2  -4  0  -2  8], [0  4  -2  6  -2  0  14])"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(1, 7, [1,1,3,-4,-3,-8,55])+sympy.Matrix(1, 7, [1,3,1,0,-3,-6,47])*(-1), \\\n",
    "sympy.Matrix(1, 7, [-1,1,-3,6,1,6,-33])+sympy.Matrix(1, 7, [1,3,1,0,-3,-6,47])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Выберем в качестве разрешающего элемента 2-ой в 2-ой строке.\n",
    " \n",
    " К 1-ой и 3-ей строке прибавим 2-ую, умноженную на -3 и -2 соответственно:\n",
    " \n",
    " $ \\left(\\begin{array}{cccccc|c}\n",
    "1 & 0 & 4 & -6 & -3 & -9 & 59 \\\\\n",
    "0 & 1 & -1 & 2 & 0 & 1 & -4 \\\\\n",
    "0 & 0 & 1 & -1 & -1 & -2 & 15 \\\\\n",
    "\\end{array}\\right)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAlIAAAAZBAMAAAAI8d3GAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVESZEM27ZiLviTJ23asWSsTfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAE4klEQVRYCe2ZX2gcRRzHv7nb+5Pc5XKILz7lwHdzGAIKkuxLwAr14ovgQzUoKqLVgA8iVHNQUBHBKKKCqCsVH2KhAUGhID2IL4JJU1vxH6kpxYJWW1sTrdZy/n4zO7u/mdu1L4e3Dx1IZn7f+c3sdz87M3vJAbgOqG1+g8GW6uaxJAODN4a1zToQsLmhJlBIcvk/a08lXS8LxkoLwMdsbpZ+BmaoAtweEsogKe2OSZU65PLhkFTlRGhZVyMHb7Fi4CMnDvOWP0zSXzxwf5KMyWlf6sM7e+iySkwl5TpZXm3LKfCpNWPU5dzP1fT9U49GKdzg4dodkwJF5SWqaU3dObXNCVG5C7vbUaAaJ+1QR+Uv8WuC7jWxWE/Qd3cqb0p59IfvfGgxlZTjpFD35KHqnb7PlzOatns/V9HLWzjUMDlUq+HKHRSpdSAfkM67r2qRKm1gZJ7UuFQTSRWaeCjOiVrVC2jNRVHc+Al4II7oVcKBFtNIuU6eAT6QU+Cob4UmsO/HqO59Gr3mY5jXTFR4uHKnSbV8jDWos5dUcQtDFjp8cm80i2iM+yKIm7lXMZZEagc4FGeFXrSYRsp18jhgX7QvpIa3UbsknTmkCgs44lN/L6mxJnIX5EgEiaTusXJkML4gI932/gCOrgidn1ooppFynfwOTNTFFH1aUxuoXZazOqTy81jj7l5SLSL1pzWynUhq5/BqQ6ZF7dzXUVM0ePk0RFzbt7oCLaaRcpx4TMp6CH1ZU+Rp9DdhTB1Gyp3efdUl/MjdvaReDlD5R458HkmkvJ0FvCLTTLuyftY0ZU1H0hPyNot++SKfUySmkXKclP8GjnTknP0iZc/Ka0q506Ry2/o8TiZFnuISJJPqArf6cZZojVtbJOzY1aicd/S90OJ/kRJOFKlAXKc/u48mPC4nNS+4vfRZih9t5bK+/15SrQA5uaZKbZvU7CaX73EFWGzEl/D2KH2FFDoj42LSvdves84pyphpazGNlOPEI2oTnXhmOvh8GZkrmVs1fSl6bDjfNKmq5jVFZaatSdHSVyR7SdE5OiRP9Jdgk1LT0K+viBQzd0plBUU5XHTP+CJ4kLwwVhLTSLlO6Jyyl6tNKpo8vNUoNo00/UmToGtO0+7UmvIupq2p4jxKclHcdO7cXz/bc6mIVqdcUyajtZ1K6m6Tw/XbwHmuSUwj5Tqhj2+LPo8xpT+khgK8YGbkmklpd4pUOXX3lZaQn5MjgXfsUEcTdE4l6MNb9u4zKTf4dElRAuB1aFGR8kQfNwt0TDhOngamrKz+kDoMfCanZVIBuwvPqUs4w91kyN3Xx7CLt4Uo9FmotxQDL+ndR88h6fMU1vyitVf3o/paKCpS7gd7NuY4Ga17b1k++kKqfHxyekNOy6SUu+jd9zl3k6HaI1d+kZl3fOG85qe6j8l+0z4w7QDVHTdOJv6FPLJ8sxmo6vLkaR9aVKRGT1jd6hE6TryD69YVz7x7smMPUpF7PyYlRS90u905k0O1SlPuNCla2RM+6fzoBl30OXW9bSMLxtQ5lafjpEHesmBIkwoySor+7suzt8yQ8joZJdXyUVnKEqmSDSoTj1DtvmfJ2KkskXJAZYcU//Nw9hop9/nYMa+pUoe0a9/N2GB6Iib1nFLp+75CBr7ve6PHIgmDN4a1b4nU+8C/MnuDvA1fb8kAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}1 & 0 & 4 & -6 & -3 & -9 & 59\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & 1 & -1 & -1 & -2 & 15\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "([1  0  4  -6  -3  -9  59], [0  0  1  -1  -1  -2  15])"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(1, 7, [1,3,1,0,-3,-6,47])+sympy.Matrix(1, 7, [0,1,-1,2,0,1,-4])*(-3), \\\n",
