Math_242_solve_systems_using_computation_and_rref.ipynb

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    "<a href=\"https://colab.research.google.com/github/nameer1811/linear_algebra_projects/blob/main/Math_242_solve_systems_using_computation_and_rref.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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   "cell_type": "markdown",
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    "# Load the preliminary code"
   ]
  },
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    "colab_type": "code",
    "id": "dhOLtwrRRc9S"
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   "outputs": [
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       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 11;\n",
       "                var nbb_unformatted_code = \"import numpy as np\\nfrom numpy import c_\\n\\nll = \\\"loaded\\\"\";\n",
       "                var nbb_formatted_code = \"import numpy as np\\nfrom numpy import c_\\n\\nll = \\\"loaded\\\"\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
       "                        }\n",
       "                        break;\n",
       "                    }\n",
       "                }\n",
       "            }, 500);\n",
       "            "
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      "text/plain": [
       "<IPython.core.display.Javascript object>"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from numpy import c_\n",
    "\n",
    "ll = \"loaded\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
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    "id": "AHRXKhal_Wcj",
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      ">> The page  loaded  correctly!\n"
     ]
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 12;\n",
       "                var nbb_unformatted_code = \"lll = \\\" correctly!\\\"\\n\\ndef rref(matrix):\\n    A = np.array(matrix, dtype=np.float64)\\n    \\n    r = 0 #row\\n    c = 0 #column\\n\\n    (maxr, maxc) = A.shape\\n\\n    prev_row_pivot = -1   # The previous row with a leading term:\\n    tmp_pivot_row = -1\\n    this_pivot_row = -1\\n\\n    for c in range(maxc):  # go through each column\\n      tmp_pivot_row = -1  # reset the tmp pivot row\\n      this_pivot_row = -1  # reset this pivot row\\n      for r in range(maxr):  # Find the leading term in this column below prev_row_pivot\\n        if ((r > prev_row_pivot) and (A[r][c] != 0.0)):\\n          tmp_pivot_row = r  # find the new pivot row\\n      if (tmp_pivot_row > -1):  # we found a candidate for the leading term:\\n        #swap row this_pivot_row with (prev_row_pivot + 1)\\n        A[[tmp_pivot_row, prev_row_pivot+1]] = A[[prev_row_pivot + 1, tmp_pivot_row]]\\n        #set this_pivot_row to prev_row_pivot+1\\n        this_pivot_row = prev_row_pivot + 1\\n        #Now, normalize the row based on the pivot value:\\n        A[ this_pivot_row ] = A[ this_pivot_row ] / A[ this_pivot_row ][c]\\n        #Then, use this row to force 0's above and below the pivot entry:\\n        #subtract multiples of of the this_pivot_row row from the others\\n        for r_ in range(len(A)):\\n            if r_ != this_pivot_row:\\n                A[r_] = A[r_] - A[this_pivot_row] * A[r_][c] / A[this_pivot_row][c]\\n        prev_row_pivot = this_pivot_row\\n    return A    \\n    \\n    \\n\\nprint(\\\">> The page \\\", ll, lll)\";\n",
