18.06-PS4.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
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   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "INFO: Removing Homework (unregistered)\n",
      "INFO: Cloning Homework from git://github.com/shashi/Homework.jl.git\n",
      "INFO: Computing changes...\n",
      "INFO: No packages to install, update or remove\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<script charset=\"utf-8\">(function ($, undefined) {\n",
       "\n",
       "    function createElem(tag, attr, content) {\n",
       "\t// TODO: remove jQuery dependency\n",
       "\tvar el = $(\"<\" + tag + \"/>\").attr(attr);\n",
       "\tif (content) {\n",
       "\t    el.append(content);\n",
       "\t}\n",
       "\treturn el[0];\n",
       "    }\n",
       "\n",
       "    // A widget must expose an id field which identifies it to the backend,\n",
       "    // an elem attribute which is will be added to the DOM, and\n",
       "    // a getState() method which returns the value to be sent to the backend\n",
       "    // a sendUpdate() method which sends its current value to the backend\n",
       "    var Widget = {\n",
       "\tid: undefined,\n",
       "\telem: undefined,\n",
       "\tlabel: undefined,\n",
       "\tgetState: function () {\n",
       "\t    return this.elem.value;\n",
       "\t},\n",
       "\tsendUpdate: undefined\n",
       "    };\n",
       "\n",
       "    var Slider = function (typ, id, init) {\n",
       "\tvar attr = { type:  \"range\",\n",
       "\t\t     value: init.value,\n",
       "\t\t     min:   init.min,\n",
       "\t\t     max:   init.max,\n",
       "\t\t     step:  init.step },\n",
       "\t    elem = createElem(\"input\", attr),\n",
       "\t    self = this;\n",
       "\n",
       "\telem.onchange = function () {\n",
       "\t    self.sendUpdate();\n",
       "\t}\n",
       "\n",
       "\tthis.id = id;\n",
       "\tthis.elem = elem;\n",
       "\tthis.label = init.label;\n",
       "\n",
       "\tInputWidgets.commInitializer(this); // Initialize communication\n",
       "    }\n",
       "    Slider.prototype = Widget;\n",
       "\n",
       "    var Checkbox = function (typ, id, init) {\n",
       "\tvar attr = { type: \"checkbox\",\n",
       "\t\t     checked: init.value },\n",
       "\t    elem = createElem(\"input\", attr),\n",
       "\t    self = this;\n",
       "\n",
       "\tthis.getState = function () {\n",
       "\t    return elem.checked;\n",
       "\t}\n",
       "\telem.onchange = function () {\n",
       "\t    self.sendUpdate();\n",
       "\t}\n",
       "\n",
       "\tthis.id = id;\n",
       "\tthis.elem = elem;\n",
       "\tthis.label = init.label;\n",
       "\n",
       "\tInputWidgets.commInitializer(this);\n",
       "    }\n",
       "    Checkbox.prototype = Widget;\n",
       "\n",
       "    var Button = function (typ, id, init) {\n",
       "\tvar attr = { type:    \"button\",\n",
       "\t\t     value:   init.label },\n",
       "\t    elem = createElem(\"input\", attr),\n",
       "\t    self = this;\n",
       "\tthis.getState = function () {\n",
       "\t    return null;\n",
       "\t}\n",
       "\telem.onclick = function () {\n",
       "\t    self.sendUpdate();\n",
       "\t}\n",
       "\n",
       "\tthis.id = id;\n",
       "\tthis.elem = elem;\n",
       "\tthis.label = init.label;\n",
       "\n",
       "\tInputWidgets.commInitializer(this);\n",
       "    }\n",
       "    Button.prototype = Widget;\n",
       "\n",
       "    var Text = function (typ, id, init) {\n",
       "\tvar attr = { type:  \"text\",\n",
       "\t\t     placeholder: init.label,\n",
       "\t\t     value: init.value },\n",
       "\t    elem = createElem(\"input\", attr),\n",
       "\t    self = this;\n",
       "\tthis.getState = function () {\n",
       "\t    return elem.value;\n",
       "\t}\n",
       "\telem.onkeyup = function () {\n",
       "\t    self.sendUpdate();\n",
       "\t}\n",
       "\n",
       "\tthis.id = id;\n",
       "\tthis.elem = elem;\n",
       "\tthis.label = init.label;\n",
       "\n",
       "\tInputWidgets.commInitializer(this);\n",
       "    }\n",
       "    Text.prototype = Widget;\n",
       "\n",
       "    var Textarea = function (typ, id, init) {\n",
       "\tvar attr = { placeholder: init.label },\n",
       "\t    elem = createElem(\"textarea\", attr, init.value),\n",
       "\t    self = this;\n",
       "\tthis.getState = function () {\n",
       "\t    return elem.value;\n",
       "\t}\n",
       "\telem.onchange = function () {\n",
       "\t    self.sendUpdate();\n",
       "\t}\n",
       "\n",
       "\tthis.id = id;\n",
       "\tthis.elem = elem;\n",
       "\tthis.label = init.label;\n",
       "\n",
       "\tInputWidgets.commInitializer(this);\n",
       "    }\n",
       "    Textarea.prototype = Widget;\n",
       "\n",
       "    // RadioButtons\n",
       "    // Dropdown\n",
       "    // HTML\n",
       "    // Latex\n",
       "\n",
       "    var InputWidgets = {\n",
       "\tSlider: Slider,\n",
       "\tCheckbox: Checkbox,\n",
       "\tButton: Button,\n",
       "\tText: Text,\n",
       "\tTextarea: Textarea,\n",
       "\tdebug: false,\n",
       "\tlog: function () {\n",
       "\t    if (InputWidgets.debug) {\n",
       "\t\tconsole.log.apply(console, arguments);\n",
       "\t    }\n",
       "\t},\n",
       "\t// a central way to initalize communication\n",
       "\t// for widgets.\n",
       "\tcommInitializer: function (widget) {\n",
       "\t    widget.sendUpdate = function () {};\n",
       "\t}\n",
       "    };\n",
       "\n",
       "    window.InputWidgets = InputWidgets;\n",
       "\n",
       "})(jQuery, undefined);\n",
       "</script>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<div id=\"interact-js-shim\">\n",
       "    <script charset=\"utf-8\">\n",
       "(function (IPython, $, _, MathJax, Widgets) {\n",
       "    $.event.special.destroyed = {\n",
       "\tremove: function(o) {\n",
       "\t    if (o.handler) {\n",
