// Numbas version: finer_feedback_settings
{"name": "Reduced Row Echelon Form; spanning a subspace", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {"diagonal": {"definition": "var m=[];\nvar l=d.length;\nfor (var i=0;i<l;i++){\n  for(var j=0;j<l;j++){\n    if(i!=j){m[i][j].push(0);} else {m[i][j].push(d[i]);}\n  }\nreturn m;}", "type": "list", "language": "javascript", "parameters": [["d", "list"]]}}, "ungrouped_variables": ["r4", "a", "b", "span", "g", "g1", "r3", "mm", "ch2", "rdformw", "ch1", "s", "r", "t", "r2", "y", "x", "w", "r1", "doornot", "d"], "name": "Reduced Row Echelon Form; spanning a subspace", "tags": ["dimension", "echelon form", "linear algebra", "linear dependence", "linear independence", "linear spaces", "reduced echelon form", "row dimension", "space of vectors", "span", "spanning set", "subspaces", "vectors"], "preamble": {"css": "", "js": ""}, "advice": "<p>a) We find that the reduced row echelon form for $S$ is $R_S=\\var{g}$.</p>\n    <p>b) We find that the reduced row echelon form for $T$ is $R_T=\\var{rdformW}$.</p>\n    <p>c) Looking at the non zero rows in $R_S$ and $R_T$ we see that they {doornot} span the same subspace of $\\mathbb{R}^5$.</p>\n    <p>&nbsp;</p>\n    <p>&nbsp;</p>", "rulesets": {}, "parts": [{"prompt": "<p>&nbsp;</p>\n        <p>Find the reduced row echelon form $R_S$ of the matrix with rows given by the vectors in $S$ &nbsp;and enter it here:</p>\n        <p><t d=\"\">[[4]]</t></p>\n        <table>\n        <tbody>\n        <tr>\n        <td rowspan=\"4\">\\[R_S=\\left( \\begin{matrix}\\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        <td>[[0]]</td>\n        <td>[[1]]</td>\n        <td>[[2]]</td>\n        <td>[[3]]</td>\n        <td>[[4]]</td>\n        <td rowspan=\"4\">\\[\\left) \\begin{matrix}\\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\\\end{matrix} \\right.\\]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>[[5]]</td>\n        <td>[[6]]</td>\n        <td>[[7]]</td>\n        <td>[[8]]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>[[9]]</td>\n        <td>[[10]]</td>\n        <td>[[11]]</td>\n        </tr>\n        </tbody>\n        </table>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "g[0][0]", "minValue": "g[0][0]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g[0][1]", "minValue": "g[0][1]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g[0][2]", "minValue": "g[0][2]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g[0][3]", "minValue": "g[0][3]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g[0][4]", "minValue": "g[0][4]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g[1][1]", "minValue": "g[1][1]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g[1][2]", "minValue": "g[1][2]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g[1][3]", "minValue": "g[1][3]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g[1][4]", "minValue": "g[1][4]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g[2][2]", "minValue": "g[2][2]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g[2][3]", "minValue": "g[2][3]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g[2][4]", "minValue": "g[2][4]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Find the reduced row echelon form $R_T$ of the matrix with rows given by the vectors in $T$ &nbsp;and enter it here:</p>\n        <table>\n        <tbody>\n        <tr>\n        <td rowspan=\"6\">\\[R_T=\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        <td>[[0]]</td>\n        <td>[[1]]</td>\n        <td>[[2]]</td>\n        <td>[[3]]</td>\n        <td>[[4]]</td>\n        <td rowspan=\"6\">\\[\\left) \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\\\end{matrix} \\right.\\]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>[[5]]</td>\n        <td>[[6]]</td>\n        <td>[[7]]</td>\n        <td>[[8]]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>[[9]]</td>\n        <td>[[10]]</td>\n        <td>[[11]]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>[[12]]</td>\n        <td>[[13]]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>[[14]]</td>\n        </tr>\n        </tbody>\n        </table>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "g1[0][0]", "minValue": "g1[0][0]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g1[0][1]", "minValue": "g1[0][1]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g1[0][2]", "minValue": "g1[0][2]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g1[0][3]", "minValue": "g1[0][3]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g1[0][4]", "minValue": "g1[0][4]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g1[1][1]", "minValue": "g1[1][1]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g1[1][2]", "minValue": "g1[1][2]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g1[1][3]", "minValue": "g1[1][3]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g1[1][4]", "minValue": "g1[1][4]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g1[2][2]", "minValue": "g1[2][2]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g1[2][3]", "minValue": "g1[2][3]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g1[2][4]", "minValue": "g1[2][4]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g1[3][3]", "minValue": "g1[3][3]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g1[3][4]", "minValue": "g1[3][4]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g1[4][4]", "minValue": "g1[4][4]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 0.4, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Do $S$ and $T$ span the same space of vectors?