       "                var nbb_formatted_code = \"lll = \\\" correctly!\\\"\\n\\n\\ndef rref(matrix):\\n    A = np.array(matrix, dtype=np.float64)\\n\\n    r = 0  # row\\n    c = 0  # column\\n\\n    (maxr, maxc) = A.shape\\n\\n    prev_row_pivot = -1  # The previous row with a leading term:\\n    tmp_pivot_row = -1\\n    this_pivot_row = -1\\n\\n    for c in range(maxc):  # go through each column\\n        tmp_pivot_row = -1  # reset the tmp pivot row\\n        this_pivot_row = -1  # reset this pivot row\\n        for r in range(\\n            maxr\\n        ):  # Find the leading term in this column below prev_row_pivot\\n            if (r > prev_row_pivot) and (A[r][c] != 0.0):\\n                tmp_pivot_row = r  # find the new pivot row\\n        if tmp_pivot_row > -1:  # we found a candidate for the leading term:\\n            # swap row this_pivot_row with (prev_row_pivot + 1)\\n            A[[tmp_pivot_row, prev_row_pivot + 1]] = A[\\n                [prev_row_pivot + 1, tmp_pivot_row]\\n            ]\\n            # set this_pivot_row to prev_row_pivot+1\\n            this_pivot_row = prev_row_pivot + 1\\n            # Now, normalize the row based on the pivot value:\\n            A[this_pivot_row] = A[this_pivot_row] / A[this_pivot_row][c]\\n            # Then, use this row to force 0's above and below the pivot entry:\\n            # subtract multiples of of the this_pivot_row row from the others\\n            for r_ in range(len(A)):\\n                if r_ != this_pivot_row:\\n                    A[r_] = A[r_] - A[this_pivot_row] * A[r_][c] / A[this_pivot_row][c]\\n            prev_row_pivot = this_pivot_row\\n    return A\\n\\n\\nprint(\\\">> The page \\\", ll, lll)\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
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       "                        break;\n",
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       "            }, 500);\n",
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    "lll = \" correctly!\"\n",
    "\n",
    "\n",
    "def rref(matrix):\n",
    "    A = np.array(matrix, dtype=np.float64)\n",
    "\n",
    "    r = 0  # row\n",
    "    c = 0  # column\n",
    "\n",
    "    (maxr, maxc) = A.shape\n",
    "\n",
    "    prev_row_pivot = -1  # The previous row with a leading term:\n",
    "    tmp_pivot_row = -1\n",
    "    this_pivot_row = -1\n",
    "\n",
    "    for c in range(maxc):  # go through each column\n",
    "        tmp_pivot_row = -1  # reset the tmp pivot row\n",
    "        this_pivot_row = -1  # reset this pivot row\n",
    "        for r in range(\n",
    "            maxr\n",
    "        ):  # Find the leading term in this column below prev_row_pivot\n",
    "            if (r > prev_row_pivot) and (A[r][c] != 0.0):\n",
    "                tmp_pivot_row = r  # find the new pivot row\n",
    "        if tmp_pivot_row > -1:  # we found a candidate for the leading term:\n",
    "            # swap row this_pivot_row with (prev_row_pivot + 1)\n",
    "            A[[tmp_pivot_row, prev_row_pivot + 1]] = A[\n",
    "                [prev_row_pivot + 1, tmp_pivot_row]\n",
    "            ]\n",
    "            # set this_pivot_row to prev_row_pivot+1\n",
    "            this_pivot_row = prev_row_pivot + 1\n",
    "            # Now, normalize the row based on the pivot value:\n",
    "            A[this_pivot_row] = A[this_pivot_row] / A[this_pivot_row][c]\n",
    "            # Then, use this row to force 0's above and below the pivot entry:\n",
    "            # subtract multiples of of the this_pivot_row row from the others\n",
    "            for r_ in range(len(A)):\n",
    "                if r_ != this_pivot_row:\n",
    "                    A[r_] = A[r_] - A[this_pivot_row] * A[r_][c] / A[this_pivot_row][c]\n",
    "            prev_row_pivot = this_pivot_row\n",
    "    return A\n",
    "\n",
    "\n",
    "print(\">> The page \", ll, lll)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "xVLAg1tp7aJX"