    "sympy.Matrix(1, 7, [0,2,-1,3,-1,0,7])+sympy.Matrix(1, 7, [0,1,-1,2,0,1,-4])*(-2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Выберем в качестве разрешающего элемента 3-ий в 3-ей строке.\n",
    " \n",
    " К 1-ой и 2-ой строке прибавим 3-ю, умноженную на -4 и 1 соответственно:\n",
    " \n",
    "$ \\left(\\begin{array}{cccccc|c}\n",
    "1 & 0 & 0 & -2 & 1 & -1 & -1 \\\\\n",
    "0 & 1 & 0 & 1 & -1 & -1 & 11 \\\\\n",
    "0 & 0 & 1 & -1 & -1 & -2 & 15 \\\\\n",
    "\\end{array}\\right)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}1 & 0 & 0 & -2 & 1 & -1 & -1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 1 & 0 & 1 & -1 & -1 & 11\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "([1  0  0  -2  1  -1  -1], [0  1  0  1  -1  -1  11])"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(1, 7, [1,0,4,-6,-3,-9,59])+sympy.Matrix(1, 7, [0,0,1,-1,-1,-2,15])*(-4), \\\n",
    "sympy.Matrix(1, 7, [0,1,-1,2,0,1,-4])+sympy.Matrix(1, 7, [0,0,1,-1,-1,-2,15])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Система уравнений примет вид:\n",
    "\n",
    "$\\begin{cases}\n",
    "x_1 - 2x_4 + x_5 - x_6 = -1 \\\\\n",
    "x_2 + x_4 - x_5 - x_6 = 11 \\\\\n",
    "x_3 - x_4 - x_5 - 2x_6 = 15 \\\\\n",
    "\\end{cases} \\iff\n",
    "\\begin{cases}\n",
    "x_1 = 2x_4 - x_5 + x_6 - 1 \\\\\n",
    "x_2 = -x_4 + x_5 + x_6 + 11 \\\\\n",
    "x_3 = x_4 + x_5 + 2x_6 + 15\\\\\n",
    "\\end{cases}$.\n",
    "\n",
    "Решение системы уравнений:\n",
    "\n",
    "$\\big(2x_4 - x_5 + x_6 - 1, \\ -x_4 + x_5 + x_6 + 11, \\ x_4 + x_5 + 2x_6 + 15, \\ x_4, \\ x_5, \\ x_6\\big)$, \n",
    "\n",
    "где $x_4, x_5, x_6$ - произвольные числа."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ x_{1} : 2 x_{4} - x_{5} + x_{6} - 1, \\quad x_{2} : - x_{4} + x_{5} + x_{6} + 11, \\quad x_{3} : x_{4} + x_{5} + 2 x_{6} + 15\\right \\}$$"
      ],
      "text/plain": [
       "{x₁: 2⋅x₄ - x₅ + x₆ - 1, x₂: -x₄ + x₅ + x₆ + 11, x₃: x₄ + x₅ + 2⋅x₆ + 15}"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x1,x2,x3,x4,x5,x6 = sympy.symbols('x1,x2,x3,x4,x5,x6')\n",
    "system = sympy.Matrix([ [1,3,1,0,-3,-6,47] , [1,1,3 ,-4,-3,-8,55], [-1,1,-3,6,1,6,-33], [1,-1,3,-6,-1,-6,33] ])\n",
    "sympy.solve_linear_system(system, x1,x2,x3,x4,x5,x6)"
   ]
  }
 ],
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