       "\t\to.handler.apply(this, arguments)\n",
       "\t    }\n",
       "\t}\n",
       "    }\n",
       "\n",
       "    var OutputArea = IPython.version >= \"4.0.0\" ? require(\"notebook/js/outputarea\").OutputArea : IPython.OutputArea;\n",
       "\n",
       "    var redrawValue = function (container, type, val) {\n",
       "\tvar selector = $(\"<div/>\");\n",
       "\tvar oa = new OutputArea(_.extend(selector, {\n",
       "\t    selector: selector,\n",
       "\t    prompt_area: true,\n",
       "\t    events: IPython.events,\n",
       "\t    keyboard_manager: IPython.keyboard_manager\n",
       "\t})); // Hack to work with IPython 2.1.0\n",
       "\n",
       "\tswitch (type) {\n",
       "\tcase \"image/png\":\n",
       "            var _src = 'data:' + type + ';base64,' + val;\n",
       "\t    $(container).find(\"img\").attr('src', _src);\n",
       "\t    break;\n",
       "\tdefault:\n",
       "\t    var toinsert = OutputArea.append_map[type].apply(\n",
       "\t\toa, [val, {}, selector]\n",
       "\t    );\n",
       "\t    $(container).empty().append(toinsert.contents());\n",
       "\t    selector.remove();\n",
       "\t}\n",
       "\tif (type === \"text/latex\" && MathJax) {\n",
       "\t    MathJax.Hub.Queue([\"Typeset\", MathJax.Hub, toinsert.get(0)]);\n",
       "\t}\n",
       "    }\n",
       "\n",
       "\n",
       "    $(document).ready(function() {\n",
       "\tWidgets.debug = false; // log messages etc in console.\n",
       "\tfunction initComm(evt, data) {\n",
       "\t    var comm_manager = data.kernel.comm_manager;\n",
       "        //_.extend(comm_manager.targets, require(\"widgets/js/widget\"))\n",
       "\t    comm_manager.register_target(\"Signal\", function (comm) {\n",
       "            comm.on_msg(function (msg) {\n",
       "                //Widgets.log(\"message received\", msg);\n",
       "                var val = msg.content.data.value;\n",
       "                $(\".signal-\" + comm.comm_id).each(function() {\n",
       "                var type = $(this).data(\"type\");\n",
       "                if (val[type]) {\n",
       "                    redrawValue(this, type, val[type], type);\n",
       "                }\n",
       "                });\n",
       "                delete val;\n",
       "                delete msg.content.data.value;\n",
       "            });\n",
       "\t    });\n",
       "\n",
       "\t    // coordingate with Comm and redraw Signals\n",
       "\t    // XXX: Test using Reactive here to improve performance\n",
       "\t    $([IPython.events]).on(\n",
       "\t\t'output_appended.OutputArea', function (event, type, value, md, toinsert) {\n",
       "\t\t    if (md && md.reactive) {\n",
       "                // console.log(md.comm_id);\n",
       "                toinsert.addClass(\"signal-\" + md.comm_id);\n",
       "                toinsert.data(\"type\", type);\n",
       "                // Signal back indicating the mimetype required\n",
       "                var comm_manager = IPython.notebook.kernel.comm_manager;\n",
       "                var comm = comm_manager.comms[md.comm_id];\n",
       "                comm.then(function (c) {\n",
       "                    c.send({action: \"subscribe_mime\",\n",
       "                       mime: type});\n",
       "                    toinsert.bind(\"destroyed\", function() {\n",
       "                        c.send({action: \"unsubscribe_mime\",\n",
       "                               mime: type});\n",
       "                    });\n",
       "                })\n",
       "\t\t    }\n",
       "\t    });\n",
       "\t}\n",
       "\n",
       "\ttry {\n",
       "\t    // try to initialize right away. otherwise, wait on the status_started event.\n",
       "\t    initComm(undefined, IPython.notebook);\n",
       "\t} catch (e) {\n",
       "\t    $([IPython.events]).on('kernel_created.Kernel kernel_created.Session', initComm);\n",
       "\t}\n",
       "    });\n",
       "})(IPython, jQuery, _, MathJax, InputWidgets);\n",
       "</script>\n",
       "    <script>\n",
       "        window.interactLoadedFlag = true\n",
       "       $(\"#interact-js-shim\").bind(\"destroyed\", function () {\n",
       "           if (window.interactLoadedFlag) {\n",
       "               console.warn(\"JavaScript required by Interact will be removed if you remove this cell or run using Interact more than once.\")\n",
       "           }\n",
       "       })\n",
       "       $([IPython.events]).on(\"kernel_starting.Kernel kernel_restarting.Kernel\", function () { window.interactLoadedFlag = false })\n",
       "   </script>\n",
       "</div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: This version of the GnuTLS library (3.2.11) is deprecated\n",
      "and contains known security vulnerabilities. Please upgrade to a\n",
      "more recent version.\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<script>(function ($) {\n",
       "        // hijack IPython's methods of all CodeCells with the required metadata\n",
       "    if (!window.Homework) {\n",
       "        var get_text = IPython.CodeCell.prototype.get_text\n",
       "        IPython.CodeCell.prototype.get_text = function () {\n",
       "            if (this.metadata && typeof(this.metadata.question) !== \"undefined\") {\n",
       "                var fn = (Homework.mode() == \"create\") ? \"new_question\" : \"attempt_prompt\";\n",
       "                return 'Homework.' + fn + '(' + [\n",
       "                                         JSON.stringify(JSON.stringify(this.metadata)),\n",
       "                                         \"begin \\n\" + get_text.call(this) +\"\\n end\"\n",
       "                                       ].join(\", \") + \")\"\n",
       "            } else {\n",
       "                console.log(\"vs\", get_text.call(this))\n",
       "                return get_text.call(this)\n",
       "            }\n",
       "        }\n",
       "\n",
       "        var cell_to_json = IPython.CodeCell.prototype.toJSON\n",