   },
   "source": [
    "# Solving Systems using Computation:  An Overview"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "2c12Csd440sV"
   },
   "source": [
    "Suppose we wanted to solve a system like:\n",
    "\n",
    "$$\n",
    "\\left\\{ \\begin{array}{rcr} 5x + 6y + 2z & = & -1\\\\ 4x + 7y + 19 z & = & 2 \\\\ 3y + 12z & = & 1 \\\\ \\end{array} \\right.\n",
    "$$\n",
    "\n",
    "First, we create a matrix $A$ to represent the matrix on matrix on the left.  We'll also create a column matrix, or vector, to represent the numbers on the right hand side.  We'll call that vector $\\vec{b}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
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    "colab_type": "code",
    "id": "uYNwPc9ISAoC"
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   "outputs": [
    {
     "data": {
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       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 13;\n",
       "                var nbb_unformatted_code = \"# create the matrix:\\nA = np.matrix([[5, 6, 2],\\n               [4, 7, 19],\\n               [0, 3, 12]])\\n\\n# create the vector for the right hand side:\\nb = np.matrix([[-1], [2], [1]])\";\n",
       "                var nbb_formatted_code = \"# create the matrix:\\nA = np.matrix([[5, 6, 2], [4, 7, 19], [0, 3, 12]])\\n\\n# create the vector for the right hand side:\\nb = np.matrix([[-1], [2], [1]])\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
       "                        }\n",
       "                        break;\n",
       "                    }\n",
       "                }\n",
       "            }, 500);\n",
       "            "
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       "<IPython.core.display.Javascript object>"
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   "source": [
    "# create the matrix:\n",
    "A = np.matrix([[5, 6, 2], [4, 7, 19], [0, 3, 12]])\n",
    "\n",
    "# create the vector for the right hand side:\n",
    "b = np.matrix([[-1], [2], [1]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "teSTnJWS4-GS"
   },
   "source": [
    "To verify that we have entered $A$ correctly, or $\\vec{b}$ correctly, simply call \n",
    "\n",
    "```\n",
    "print(A)\n",
    "```\n",
    "\n",
    " or \n",
    " \n",
    " \n",
    "\n",
    "```\n",
    "print(b)\n",
    "```\n",
    "\n",
    "and the system will display that element:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 68
    },
    "colab_type": "code",
    "id": "buHLWqEJ5Hds",
    "outputId": "08211b20-f92e-4799-8dad-6d102c387798"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 5  6  2]\n",
      " [ 4  7 19]\n",
      " [ 0  3 12]]\n"
     ]
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 14;\n",
       "                var nbb_unformatted_code = \"print(A)\";\n",
       "                var nbb_formatted_code = \"print(A)\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
       "                        }\n",
       "                        break;\n",
       "                    }\n",
       "                }\n",
       "            }, 500);\n",
       "            "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
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     "output_type": "display_data"
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   "source": [
    "print(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "EDZT8emc5iGZ"
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   "source": [
    "We want to solve the augmented matrix  $\\left( A \\, | \\, \\vec{b} \\right)$, so we need to create an augmented matrix.  We use:\n",
    "\n",
    "```\n",
    " np.c_(A, b)\n",