</p>", "matrix": "mm", "shuffleChoices": false, "variableReplacements": [], "choices": ["<p>Yes</p>", "<p>No</p>"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "marks": 0, "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}], "extensions": [], "statement": "<p><strong>Reduced row echelon form.</strong></p>\n    <p>Let $S=\\{v_1,\\;v_2,\\;v_3\\}$ be the set of vectors given by:</p>\n    <p>$v_1=\\var{rowvector(list(a[0]))},\\;\\;v_2=\\var{rowvector(list(a[1]))},\\;\\;v_3=\\var{rowvector(list(a[2]))}$</p>\n    <p>Let $T=\\{w_1,\\;w_2,\\;w_3,\\;w_4,\\;w_5\\}$ be the set of vectors given by:</p>\n    <p>$w_1=\\var{rowvector(list(b[0]))},\\;\\;w_2=\\var{rowvector(list(b[1]))},\\;\\;w_3=\\var{rowvector(list(b[2]))}$</p>\n    <p>$w_4=\\var{rowvector(list(b[3]))},\\;\\;w_5=\\var{rowvector(list(b[4]))}$</p>\n    <p>Find the reduced row echelon forms of the two matrices with rows given by $S$ and $T$, respectively. Determine whether or not $S$ and $T$ span the same vector subspace of $\\mathbb{R}^5$.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"r4": {"definition": "-random(1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "r4", "description": ""}, "a": {"definition": "r*g", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "s*g1", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "span": {"definition": "random(0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "span", "description": ""}, "g": {"definition": "\n    matrix([if(t>7,0,1),\n    if(t=6 or t=3 or t=7,random(-3..3), if(t=8,1,0)),\n    if(t=5 or t=8 ,random(-3..3), if(t=9,1,0)),\n    if(t=2 or t=4 or t=6 or t=7 or t=9,random(-3..3), 0),\n    if(t=1 or t=4 or t=5 or t=7 or t=8,random(-3..3), 0)],\n    [0,if(t=2 or t=4 or t=1 or t=5,1, 0),\n    if(t=2 or t=4 or t=1or t=8 or t=9,0, if(t=5,random(-3..3),1)),\n    if(t=2 or t=4 or t=6 or t=7 ,random(-3..3), if(t=8,1,0)),\n    if(t=1 or t=4 or t=5 or t=7 ,random(-3..3), if(t=9,1,0))],\n    [0,0,if(t=1 or t=2 or t=4,1,0),\n    if(t=2 or t=4  ,random(-3..3), if(t=3 or t=5,1,0)),\n    if(t=1 or t=5 or t=4,random(-3..3),if(t=6,1,0))])\n       \n    ", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "g1": {"definition": "g+matrix([0,0,0,ch1-g[0][3],ch2-g[0][4]],[0,0,0,0,0],[0,0,0,0,0])\n    ", "templateType": "anything", "group": "Ungrouped variables", "name": "g1", "description": ""}, "r3": {"definition": "random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "r3", "description": ""}, "mm": {"definition": "switch(span=1,[0,1],[1,0])", "templateType": "anything", "group": "Ungrouped variables", "name": "mm", "description": ""}, "s": {"definition": "matrix([w[0],w[1],w[2]],[w[1],w[0],w[2]],[w[2],w[1],w[0]],x,y)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "d": {"definition": "switch(t<4,4,t<7,3,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "rdformw": {"definition": "matrix(list(g1[0]),list(g1[1]),list(g1[2]),[0,0,0,0,0],[0,0,0,0,0])", "templateType": "anything", "group": "Ungrouped variables", "name": "rdformw", "description": ""}, "ch1": {"definition": "if(span=1,if(t=8 or t=5,0,g[0][3]-1),g[0][3])", "templateType": "anything", "group": "Ungrouped variables", "name": "ch1", "description": ""}, "ch2": {"definition": "if(span=1,if(t=8 or t=5,g[0][4]-1,g[0][4]),g[0][4])", "templateType": "anything", "group": "Ungrouped variables", "name": "ch2", "description": ""}, "r": {"definition": "\n    matrix(\n      [r1,r2,r3],\n      [r2,r4,r3],\n      [r3,r4,r2]\n      )\n      \n      \n    ", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "t": {"definition": "random(4..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "r2": {"definition": "-random(-2..2)", "templateType": "anything", "group": "Ungrouped variables", "name": "r2", "description": ""}, "y": {"definition": "list(vector(w)+vector(1,1,1))", "templateType": "anything", "group": "Ungrouped variables", "name": "y", "description": ""}, "x": {"definition": "[random(-1,1),random(1,-1),random(-1,1)]", "templateType": "anything", "group": "Ungrouped variables", "name": "x", "description": ""}, "w": {"definition": "[random(2,3),-random(3,4),random(-2,2)]", "templateType": "anything", "group": "Ungrouped variables", "name": "w", "description": ""}, "doornot": {"definition": "if(span=1, 'do not','')", "templateType": "anything", "group": "Ungrouped variables", "name": "doornot", "description": ""}, "r1": {"definition": "1", "templateType": "anything", "group": "Ungrouped variables", "name": "r1", "description": ""}}, "metadata": {"description": "<p>Given two ordered sets of vectors $S,\\;T$ in $\\mathbb{R^5}$ find the reduced echelon form of the matrices given by $S$ and $T$ and hence determine whether or not they span the same subspace.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Lois Rollings", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/326/"}]}]}], "contributors": [{"name": "Lois Rollings", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/326/"}]}