       "        IPython.CodeCell.prototype.toJSON = function () {\n",
       "            if (this.metadata && typeof(this.metadata.question) !== \"undefined\") {\n",
       "                data = cell_to_json.call(this)\n",
       "                data.source = get_text.call(this)\n",
       "                return data\n",
       "            } else {\n",
       "                return cell_to_json.call(this)\n",
       "            }\n",
       "        }\n",
       "\n",
       "        var rename_keys = IPython.OutputArea.prototype.rename_keys;\n",
       "        IPython.OutputArea.prototype.rename_keys = function (data, key_map) {\n",
       "            data = rename_keys(data, key_map)\n",
       "\n",
       "            if (data.metadata && data.metadata.reactive) {\n",
       "                var cls = \".signal-\" + data.metadata.comm_id\n",
       "                data.html = $(cls).eq(0).html()\n",
       "            }\n",
       "            return data\n",
       "        }\n",
       "\n",
       "        var delete_cell = IPython.Notebook.prototype.delete_cell\n",
       "        IPython.Notebook.prototype.delete_cell = function (index) {\n",
       "            var i = this.index_or_selected(index);\n",
       "            var cell = this.get_selected_cell();\n",
       "            if (cell.metadata && typeof(cell.metadata.question) !== \"undefined\") {\n",
       "                alert(\"Cannot delete this cell. You need to code the answer to Question \" + \n",
       "                        cell.metadata.question + \" here.\")\n",
       "            } else {\n",
       "                // go on and delete\n",
       "                delete_cell.call(this, i)\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "\n",
       "\n",
       "    function get_question(q) {\n",
       "        var i = 0\n",
       "        var cell = IPython.notebook.get_cell(i)\n",
       "        while(cell !== null) {\n",
       "            if (cell.metadata && cell.metadata.question === q) {\n",
       "                return cell\n",
       "            }\n",
       "            cell = IPython.notebook.get_cell(i)\n",
       "            i+=1\n",
       "        }\n",
       "        return null\n",
       "    }\n",
       "\n",
       "    function mount_message(cell) {\n",
       "        if (!cell) { return }\n",
       "        var meta = cell.metadata,\n",
       "            msg = meta.msg || \"<span class='icon-terminal'></span> &nbsp; Code your answer above, hit &lt;Shift+Enter&gt;, then Submit. <span style='float: right'>\",\n",
       "            score = meta.score || 0,\n",
       "            max_score = meta.max_score || 0,\n",
       "            attempts = meta.attempts || 0,\n",
       "            max_attempts = meta.max_attempts || 0\n",
       "\n",
       "        msg += \"<span style='position:absolute; top: 0.5em; right:0.5em' class='label label-info'> Score: \"\n",
       "        if (Homework.mode() === \"answering\") {\n",
       "            msg += \"<b>\" + score + \"</b> / <b>\" + max_score + \"</b>\"\n",
       "        } else if (Homework.mode() === \"create\") {\n",
       "            msg += \"<b>\" + max_score + \"</b>\"\n",
       "        }\n",
       "\n",
       "        msg += \" &middot; Attempts: \"\n",
       "        if (Homework.mode() === \"answering\") {\n",
       "            msg += \"<b>\" + attempts + \"</b>\"\n",
       "            if (max_attempts != 0) {\n",
       "                msg += \" / <b>\" + max_attempts + \"</b>\"\n",
       "            }\n",
       "        } else if (Homework.mode() === \"create\") {\n",
       "            msg += \"<b>\" + max_attempts + \"</b>\"\n",
       "        }\n",
       "        msg += \"</span>\"\n",
       "\n",
       "        var level = meta.alert || \"info\"\n",
       "\n",
       "        msg = \"<div class='hw-msg alert alert-\" + level + \"' id='hw-msg-\" + meta.question +\n",
       "               \"' style='padding: 0.5em; margin: 0; position:relative; border-radius: 0'>\" +\n",
       "               msg + \"</div>\"\n",
       "\n",
       "        $(cell.element).find(\".hw-msg\").remove()\n",
       "        $(cell.element).find(\".input_area\").eq(0).append(msg)\n",
       "        if (cell.metadata.finished) { $(cell.element).find(\".widget-area\").hide() }\n",
       "    }\n",
       "\n",
       "    function set_meta(question, extension) {\n",
       "        console.log(\"set meta\", question, extension)\n",
       "        var cell = get_question(question)\n",
       "        for (var key in extension) {\n",
       "            if (extension.hasOwnProperty(key)) {\n",
       "                cell.metadata[key] = extension[key]\n",
       "            }\n",
       "        }\n",
       "        mount_message(cell)\n",
       "    }\n",
       "\n",
       "    function get_question_cells() {\n",
       "        var cell = IPython.notebook.get_cell(0),\n",
       "            i = 0,\n",
       "            cells = []\n",
       "        while(cell !== null) {\n",
       "            if (cell.metadata && typeof(cell.metadata.question) !== \"undefined\") {\n",
       "                var q = cell.metadata.question\n",
       "                cells.push(cell)\n",
       "            }\n",
       "            i+=1\n",
       "            cell = IPython.notebook.get_cell(i)\n",
       "        }\n",
       "        return cells\n",
       "    }\n",
       "\n",
       "    function hw_create(host, course, s, f) {\n",
       "        if(!s) {\n",
       "            s = function(result){\n",
       "                if(result.code == 0) {\n",
       "                    IPython.dialog.modal({\n",
       "                        title: 'Problemset created',\n",
       "                        body: \"The problem set was created. Now run <code>Homework.clear_answers()</code> to get the notebook ready for distribution.\"\n",
       "                    })\n",
       "                }\n",
       "                else {\n",
       "                    IPython.dialog.modal({\n",