    "```\n",
    "to create the augmented matrix by putting the $\\vec{b}$ vector on the right of the $A$ matrix.  We'll save this new matrix as $M$:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 68
    },
    "colab_type": "code",
    "id": "VFHQRedm5920",
    "outputId": "0d731c34-e3ec-4269-d994-95c88f607735"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 5  6  2 -1]\n",
      " [ 4  7 19  2]\n",
      " [ 0  3 12  1]]\n"
     ]
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 15;\n",
       "                var nbb_unformatted_code = \"M = np.c_[A,b]\\nprint(M)\";\n",
       "                var nbb_formatted_code = \"M = np.c_[A, b]\\nprint(M)\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
       "                        }\n",
       "                        break;\n",
       "                    }\n",
       "                }\n",
       "            }, 500);\n",
       "            "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "M = np.c_[A, b]\n",
    "print(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "0iuR27cy86S6"
   },
   "source": [
    "Lastly, we'll use te function \n",
    "\n",
    "```\n",
    "rref(M)\n",
    "```\n",
    "\n",
    "to provide the reduced row echelon form of the matrix $M$.  Let's print out the result:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 68
    },
    "colab_type": "code",
    "id": "9Qhe2zZjYAVL",
    "outputId": "0738eed7-0cf3-44d0-eae8-4d1e629040f7"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.          0.          0.          0.45736434]\n",
      " [ 0.          1.          0.         -0.62790698]\n",
      " [-0.         -0.          1.          0.24031008]]\n"
     ]
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 16;\n",
       "                var nbb_unformatted_code = \"print(rref(M))\";\n",
       "                var nbb_formatted_code = \"print(rref(M))\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
       "                        }\n",
       "                        break;\n",
       "                    }\n",
       "                }\n",
       "            }, 500);\n",
       "            "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
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     "metadata": {},
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   ],
   "source": [
    "print(rref(M))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "tL_FR5pV9MIQ"
   },
   "source": [
    "# Use Computation to check prior homework"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "SLu14LMjXgAj"
   },
   "source": [
    "We'll use Google Colab to check our solutions to the six systems below, which we solved previously using row operations:\n",
    "\n",
    "$$\n",
    "(a) \\hspace{.5cm}\n",
    "\\left\\{ \\begin{array}{rcr} 2x + 2y & = & 5 \\\\ x - 4y & = & 0 \\\\ \\end{array} \\right.\n",
    "\\hspace{1cm}\n",
    "(b) \\hspace{.5cm}\n",
    "\\left\\{ \\begin{array}{rcr} -x + y &=& 1 \\\\ x + y &=& 2 \\\\ \\end{array}\n",
    "\\right.\n",
    "\\hspace{1cm}\n",
    "(c) \\hspace{.5cm}\n",
    "\\left\\{ \\begin{array}{rcr} x - 3y + z & = 1 \\\\ x + y + 2z & = & 14 \\\\ \\end{array}\n",
    "\\right.\n",
    "\\hspace{1cm}\n",
    "(d) \\hspace{.5cm}\n",
    "\\left\\{ \\begin{array}{rcr} -x - y &=& 1 \\\\ -3x -3 y &=& 2 \\\\ \\end{array}\n",
    "\\right.\n",
    "$$\n",
    "\n",
    "\n",
    "$$\n",
    "(e) \\hspace{.5cm}\n",
    "\\left\\{ \\begin{array}{rcr} 4y + z & = & 20 \\\\ 2x - 2y + z & = & 0 \\\\ x + z & = & 5 \\\\ x + y - z & = & 10 \\\\ \\end{array} \\right.\n",
    "\\hspace{1cm}\n",
    "(f) \\hspace{.5cm}\n",
