08.2.2 Gauss-Jordan with Appended System.ipynb

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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "8.2.2 Gauss-Jordan with Appended System"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<font color=red> Copy this notebook first! At each step, if you make a mistake, rerun all the cells above the current cell!!!</font>\n",
      "\n",
      "<br>\n",
      "\n",
      "With this notebook, we walk you through both Homework 8.2.1.1 and Homework 8.2.2.1.  Let's start with the first step in Homework 8.2.1.1:\n",
      "\n",
      "<img src=\"https://studio.edx.org/c4x/UTAustinX/UT.5.01x/asset/8_2_1_1_QuestionA.png\" width=\"75%\"> </img>\n",
      "\n",
      "Let's set this up as an appended matrix and the application of a Gauss transform.  Here matrix $ A $ holds the appended system\n",
      "$ \n",
      "\\left( \\begin{array}{ r r r | r }\n",
      "-2 & 2 & -5 & -7 \\\\\n",
      "2 & -3 & 7 & 11 \\\\\n",
      "-4 & 3 & -7 & -9\n",
      "\\end{array} \\right)\n",
      "$\n",
      "Now, fill in the ? in $ G_0 $ with appropriate values to introduce zeroes in the off-diagonal elements of the first column of $ A $:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "\n",
      "A = np.matrix( '-2,  2, -5, -7;\\\n",
      "                 2, -3,  7, 11;\\\n",
      "                -4,  3, -7, -9' )\n",
      "\n",
      "print( 'A = ' )\n",
      "print( A )\n",
      "\n",
      "G0 = np.matrix( ' 1,  0,  0;\\\n",
      "                  ?,  1,  0;\\\n",
      "                  ?,  0,  1' );\n",
      "\n",
      "A0 = G0 * A\n",
      "\n",
      "print( 'A0 = ' )\n",
      "print( A0 )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If you did this right, the result matrix $ A_0 = G_0 A $ has zeroes in the off-diagonal entries of the first column, which then corresponds to the below answer to the first step in Homework 8.2.2.1:\n",
      "\n",
      "<img src=\"https://studio.edx.org/c4x/UTAustinX/UT.5.01x/asset/8_2_2_1_QuestionA.png\" width=\"75%\"> </img>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This also answers the question for Step 1 in Homework 8.2.1.1.\n",
      "\n",
      "Now, let's look at the next step in Homework 8.2.1.1:\n",
      "\n",
      "<img src=\"https://studio.edx.org/c4x/UTAustinX/UT.5.01x/asset/8_2_1_1_QuestionB.png\" width=\"75%\"> </img>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The appended system is now represented by <code> A0 </code>:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print( 'A0 =' )\n",
      "print( A0 )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, fill in the ? in $ G_1 $ with appropriate values to introduce zeroes in the off-diagonal elemental of the second column of $ A_0 $:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "G1 = np.matrix( ' 1,  ?,  0;\\\n",
      "                  0,  1,  0;\\\n",
      "                  0,  ?,  1' );\n",
      "\n",
      "A1 = G1 * A0\n",
      "\n",
      "print( 'A1 = ' )\n",
      "print( A1 )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If you did this right, the result matrix $ A_1 = G_1 A_0 = G_1 G_0 A $ has zeroes in the off-diagonal entries of the first column and the second column, which then corresponds to the below answer to the second step in Homework 8.2.2.1:\n",
      "\n",
      "<img src=\"https://studio.edx.org/c4x/UTAustinX/UT.5.01x/asset/8_2_2_1_QuestionB.png\" width=\"75%\"> </img>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This also answers the question for Step 2 in Homework 8.2.1.1.\n",
      "\n",
      "Now, let's look at the third step in Homework 8.2.1.1:\n",
      "\n",
      "<img src=\"https://studio.edx.org/c4x/UTAustinX/UT.5.01x/asset/8_2_1_1_Questionc.png\" width=\"75%\"> </img>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The appended system is now represented by <code> A1 </code>:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print( 'A1 =' )\n",
      "print( A1 )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, fill in the ? in $ G_2 $ with appropriate values to introduce zeroes in the off-diagonal elemental of the third column of $ A_1 $:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "G2 = np.matrix( ' 1,  0,  ?;\\\n",
      "                  0,  1,  ?;\\\n",
      "                  0,  0,  1' );\n",
      "\n",
      "A2 = G2 * A1\n",
      "\n",
      "print( 'A2 = ' )\n",
      "print( A2 )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If you did this right, the result matrix $ A_2 = G_2 A_1 = G_2 G_1 A_0 = G_2 G_1 G_0 A $ has zeroes in the off-diagonal entries of the first, second, and third column, which then corresponds to the below answer to the third step in Homework 8.2.2.1:\n",
      "\n",
      "<img src=\"https://studio.edx.org/c4x/UTAustinX/UT.5.01x/asset/8_2_2_1_QuestionC.png\" width=\"75%\"> </img>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This also answers the question for Step 3 in Homework 8.2.1.1.\n",
      "\n",
      "Now, let's look at the fourth step in Homework 8.2.1.1:\n",
      "\n",
      "<img src=\"https://studio.edx.org/c4x/UTAustinX/UT.5.01x/asset/8_2_1_1_QuestionD.png\" width=\"75%\"> </img>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The appended system is now represented by <code> A2 </code>:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print( 'A2 =' )\n",
      "print( A2 )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, fill in the ? in $ D $ with appropriate values to set the diagonal elements in the appended system to one, making the first three columns into an identity matrix:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "D = np.matrix( '  ?,  0,  0;\\\n",
      "                  0,  ?,  0;\\\n",
      "                  0,  0,  ?' );\n",
      "\n",
      "A3 = D * A2\n",
      "\n",
      "print( 'A3 = ' )\n",
      "print( A3 )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If you did this right, the result matrix \n",
      "\n",
      "$ A_3 = D A_2 = D G_2 A_1 = D G_2 G_1 A_0 = D G_2 G_1 G_0 A $ \n",
      "\n",
      "has an identity matrix in the first three columns, which then corresponds to the below answer to the fourth step in Homework 8.2.2.1:\n",
      "\n",
      "<img src=\"https://studio.edx.org/c4x/UTAustinX/UT.5.01x/asset/8_2_2_1_QuestionD.png\" width=\"75%\"> </img>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This also answers the question for Step 4 in Homework 8.2.1.1.\n",
      "\n",
      "Now, the answer above is an appended system that represent the system of linear equations\n",
      "\n",
      "$ \\begin{array}{c c c c r}\n",
      "\\chi_0 & & & = & -1 \\\\\n",
      "& \\chi_1 & & = & -2 \\\\\n",
      "& & \\chi_2 & = & 1\n",
      "\\end{array}\n",
      " $\n",
      "\n",
      "which provides the answer to the linear system with which we started:\n",
      "\n",
      "<img src=\"https://studio.edx.org/c4x/UTAustinX/UT.5.01x/asset/8_2_2_1_QuestionE.png\" width=\"75%\"> </img>\n",
      "\n",
      "We can check that we've just done this:\n",
      "\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = np.matrix( '-2,  2, -5;\\\n",
      "                 2, -3,  7;\\\n",
      "                -4,  3, -7' )\n",
      "\n",
      "print( 'A = ' )\n",
      "print( A )\n",
      "\n",
      "x = np.matrix( '-1;\\\n",
      "                -2;\\\n",
      "                 1' )\n",
      "\n",
      "print( 'x = ' )\n",
      "print( x )\n",
      "\n",
      "print( ' A * x = ' )\n",
      "print( A * x )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
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