       "                        title: 'Problemset NOT created',\n",
       "                        body: \"Non-zero result code. There was a problem saving the problem set!\"\n",
       "                    })\n",
       "                }\n",
       "            };\n",
       "        };\n",
       "        if(!f) {\n",
       "            f = function() {\n",
       "                    IPython.dialog.modal({\n",
       "                        title: 'Failure',\n",
       "                        body: \"Failed to create problem set. Are you sure you have permission to edit the course? Unexpected server error.\"\n",
       "                    })\n",
       "            };\n",
       "        };\n",
       "        console.log({\n",
       "            'mode': 'create',\n",
       "            'params': JSON.stringify(course)\n",
       "        })\n",
       "        $.ajax({\n",
       "            url: host + '/jboxplugin/hw/',\n",
       "            method: 'POST',\n",
       "            data: {\n",
       "                'mode': 'create',\n",
       "                'params': JSON.stringify(course)\n",
       "            },\n",
       "            success: s,\n",
       "            failure: f\n",
       "        })\n",
       "    }\n",
       "\n",
       "    function create_problemset(config) {\n",
       "        config = config || IPython.notebook.metadata.homework\n",
       "\n",
       "        var cells = get_question_cells()\n",
       "\n",
       "        if (!config.host) { config.host == \"\" }\n",
       "        var questions = []\n",
       "        for (var i=0, l=cells.length; i < l; i++) {\n",
       "            questions.push({\n",
       "                id:    String(cells[i].metadata.question),\n",
       "                score: cells[i].metadata.max_score,\n",
       "                attempts: cells[i].metadata.max_attempts,\n",
       "                ans:   cells[i].metadata.answer,\n",
       "                explanation: cells[i].metadata.explanation || \"\"\n",
       "            })\n",
       "        }\n",
       "\n",
       "        var course = {\n",
       "             admins: config.admins || [],\n",
       "             id: config.course,\n",
       "             problemsets: [{\n",
       "                 id: config.problemset,\n",
       "                 questions: questions\n",
       "             }]\n",
       "        }\n",
       "\n",
       "        console.log(course)\n",
       "\n",
       "        try {\n",
       "            hw_create(config.host || \"\", course)\n",
       "        } catch (e) {\n",
       "            alert(\"An error occured. Problem set not saved!\")\n",
       "        }\n",
       "   }\n",
       "\n",
       "   function clear_admin_metadata() {\n",
       "       var cells = get_question_cells()\n",
       "       for (var i=0, l=cells.length; i < l; i++) {\n",
       "           var cell = cells[i]\n",
       "           delete cell.metadata.answer\n",
       "           delete cell.metadata.msg\n",
       "           cell.metadata.finished && delete cell.metadata.finished\n",
       "           delete cell.metadata.alert\n",
       "       }\n",
       "       IPython.notebook.metadata.homework.mode = \"answering\"\n",
       "       Homework.refresh_messages()\n",
       "   }\n",
       "\n",
       "   function clear_answers() {\n",
       "       var cells = get_question_cells()\n",
       "       for (var i=0, l=cells.length; i < l; i++) {\n",
       "           var cell = cells[i]\n",
       "           cell.clear_input()\n",
       "           cell.clear_output()\n",
       "       }\n",
       "       IPython.notebook.metadata.homework.mode = \"answering\"\n",
       "       Homework.refresh_messages()\n",
       "   }\n",
       "\n",
       "    function refresh_messages(q) {\n",
       "        _.map(q ? [get_question(q)] : get_question_cells(), mount_message)\n",
       "    }\n",
       "\n",
       "    $([IPython.events]).on('notebook_loaded.Notebook', refresh_messages)\n",
       "    $([IPython.events]).on('select.Cell', refresh_messages)\n",
       "\n",
       "    setTimeout(refresh_messages, 100)\n",
       "    setTimeout(refresh_messages, 200)\n",
       "    setTimeout(refresh_messages, 400)\n",
       "    setTimeout(refresh_messages, 500)\n",
       "    setTimeout(refresh_messages, 1000)\n",
       "    setTimeout(refresh_messages, 2000)\n",
       "    setTimeout(refresh_messages, 4000)\n",
       "\n",
       "    $([IPython.events]).on('status_started.Kernel', function (data) {\n",
       "        var k = data.kernel || IPython.notebook.kernel\n",
       "        k.comm_manager.register_target(\"HomeworkData\", function (comm) {})\n",
       "    })\n",
       "    IPython.notebook && IPython.notebook.kernel &&\n",
       "        IPython.notebook.kernel.comm_manager.register_target(\"HomeworkData\", function (comm) {})\n",
       "\n",
       "    window.Homework = {\n",
       "        config: {},\n",
       "        metadata: function (field) {\n",
       "            return IPython.notebook.metadata.homework[field]\n",
       "        },\n",
       "        mode: function () {\n",
       "            return Homework.metadata(\"mode\") || \"answering\"\n",
       "        },\n",
       "        set_meta: set_meta,\n",
       "        mount_message: mount_message,\n",
       "        clear_answers: clear_answers,\n",
       "        clear_admin_metadata: clear_admin_metadata,\n",
       "        get_question: get_question,\n",
       "        get_question_cells: get_question_cells,\n",
       "        refresh_messages: refresh_messages,\n",
       "        create_problemset: create_problemset\n",
       "    }\n",
       "\n",
       "})(jQuery)\n",
       "</script>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<script>IPython.notebook.kernel.comm_manager.comms[\"cafa1f12-7b24-4e48-b216-4e8c3f492da4\"].then(\n",
       "    function (comm) {\n",
       "        comm.send({\n",
       "            cookie: document.cookie,\n",