    "\\left\\{ \\begin{array}{rcr} 2x + z + w & = & 5 \\\\ y - w & = & -1 \\\\ 3x - z - w & = & 0 \\\\ 4x + y + 2z + w & = & 9 \\\\ \\end{array}\n",
    "\\right.\n",
    "$$\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "XNUulSKp9TgA"
   },
   "source": [
    "The main steps are to:\n",
    "\n",
    "1) Create the matrix $A$ and column vector $\\vec{b}$ for the system using\n",
    "\n",
    "\n",
    "\n",
    "```\n",
    "# create the matrix:\n",
    "A = np.matrix([[5, 6, 2],\n",
    "               [4, 7, 19],\n",
    "               [0, 3, 12]])\n",
    "\n",
    "# create the vector for the right hand side:\n",
    "b = np.matrix([[-1], [2], [1]])\n",
    "```\n",
    "\n",
    "\n",
    "2) Create the augmented matrix $\\left( A \\, | \\, \\vec{b} \\right)$ using\n",
    "\n",
    "\n",
    "\n",
    "```\n",
    "M = np.c_[A,b]\n",
    "```\n",
    "\n",
    "3)  Display the reduced row echelon form of $M$ using\n",
    "\n",
    "\n",
    "\n",
    "```\n",
    "print(rref(M))\n",
    "```\n",
    "\n",
    "You'll do this for each of the six systems of equations from the homework:\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "ieV1CTIh-Lzd"
   },
   "source": [
    "$$\n",
    "(a) \\hspace{.5cm}\n",
    "\\left\\{ \\begin{array}{rcr} 2x + 2y & = & 5 \\\\ x - 4y & = & 0 \\\\ \\end{array} \\right.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "uEUzkFMc-NrD"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "This is the left hand side\n",
      " [[ 2  2]\n",
      " [ 1 -4]]\n",
      "This is the right hand side\n",
      " [[5]\n",
      " [0]]\n",
      "This is the augmented matrix\n",
      " [[ 2  2  5]\n",
      " [ 1 -4  0]]\n",
      "This is the rref answer\n",
      " [[1.  0.  2. ]\n",
      " [0.  1.  0.5]]\n"
     ]
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 22;\n",
       "                var nbb_unformatted_code = \"A = np.matrix([[2, 2], [1, -4]])\\nprint(\\\"This is the left hand side\\\\n\\\", A)\\nb = np.matrix([[5],[0]])\\nprint(\\\"This is the right hand side\\\\n\\\", b)\\n\\nM = np.c_[A, b]\\nprint(\\\"This is the augmented matrix\\\\n\\\", M)\\n\\nprint(\\\"This is the rref answer\\\\n\\\", rref(M))\";\n",
       "                var nbb_formatted_code = \"A = np.matrix([[2, 2], [1, -4]])\\nprint(\\\"This is the left hand side\\\\n\\\", A)\\nb = np.matrix([[5], [0]])\\nprint(\\\"This is the right hand side\\\\n\\\", b)\\n\\nM = np.c_[A, b]\\nprint(\\\"This is the augmented matrix\\\\n\\\", M)\\n\\nprint(\\\"This is the rref answer\\\\n\\\", rref(M))\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
       "                        }\n",
       "                        break;\n",
       "                    }\n",
       "                }\n",
       "            }, 500);\n",
       "            "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = np.matrix([[2, 2], [1, -4]])\n",
    "print(\"This is the left hand side\\n\", A)\n",
    "b = np.matrix([[5], [0]])\n",
    "print(\"This is the right hand side\\n\", b)\n",
    "\n",
    "M = np.c_[A, b]\n",
    "print(\"This is the augmented matrix\\n\", M)\n",
    "\n",
    "print(\"This is the rref answer\\n\", rref(M))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "l_5PUxlo-OnH"
   },
   "source": [
    "$$\n",
    "(b) \\hspace{.5cm}\n",
    "\\left\\{ \\begin{array}{rcr} -x + y &=& 1 \\\\ x + y &=& 2 \\\\ \\end{array}\n",
    "\\right.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "rgpL6ig2-QJf"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "This is the left hand side\n",
      " [[-1  1]\n",
      " [ 1  1]]\n",
      "This is the right hand side\n",
      " [[1]\n",
      " [2]]\n",
      "This is the augmented matrix\n",
      " [[-1  1  1]\n",
      " [ 1  1  2]]\n",