       "            host: IPython.notebook.metadata.homework.host,\n",
       "            course: IPython.notebook.metadata.homework.course,\n",
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       "            problemset: IPython.notebook.metadata.homework.problemset\n",
       "        })\n",
       "    }\n",
       ")</script>"
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       "Html(\"<script>IPython.notebook.kernel.comm_manager.comms[\\\"cafa1f12-7b24-4e48-b216-4e8c3f492da4\\\"].then(\\n    function (comm) {\\n        comm.send({\\n            cookie: document.cookie,\\n            host: IPython.notebook.metadata.homework.host,\\n            course: IPython.notebook.metadata.homework.course,\\n            mode: IPython.notebook.metadata.homework.mode,\\n            problemset: IPython.notebook.metadata.homework.problemset\\n        })\\n    }\\n)</script>\")"
      ]
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     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<iframe src=\"https://docs.google.com/forms/d/1uPqGeJChe0FFOsuUkdeLVbgPo9LzQEbRnS0xIzE-VGo/viewform?embedded=true\" width=\"720\" height=\"570\" frameborder=\"0\" marginheight=\"0\" marginwidth=\"0\">Loading...</iframe>"
      ]
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "\n",
    "# Running this cell will set up auto-grading\n",
    "# It might take a while to execute.\n",
    "\n",
    "# Please fill in the MIT id form that appears below. You only need to do this once.\n",
    "\n",
    "Pkg.rm(\"Homework\")\n",
    "Pkg.clone(\"git://github.com/shashi/Homework.jl.git\")\n",
    "\n",
    "using Homework\n",
    "\n",
    "Homework.show_mit_form()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [],
      "text/plain": [
       "Options{:ToggleButtons,ASCIIString}([Input{ASCIIString}] score,\"report\",\"score\",\"Score\",OptionDict({\"Score\",\"Incorrect attempts\"},{\"Score\"=>\"score\",\"Incorrect attempts\"=>\"attempts\"}),None[],None[])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><tr><th></th><th>Names</th><th>total</th><th>0a</th><th>0b</th><th>0c.1</th><th>0c.2</th><th>1a</th><th>1b</th><th>2a</th><th>2b</th><th>2c</th><th>2d</th><th>3a</th><th>3b</th><th>4a</th><th>4b</th><th>5a</th><th>5b</th></tr><tr><th>1</th><td>MAX</td><td><span style=\"color: black\">50.0</span></td><td><span style=\"color: black\">1.0</span></td><td><span style=\"color: black\">2.0</span></td><td><span style=\"color: black\">2.0</span></td><td><span style=\"color: black\">2.0</span></td><td><span style=\"color: black\">4.0</span></td><td><span style=\"color: black\">3.0</span></td><td><span style=\"color: black\">2.0</span></td><td><span style=\"color: black\">2.0</span></td><td><span style=\"color: black\">2.0</span></td><td><span style=\"color: black\">2.0</span></td><td><span style=\"color: black\">5.0</span></td><td><span style=\"color: black\">5.0</span></td><td><span style=\"color: black\">4.0</span></td><td><span style=\"color: black\">4.0</span></td><td><span style=\"color: black\">5.0</span></td><td><span style=\"color: black\">5.0</span></td></tr></table>"
      ],
      "text/plain": [
       "1x18 DataFrame\n",
       "| Row | Names | total                 | 0a                     |\n",
       "|-----|-------|-----------------------|------------------------|\n",
       "| 1   | \"MAX\" | Colored(\"black\",50.0) | Colored(\"black\",\"1.0\") |\n",
       "\n",
       "| Row | 0b                     | 0c.1                   |\n",
       "|-----|------------------------|------------------------|\n",
       "| 1   | Colored(\"black\",\"2.0\") | Colored(\"black\",\"2.0\") |\n",
       "\n",
       "| Row | 0c.2                   | 1a                     |\n",
       "|-----|------------------------|------------------------|\n",
       "| 1   | Colored(\"black\",\"2.0\") | Colored(\"black\",\"4.0\") |\n",
       "\n",
       "| Row | 1b                     | 2a                     |\n",
       "|-----|------------------------|------------------------|\n",
       "| 1   | Colored(\"black\",\"3.0\") | Colored(\"black\",\"2.0\") |\n",
       "\n",
       "| Row | 2b                     | 2c                     |\n",
       "|-----|------------------------|------------------------|\n",
       "| 1   | Colored(\"black\",\"2.0\") | Colored(\"black\",\"2.0\") |\n",
       "\n",
       "| Row | 2d                     | 3a                     |\n",
       "|-----|------------------------|------------------------|\n",
       "| 1   | Colored(\"black\",\"2.0\") | Colored(\"black\",\"5.0\") |\n",
       "\n",
       "| Row | 3b                     | 4a                     |\n",
       "|-----|------------------------|------------------------|\n",
       "| 1   | Colored(\"black\",\"5.0\") | Colored(\"black\",\"4.0\") |\n",
       "\n",
       "| Row | 4b                     | 5a                     |\n",
       "|-----|------------------------|------------------------|\n",
       "| 1   | Colored(\"black\",\"4.0\") | Colored(\"black\",\"5.0\") |\n",
       "\n",
       "| Row | 5b                     |\n",
       "|-----|------------------------|\n",
       "| 1   | Colored(\"black\",\"5.0\") |"
      ]
     },
     "execution_count": 62,
     "metadata": {
      "comm_id": "ebfaf1f0-fd1c-4b1c-bde2-40c64b080598",
      "reactive": true
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# This might take more than one attempt to work...\n",
    "\n",
    "Homework.progress()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Cheatsheet: How to enter vectors and matrices\n",
    "\n",
    "v=[1,2,3]; # vector  (no row vectors/column vectors, just vectors!)\n",
    "A=[[1 2 3],[4 5 6],[7 8 9]] # matrix by rows .. or\n",
    "A=[1 2 3\n",
    "   4 5 6\n",
    "   7 8 9]\n",
    "B=[[1,2,3] [4,5,6] [7,8,9]] # matrix by columns;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 0\n",
    "\n",
    "### Problem 0a (1 point)\n",
    "\n",
    "If you exchange the first two rows of a matrix $A$, which of the four subspaces stay the same?\n",
    "\n",
    "Select the answers that match.\n",
    "\n",
    "1. Row space\n",
    "2. Column space\n",
    "3. Null space\n",
    "4. Left null space\n",
    "5. All 4 spaces\n",
    "\n",
    "Enter your answer as a vector of answer numbers: e.g. [1,2] if both 1 and 2 are correct"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "[1.0,3.0]",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 1,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 0a",
    "question": "0a"
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem 0b (2 points)\n",
    "If you exchange the first two rows of $A$, and $v=(1,2,3,4)$ is in the left nullspace of $A$, write down a nonzero vector in the left nullspace of the new matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "[2.0,1.0,3.0,4.0]",
    "collapsed": false,
    "explanation": "A vector in the left null space tells you a combination of the rows which is zero. If you switch two rows, you can find a combination which is zero by switching the first two entries of the vector.",
    "max_attempts": 1000,
    "max_score": 2,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 0b",
    "question": "0b"
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem 0c (4 points)\n",
    "\n",
    "$A^T y=d$ is solvable when $d$ is in which of the four subspaces?\n",
    "\n",
    "Choose one of the four subspaces.\n",
    "\n",
    "1. Row space of A\n",
    "2. Column space of A\n",
    "3. Null space of A\n",
    "4. Left null space of A\n",
    "\n",
    "(Answer format: integer 1-4 : the answer)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "1.0",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 2,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 0c.1",
    "question": "0c.1"
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The solution $y$ is unique when which of the four subspaces contains only the zero vector?\n",
    "\n",
    "Choose one of the four subspaces.\n",
    "\n",
    "1. Row space of A\n",
    "2. Column space of A\n",
    "3. Null space of A\n",
    "4. Left null space of A\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "4.0",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 2,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 0c.2",
    "question": "0c.2"
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 1 *(7 points)*\n",
    "\n",
    "### Problem 1a *(4 point)*\n",
    "\n",
    "**The Incidence matrix for a triangle graph**\n",
    "\n",
    "<img src=\"http://web.mit.edu/18.06/www/Spring15/ps4/TriangleGraphP428.png\">\n",
    "\n",
    "Enter the 3 by 3 incidence matrix A for the triangle graph on the picture. The first row has `−1` in column `1` and `+1` in column 2.\n",
    "\n",
    "Easiest format: `[[-1 1 ?],[? ? ?],[? ? ?]]`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "[[-1.0,-1.0,0.0],[1.0,0.0,-1.0],[0.0,1.0,1.0]]",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 4,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 1a",
    "question": "1a"
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem 1b. (3 point)\n",
    "Give the vector in A's nullspace whose first entry is 5. Format idea: [5, ?, ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "[5.0,5.0,5.0]",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 3,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 1b",
    "question": "1b"
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 2 (8 points)\n",
    "### Problem 2a. (2 point) \n",
    "\n",
    "<img src=\"http://web.mit.edu/18.06/www/Spring15/ps4/SquareGraphP428.png\"> <br>\n",
    "\n",
    "Enter  the 5 by 4 incidence matrix A for the square graph in the picture. <br>\n",
    "[[a b c d],[e f g h], ...  ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "[[-1.0,-1.0,0.0,0.0,0.0],[1.0,0.0,-1.0,-1.0,0.0],[0.0,1.0,1.0,0.0,-1.0],[0.0,0.0,0.0,1.0,1.0]]",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 2,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 2a",
    "question": "2a"
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem 2b *(2 point) *\n",
    "\n",
    "Find a solution to Ax=0 whose first element is 5. [a, b, c, ...]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "[5.0,5.0,5.0,5.0]",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 2,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 2b",
    "question": "2b"
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem 2c *(2 point) *\n",
    "\n",
    "Find two solutions with x₃ = 1 (third entry is 1) to Aᵀx = 0 that correspond to triangular loops. <br>\n",
    "The \"upper left loop\" solution is: [a,b,...]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "[1.0,-1.0,1.0,0.0,0.0]",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 2,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 2c",
    "question": "2c"
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem 2d *(2 point) *\n",
    "The \"lower right solution is\": [a,b,...]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "[0.0,0.0,1.0,-1.0,1.0]",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 2,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 2d",
    "question": "2d"
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem 3a *(10 points)*\n",
    "\n",
    "Find the incidence matrix A of the directed complete graph on 5 vertices. The vertices and edges are labeled for you.\n",
    "\n",
    "Recall that the columns of the incidence matrix are indexed by the vertices. The rows are indexed by the edges. Each row has entries which are either -1, 1, or 0. The -1 occurs at the vertex position where the edge begins, and the 1 occurs at the vertex position where the edge terminates.\n",
    "\n",
    "<img src=\"http://web.mit.edu/18.06/www/Spring15/ps4/images_k5graph.png\">\n",
    "\n",
    "Write the incidence matrix A as an Integer matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "[[-1.0,-1.0,-1.0,-1.0,0.0,0.0,0.0,0.0,0.0,0.0],[1.0,0.0,0.0,0.0,-1.0,-1.0,-1.0,0.0,0.0,0.0],[0.0,1.0,0.0,0.0,1.0,0.0,0.0,-1.0,-1.0,0.0],[0.0,0.0,1.0,0.0,0.0,1.0,0.0,1.0,0.0,-1.0],[0.0,0.0,0.0,1.0,0.0,0.0,1.0,0.0,1.0,1.0]]",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 5,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 3a",