      "This is the rref answer\n",
      " [[1.  0.  0.5]\n",
      " [0.  1.  1.5]]\n"
     ]
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 25;\n",
       "                var nbb_unformatted_code = \"A = np.matrix([[-1, 1], [1, 1]])\\nprint(\\\"This is the left hand side\\\\n\\\", A)\\nb = np.matrix([[1], [2]])\\nprint(\\\"This is the right hand side\\\\n\\\", b)\\n\\nM = np.c_[A, b]\\nprint(\\\"This is the augmented matrix\\\\n\\\", M)\\n\\nprint(\\\"This is the rref answer\\\\n\\\", rref(M))\";\n",
       "                var nbb_formatted_code = \"A = np.matrix([[-1, 1], [1, 1]])\\nprint(\\\"This is the left hand side\\\\n\\\", A)\\nb = np.matrix([[1], [2]])\\nprint(\\\"This is the right hand side\\\\n\\\", b)\\n\\nM = np.c_[A, b]\\nprint(\\\"This is the augmented matrix\\\\n\\\", M)\\n\\nprint(\\\"This is the rref answer\\\\n\\\", rref(M))\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
       "                        }\n",
       "                        break;\n",
       "                    }\n",
       "                }\n",
       "            }, 500);\n",
       "            "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = np.matrix([[-1, 1], [1, 1]])\n",
    "print(\"This is the left hand side\\n\", A)\n",
    "b = np.matrix([[1], [2]])\n",
    "print(\"This is the right hand side\\n\", b)\n",
    "\n",
    "M = np.c_[A, b]\n",
    "print(\"This is the augmented matrix\\n\", M)\n",
    "\n",
    "print(\"This is the rref answer\\n\", rref(M))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "6pw-7ZxU-QuE"
   },
   "source": [
    "$$\n",
    "(c) \\hspace{.5cm}\n",
    "\\left\\{ \\begin{array}{rcr} x - 3y + z & = 1 \\\\ x + y + 2z & = & 14 \\\\ \\end{array}\n",
    "\\right.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "4Fm9M0Pg-S_F"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "This is the left hand side\n",
      " [[ 1 -3  1]\n",
      " [ 1  1  2]]\n",
      "This is the right hand side\n",
      " [[ 1]\n",
      " [14]]\n",
      "This is the augmented matrix\n",
      " [[ 1 -3  1  1]\n",
      " [ 1  1  2 14]]\n",
      "This is the rref answer\n",
      " [[ 1.    0.    1.75 10.75]\n",
      " [-0.    1.    0.25  3.25]]\n"
     ]
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 26;\n",
       "                var nbb_unformatted_code = \"A = np.matrix([[1, -3, 1], [1, 1, 2]])\\nprint(\\\"This is the left hand side\\\\n\\\", A)\\nb = np.matrix([[1], [14]])\\nprint(\\\"This is the right hand side\\\\n\\\", b)\\n\\nM = np.c_[A, b]\\nprint(\\\"This is the augmented matrix\\\\n\\\", M)\\n\\nprint(\\\"This is the rref answer\\\\n\\\", rref(M))\";\n",
       "                var nbb_formatted_code = \"A = np.matrix([[1, -3, 1], [1, 1, 2]])\\nprint(\\\"This is the left hand side\\\\n\\\", A)\\nb = np.matrix([[1], [14]])\\nprint(\\\"This is the right hand side\\\\n\\\", b)\\n\\nM = np.c_[A, b]\\nprint(\\\"This is the augmented matrix\\\\n\\\", M)\\n\\nprint(\\\"This is the rref answer\\\\n\\\", rref(M))\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
       "                        }\n",
       "                        break;\n",
       "                    }\n",
       "                }\n",
       "            }, 500);\n",
       "            "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = np.matrix([[1, -3, 1], [1, 1, 2]])\n",
    "print(\"This is the left hand side\\n\", A)\n",
    "b = np.matrix([[1], [14]])\n",
    "print(\"This is the right hand side\\n\", b)\n",
    "\n",
    "M = np.c_[A, b]\n",
    "print(\"This is the augmented matrix\\n\", M)\n",
    "\n",
    "print(\"This is the rref answer\\n\", rref(M))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "KiRFFlCt-Tr2"