    "question": "3a"
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem 3b *(5 points)*\n",
    "Determine a basis for the row space. <br> Enter your basis as a matrix  whose rows each form a basis element for the row space of A.  <br>\n",
    "\n",
    "(Format: the answer we are checking for takes some of the rows of A in the order in which they appear in A. More precisely please create a matrix that has the first k rows of A, where k is the number of basis vectors.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "[[-1.0,-1.0,-1.0,-1.0],[1.0,0.0,0.0,0.0],[0.0,1.0,0.0,0.0],[0.0,0.0,1.0,0.0],[0.0,0.0,0.0,1.0]]",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 5,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 3b",
    "question": "3b"
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Problem 4\n",
    "\n",
    "**About Stoichiometric Matrices**\n",
    "\n",
    "\n",
    "[ Quick Summary for those\n",
    "who do not want to read everything: chemical reactions turn into matrices by turning \n",
    "coefficients into matrix elements. Every reaction is a column in the matrix. The coefficients on the left side of the\n",
    "chemical reaction appear with negative signs.\n",
    "The nullspace corresponds to the reactions' steady state with negative signs\n",
    "corresponds to backwards reactions.]\n",
    "\n",
    "\n",
    "Suppose that you have a system of chemical reactions. You know the velocity, or rate of each individual reaction, and you know which species are involved in each reaction. You can think of the reactants and reaction rates as the vertices and edges in a network, or graph. We create something very similar to the incidence matrix from a graph, and use it to discover the steady state solutions to a system of reactions.\n",
    "\n",
    "<b> Basics of chemical reactions </b>\n",
    "\n",
    "Recall the following basics about chemical reactions.\n",
    "\n",
    "If A is a molecule, then we use [A] to denote the concentration of that molecular species. The concentration is a function of time, but we typically are interested in the concentration at steady state (no time dependence).\n",
    "\n",
    "If A→B is a unimolecular reaction, then the velocity or reaction rate, v, is written as\n",
    "\n",
    "$$ v=\\frac{d[B]}{dt}=−\\frac{d[A]}{dt}=k[A].$$\n",
    "If A+B→P+Q is a bimolecular chemical reaction, then the reaction rate is\n",
    "\n",
    "$$v=\\frac{d[P]}{dt}=\\frac{d[Q]}{dt}=−\\frac{d[A]}{dt}=−\\frac{d[B]}{dt}=k[A][B].$$\n",
    "\n",
    "<b> Example 1: Simple Stoichiometric Matrix </b>\n",
    "\n",
    "Consider the single reaction with three different chemical species.\n",
    "\n",
    "2NO+O2⇌2NO2.\n",
    "<br>\n",
    "\n",
    "The stoichiometric matrix is a column vector:\n",
    "\n",
    "$$\\begin{pmatrix} 2 \\\\ -1 \\\\ 2 \\end{pmatrix}$$\n",
    "such that\n",
    "\n",
    " [[NO] [O$_2$] [NO$_2$]] $\\begin{pmatrix} 2 \\\\ -1 \\\\ 2 \\end{pmatrix} =0. $\n",
    "\n",
    "\n",
    "Note the convention that reactants have negative sign, and products have positive sign.\n",
    "\n",
    "<b> Example 2: Input and output </b>\n",
    "\n",
    "Now consider the same reaction that is no longer closed. We will allow that oxygen is being continuously added into the system, and we are taking out nitrogen dioxide. The reaction is now the following system:\n",
    "\n",
    "r1: 2NO+O$_2$⇌2NO$_2$ <br>\n",
    "e1: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; →O$_2$ <br>\n",
    "e2: &nbsp;&nbsp;&nbsp;NO$_2$→\n",
    "\n",
    "\n",
    "\n",
    "                \n",
    "\n",
    "The stoichiometric matrix is\n",
    "\n",
    "\n",
    "$\\begin{matrix} & r_1 & e_1 &  e_2 \\\\\n",
    "NO & -2 & 0 & 0 \\\\\n",
    "O_2 & -1 & 1 & 0 \\\\\n",
    "NO_2 & 2 & 0 & -1\n",
    "\\end{matrix}$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    " \t \t \t \n",
    "<b> Example 3: Closed System of Reactions </b>\n",
    "\n",
    "Suppose we have the following closed system of chemical reactions. It is closed because no new reactants are entering the system, and no products are leaving the system.\n",
    "\n",
    "r1: H$_2$O + CO ⇌CO$_2$  + H$_2$<br>\n",
    "r2: H$_2$O + H  ⇌ H$_2$ + OH <br>\n",
    "r3: OH + CO   ⇌ CO$_2$  + H\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "The  following molecular species can be designated:\n",
    "\n",
    " X$_1$\t= H$_2$O\t \t\n",
    " X$_2$\t= CO\t \t \n",
    " X$_3$\t= CO$_2$\t  \n",
    " X$_4$\t= H$_2$\t \t \n",
    " X$_5$\t= H\t \t \n",
    " X$_6$\t= OH\t \t\n",
    "    \n",
    "Then the coefficient on every term in this reaction is 1, the entries will all be +1, −1, or 0. The convention is that the reactants have negative, and the products have positive sign.\n",
    "\n",
    "The stoichiometric matrix is given by the following, where the columns each designate a particular reaction, and the coefficients determine the amount of each species present.\n",
    "\n",
    "$\\begin{matrix} & r_1 & e_1 &  e_2 \\\\\n",
    "X_1=H_20 & -1 & -1 & 0 \\\\\n",
    "X_2=CO & -1 & 0 & -1 \\\\\n",
    "X_3=CO_2  & 1 & 0 &  1 \\\\\n",
    "X_4 = H_2 & 1 & 1 & 0 \\\\\n",
    "X_5 =H & 0 & -1 & 1 \\\\\n",
    "X_6 = OH & 0 & 1 & -1\n",
    "\\end{matrix}$\n",
    "\n",
    "\n",
    "\n",
    "Let $v_1$ is the velocity of reaction 1, $v_2$ is the velocity of reaction 2, and $v_3$ is the velocity of reaction 3, then\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "\\frac{d[X_1]}{dt} \\\\\n",
    "\\frac{d[X_2]}{dt} \\\\\n",
    "\\frac{d[X_3]}{dt} \\\\\n",
    "\\frac{d[X_4]}{dt} \\\\\n",
    "\\frac{d[X_5]}{dt} \\\\\n",
    "\\frac{d[X_6]}{dt} \n",
    "\\end{pmatrix}=\n",
    "\\begin{pmatrix}\n",
    "-1 & -1 & 0 \\\\\n",
    "-1 & 0 & -1 \\\\\n",
    "1 & 0 & 1\\\\\n",
    "1 & 1 & 0 \\\\\n",
    "0 & -1 & 1 \\\\\n",
    "0 & 1 & -1\n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "v_1 \\\\\n",
    "v_3 \\\\\n",
    "v_3\n",