   },
   "source": [
    "$$\n",
    "(d) \\hspace{.5cm}\n",
    "\\left\\{ \\begin{array}{rcr} -x - y &=& 1 \\\\ -3x -3 y &=& 2 \\\\ \\end{array}\n",
    "\\right.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "aoJNak7k-VlQ"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "This is the left hand side\n",
      " [[-1 -1]\n",
      " [-3 -3]]\n",
      "This is the right hand side\n",
      " [[1]\n",
      " [2]]\n",
      "This is the augmented matrix\n",
      " [[-1 -1  1]\n",
      " [-3 -3  2]]\n",
      "This is the rref answer\n",
      " [[1. 1. 0.]\n",
      " [0. 0. 1.]]\n"
     ]
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 27;\n",
       "                var nbb_unformatted_code = \"A = np.matrix([[-1, -1], [-3, -3]])\\nprint(\\\"This is the left hand side\\\\n\\\", A)\\nb = np.matrix([[1], [2]])\\nprint(\\\"This is the right hand side\\\\n\\\", b)\\n\\nM = np.c_[A, b]\\nprint(\\\"This is the augmented matrix\\\\n\\\", M)\\n\\nprint(\\\"This is the rref answer\\\\n\\\", rref(M))\";\n",
       "                var nbb_formatted_code = \"A = np.matrix([[-1, -1], [-3, -3]])\\nprint(\\\"This is the left hand side\\\\n\\\", A)\\nb = np.matrix([[1], [2]])\\nprint(\\\"This is the right hand side\\\\n\\\", b)\\n\\nM = np.c_[A, b]\\nprint(\\\"This is the augmented matrix\\\\n\\\", M)\\n\\nprint(\\\"This is the rref answer\\\\n\\\", rref(M))\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
       "                        }\n",
       "                        break;\n",
       "                    }\n",
       "                }\n",
       "            }, 500);\n",
       "            "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = np.matrix([[-1, -1], [-3, -3]])\n",
    "print(\"This is the left hand side\\n\", A)\n",
    "b = np.matrix([[1], [2]])\n",
    "print(\"This is the right hand side\\n\", b)\n",
    "\n",
    "M = np.c_[A, b]\n",
    "print(\"This is the augmented matrix\\n\", M)\n",
    "\n",
    "print(\"This is the rref answer\\n\", rref(M))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "wyc1fCaY-WFu"
   },
   "source": [
    "$$\n",
    "(e) \\hspace{.5cm}\n",
    "\\left\\{ \\begin{array}{rcr} 4y + z & = & 20 \\\\ 2x - 2y + z & = & 0 \\\\ x + z & = & 5 \\\\ x + y - z & = & 10 \\\\ \\end{array} \\right.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "98NNaKMX-Xiq"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "This is the left hand side\n",
      " [[ 0  4  1]\n",
      " [ 2 -2  1]\n",
      " [ 1  0  1]\n",
      " [ 1  1 -1]]\n",
      "This is the right hand side\n",
      " [[20]\n",
      " [ 0]\n",
      " [ 5]\n",
      " [10]]\n",
      "This is the augmented matrix\n",
      " [[ 0  4  1 20]\n",
      " [ 2 -2  1  0]\n",
      " [ 1  0  1  5]\n",
      " [ 1  1 -1 10]]\n",
      "This is the rref answer\n",
      " [[1. 0. 0. 5.]\n",
      " [0. 1. 0. 5.]\n",
      " [0. 0. 1. 0.]\n",
      " [0. 0. 0. 0.]]\n"
     ]
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 28;\n",
       "                var nbb_unformatted_code = \"A = np.matrix([[0, 4, 1], [2, -2, 1], [1, 0, 1], [1, 1, -1]])\\nprint(\\\"This is the left hand side\\\\n\\\", A)\\nb = np.matrix([[20], [0], [5], [10]])\\nprint(\\\"This is the right hand side\\\\n\\\", b)\\n\\nM = np.c_[A, b]\\nprint(\\\"This is the augmented matrix\\\\n\\\", M)\\n\\nprint(\\\"This is the rref answer\\\\n\\\", rref(M))\";\n",
       "                var nbb_formatted_code = \"A = np.matrix([[0, 4, 1], [2, -2, 1], [1, 0, 1], [1, 1, -1]])\\nprint(\\\"This is the left hand side\\\\n\\\", A)\\nb = np.matrix([[20], [0], [5], [10]])\\nprint(\\\"This is the right hand side\\\\n\\\", b)\\n\\nM = np.c_[A, b]\\nprint(\\\"This is the augmented matrix\\\\n\\\", M)\\n\\nprint(\\\"This is the rref answer\\\\n\\\", rref(M))\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
       "                        }\n",
       "                        break;\n",
       "                    }\n",
       "                }\n",
       "            }, 500);\n",
       "            "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = np.matrix([[0, 4, 1], [2, -2, 1], [1, 0, 1], [1, 1, -1]])\n",
    "print(\"This is the left hand side\\n\", A)\n",
    "b = np.matrix([[20], [0], [5], [10]])\n",
    "print(\"This is the right hand side\\n\", b)\n",
    "\n",
    "M = np.c_[A, b]\n",
    "print(\"This is the augmented matrix\\n\", M)\n",
    "\n",
    "print(\"This is the rref answer\\n\", rref(M))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "22QYFvJX-a6O"
   },
   "source": [
    "$$\n",
    "(f) \\hspace{.5cm}\n",
    "\\left\\{ \\begin{array}{rcr} 2x + z + w & = & 5 \\\\ y - w & = & -1 \\\\ 3x - z - w & = & 0 \\\\ 4x + y + 2z + w & = & 9 \\\\ \\end{array}\n",
    "\\right.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "r3WB6JRn-b5W"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "This is the left hand side\n",
      " [[ 2  0  1  1]\n",
      " [ 0  1  0 -1]\n",
      " [ 3  0 -1 -1]\n",
      " [ 4  1  2  1]]\n",
      "This is the right hand side\n",
      " [[ 5]\n",
      " [-1]\n",
      " [ 0]\n",
      " [ 9]]\n",
      "This is the augmented matrix\n",
      " [[ 2  0  1  1  5]\n",
      " [ 0  1  0 -1 -1]\n",
      " [ 3  0 -1 -1  0]\n",
      " [ 4  1  2  1  9]]\n",
      "This is the rref answer\n",
      " [[ 1.  0.  0.  0.  1.]\n",
      " [-0.  1.  0. -1. -1.]\n",
      " [-0. -0.  1.  1.  3.]\n",
      " [ 0.  0.  0.  0.  0.]]\n"
     ]
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 29;\n",
       "                var nbb_unformatted_code = \"A = np.matrix([[2, 0, 1, 1], [0, 1, 0, -1], [3, 0, -1, -1], [4, 1, 2, 1]])\\nprint(\\\"This is the left hand side\\\\n\\\", A)\\nb = np.matrix([[5], [-1], [0], [9]])\\nprint(\\\"This is the right hand side\\\\n\\\", b)\\n\\nM = np.c_[A, b]\\nprint(\\\"This is the augmented matrix\\\\n\\\", M)\\n\\nprint(\\\"This is the rref answer\\\\n\\\", rref(M))\";\n",
       "                var nbb_formatted_code = \"A = np.matrix([[2, 0, 1, 1], [0, 1, 0, -1], [3, 0, -1, -1], [4, 1, 2, 1]])\\nprint(\\\"This is the left hand side\\\\n\\\", A)\\nb = np.matrix([[5], [-1], [0], [9]])\\nprint(\\\"This is the right hand side\\\\n\\\", b)\\n\\nM = np.c_[A, b]\\nprint(\\\"This is the augmented matrix\\\\n\\\", M)\\n\\nprint(\\\"This is the rref answer\\\\n\\\", rref(M))\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
       "                        }\n",
       "                        break;\n",
       "                    }\n",
       "                }\n",
       "            }, 500);\n",
       "            "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = np.matrix([[2, 0, 1, 1], [0, 1, 0, -1], [3, 0, -1, -1], [4, 1, 2, 1]])\n",
    "print(\"This is the left hand side\\n\", A)\n",
    "b = np.matrix([[5], [-1], [0], [9]])\n",
    "print(\"This is the right hand side\\n\", b)\n",
    "\n",
    "M = np.c_[A, b]\n",
    "print(\"This is the augmented matrix\\n\", M)\n",
    "\n",
    "print(\"This is the rref answer\\n\", rref(M))"
   ]
  }
 ],
 "metadata": {
  "colab": {
   "authorship_tag": "ABX9TyOwwAg31yhYOUQwpUs7IHyH",
   "include_colab_link": true,
   "name": "Math_242_solve_systems_using_computation_and_rref.ipynb",
   "provenance": []
  },
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.4"
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 },
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