    "\\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "To find the steady state solution, we are interested in finding the velocities such that each concentration is steady, that is the derivative is zero:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "0 \\\\ 0 \\\\ 0 \\\\\n",
    "0 \\\\ 0 \\\\ 0 \n",
    "\\end{pmatrix}=\n",
    "\\begin{pmatrix}\n",
    "-1 & -1 & 0 \\\\\n",
    "-1 & 0 & -1 \\\\\n",
    "1 & 0 & 1\\\\\n",
    "1 & 1 & 0 \\\\\n",
    "0 & -1 & 1 \\\\\n",
    "0 & 1 & -1\n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "v_1 \\\\\n",
    "v_3 \\\\\n",
    "v_3\n",
    "\\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "This is exactly the nullspace of the matrix. In this case the nullspace, which corresponds to the steady state is:\n",
    "\n",
    "$$\\begin{pmatrix}\n",
    "v_1 \\\\\n",
    "v_3 \\\\\n",
    "v_3\n",
    "\\end{pmatrix}=\n",
    "\\begin{pmatrix}\n",
    "-1 \\\\\n",
    "1 \\\\\n",
    "1\n",
    "\\end{pmatrix} .\n",
    "$$\n",
    "This is telling us that at steady state, the first reaction will be proceeding in the backwards direction, and the other two in the forwards direction, all with the same velocity, or reaction rate k.\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem 4a *(5 pts)*\n",
    "\n",
    "**Closed Stoichiometric Matrix**\n",
    "\n",
    "Find the stoichiometric matrix A for the follow system of reactions. Please order your rows such that the species occur in alphabetical order according to the given variable names (A,B,C,D,E,X,Y).\n",
    "\n",
    "In this problem, we assume that we are interested in creating a product E with input A. Y and X are enzymes that will further the reaction rates.\n",
    "\n",
    "$r1: A ⇌ B\\\\\n",
    "r2: 2B  ⇌ C+Y\\\\\n",
    "r3: 2B+X  ⇌ D+Y\\\\\n",
    "r4: D ⇌ E+X\\\\\n",
    "r5: C+X ⇌ D\\\\\n",
    "r6: C⇌ E\n",
    "$\n",
    "\n",
    "The stoichiometric matrix A (Int64 entries is)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "[[-1.0,1.0,0.0,0.0,0.0,0.0,0.0],[0.0,-2.0,1.0,0.0,0.0,0.0,1.0],[0.0,-2.0,0.0,1.0,0.0,-1.0,1.0],[0.0,0.0,0.0,-1.0,1.0,1.0,0.0],[0.0,0.0,-1.0,1.0,0.0,-1.0,0.0],[0.0,0.0,-1.0,0.0,1.0,0.0,0.0]]",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 4,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 4a",
    "question": "4a"
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use Julia's rref command to find the nullspace of A. \n",
    "Directly reading the information in the rref(A)\n",
    "create a flux matrix\n",
    "that has the basis vectors for the nullspace of A as rows. \n",
    "<br>\n",
    "While there are many bases for the nullspace, the autograder is looking\n",
    "for the one that would naturally be produced by pencil and paper.\n",
    "(Format: Int64, basis vectors listed in the rows)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "[[0.0,0.0],[1.0,1.0],[-1.0,-1.0],[0.0,-1.0],[1.0,0.0],[0.0,1.0]]",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 4,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 4b",
    "question": "4b"
   },
   "outputs": [],
   "source": [
    "A = [? 0 0 0 0 0 \n",
    "     ? ? ? 0 0 0\n",
    "     0 ? 0 0 ? ?\n",
    "     0 0 ? ? ? 0\n",
    "     0 0 0 ? 0 ?\n",
    "     0 0 ? ? ? 0\n",
    "     0 ? ? 0 0 0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 5 *(8 points)*\n",
    "\n",
    "Find the stoichiometric matrix A for the follow system of reactions. Please order your rows such that the species occur in alphabetical order according to the given variable names (A,B,C,D,F,X,Y).\n",
    "\n",
    "In this problem, we assume that we are interested in creating a product F with input A. Y and X are enzymes that will further the reaction rates. We continue to input species A into the reaction and siphon off species F and the extra enzyme Y.\n",
    "\n",
    "$r1: A ⇌ B\\\\\n",
    "r2: 2B  ⇌ C+Y \\\\\n",
    "r3: 2B+X  ⇌ D+Y \\\\\n",
    "r4: D ⇌ F+X \\\\\n",
    "r5: C+X ⇌ D \\\\\n",
    "r6: C⇌ F \\\\\n",
    "e1:→A \\\\\n",
    "e2:F→\\\\\n",
    "e3:Y→\\\\\n",
    "$\n",
    "\n",
    "Use Julia's `rref` command to find the nullspace of A. \n",
    "Directly reading the information in the rref(A)\n",
    "create a flux matrix\n",
    "that has the basis vectors for the nullspace of A as rows. \n",
    "<br>\n",
    "While there are many bases for the nullspace, the autograder is looking\n",
    "for the one that would naturally be produced by pencil and paper.\n",
    "(Format: Int64, basis vectors listed in the rows)\n",
    "\n",
    "This might be best done manually, as on page 147 of your text."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "[[-1.0,1.0,0.0,0.0,0.0,0.0,0.0],[0.0,-2.0,1.0,0.0,0.0,0.0,1.0],[0.0,-2.0,0.0,1.0,0.0,-1.0,1.0],[0.0,0.0,0.0,-1.0,1.0,1.0,0.0],[0.0,0.0,-1.0,1.0,0.0,-1.0,0.0],[0.0,0.0,-1.0,0.0,1.0,0.0,0.0],[1.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,-1.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,-1.0]]",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 5,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 5a",
    "question": "5a"
   },
   "outputs": [],
   "source": [
    "A = [ ? 0 0 0 0 0 ? 0 0\n",
    "      1 ? ? 0 0 0 0 0 0\n",
    "      0 1 0 0 ? ? 0 0 0 \n",
    "      0 0 ? ? ? 0 0 0 0 \n",
    "      0 0 0 1 0 1 0 ? 0 \n",
    "      0 0 ? ? ? 0 0 0 0 \n",
    "      0 ? ? 0 0 0 0 0 ?]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use the RREF to obtain the nullspace of A as rows of an int64 matrix. <br>\n",
    "This might be best done manually, as on page 147 of your text."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "alert": "info",
    "answer": "[[0.0,0.0,2.0],[1.0,1.0,0.0],[-1.0,-1.0,1.0],[0.0,-1.0,1.0],[1.0,0.0,0.0],[0.0,1.0,0.0],[0.0,0.0,2.0],[0.0,0.0,1.0],[0.0,0.0,1.0]]",
    "collapsed": false,
    "max_attempts": 1000,
    "max_score": 5,
    "msg": "<span class='icon-info'></span> Will commit last result as the answer to question 5b",
    "question": "5b"
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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