// Numbas version: finer_feedback_settings
{"name": "Simon's copy of Row reducing a matrix and finding its rank and nullity-MA2223", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>Reduce a 5x6 matrix to row reduced form and using this find&nbsp;rank and nullity.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variablesTest": {"condition": "deter<>0\nand\nmax(map(testing(record[0][x]),x,0..length(record[0])-1))=0", "maxRuns": "200"}, "preamble": {"css": "", "js": "question.functions={};\nquestion.functions.ref=function(m){\n  \n  m = util.copyarray(m,true);\n  \n  function finish() {\n\n    return rank;\n  }\n\n\n\n  var lead = 0;\n  var rank = 0;\n  var echelon=[];\n  var rows = m.length;\n  for(var i=0;i<rows;i++){echelon.push(0);}\n  var columns=m[0].length;\n  for (var r = 0; r < rows; r++) {\n    if (columns <= lead) {\n      return finish();\n    }\n    var i = r;\n    while (m[i][lead] == 0) {\n      i++;\n      if (rows == i) {\n\n        i = r;\n        lead++;\n        if (columns == lead) {\n          return finish();\n        }\n      }\n    }\n\n    var tmp = m[i];\n    m[i] = m[r];\n    m[r] = tmp;\n    echelon[r]=lead+1;\n    rank+=1;\n    //var val = m[r][lead];\n    //var inval;\n    //nval=1/val;\n\n    // divide this row by the leading value\n    /*\n        for (var j = 0; j < columns; j++) {\n            m[r][j] = m[r][j]*inval;\n        }\n      */\n\n    // for each row, find a linear combination of m[r] and m[i] so that m[i][lead] becomes 0\n    for (var i = 0; i < rows; i++) {\n      if (i == r) continue;\n      var v_i = m[i][lead];\n      if(v_i!=0) {\n        var v_r = m[r][lead];\n        var lcm = Numbas.math.lcm(Math.abs(v_i),Math.abs(v_r));\n        var a = lcm/Math.abs(v_i);\n        var b = -Numbas.math.sign(v_i*v_r)*lcm/Math.abs(v_r);\n        for (var j = 0; j < columns; j++) {\n          m[i][j] = a*m[i][j] + b*m[r][j];\n        }\n      }\n    }\n    lead++;\n  }\n  return finish();\n}\n\n"}, "tags": [], "variables": {"rows": {"definition": "6", "name": "rows", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "rref1": {"definition": "rref1(list(testmatrix))[0]", "name": "rref1", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "maxint": {"definition": "4", "name": "maxint", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "nullity": {"definition": "5-rank", "name": "nullity", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "sum": {"definition": "sum(echform)", "name": "sum", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "echelonform": {"definition": "echelon_form_matrix(rows,rank,columns,maxint)", "name": "echelonform", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "ec": {"definition": "echmatrix", "name": "ec", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "echmatrix": {"definition": "matrix(ech)", "name": "echmatrix", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "record_ops_matrix": {"definition": "record[0]", "name": "record_ops_matrix", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "columns": {"definition": "5", "name": "columns", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "ech": {"definition": "echelonform[0]", "name": "ech", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "rank": {"definition": "random(2,3,4)\n", "name": "rank", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "null_basis": {"definition": "switch(sum=6,[[ec[0][2]*ec[2][3]-ec[0][3],ec[1][2]*ec[2][3]-ec[1][3],-ec[2][3],1,0],\n    [ec[0][2]*ec[2][4]-ec[0][4],ec[1][2]*ec[2][4]-ec[1][4],-ec[2][4],0,1]],\n  sum=7,[[-ec[0][2],-ec[1][2],1,0,0],[-ec[0][4],-ec[1][4],0,-ec[2][4],1]],\n  (sum=8) and(5 in echform),[[-ec[0][2],-ec[1][2],1,0,0],[-ec[0][3],-ec[1][3],0,1,0]],\n  sum=8,[[-ec[0][1],1,0,0,0],[-ec[0][4],0,-ec[1][4],-ec[2][4],1]],\n  sum=9,[[-ec[0][1],1,0,0,0],[-ec[0][3],0,-ec[1][3],1,0]],\n  [[-ec[0][1],1,0,0,0],[-ec[0][2],0,1,0,0]])", "name": "null_basis", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "testmatrix": {"definition": "matrix(rdmat)*matrix(ech)", "name": "testmatrix", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "record": {"definition": "rref(list(testmatrix),0)", "name": "record", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "chosen": {"definition": "[true,true,false,true,false]", "name": "chosen", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "deter": {"definition": "rref(rdmat,0)[3]", "name": "deter", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "record_ops_message": {"definition": "record[1]", "name": "record_ops_message", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "student_cols_matrix": {"definition": "transpose(matrix(map(transpose(echmatrix)[n],n,filter(chosen[j],j,0..4))))", "name": "student_cols_matrix", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "echform": {"definition": "map(x+1,x,echelonform[1])", "name": "echform", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "trtestmatrix": {"definition": "transpose(testmatrix)", "name": "trtestmatrix", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "column_basis": {"definition": "set(map(trtestmatrix[x],x,echelonform[1]))", "name": "column_basis", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "not_chosen_col": {"definition": "transpose(matrix(map(transpose(echmatrix)[n],n,filter(!chosen[j],j,0..4)))[0])", "name": "not_chosen_col", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "steps": {"definition": "rref(list(testmatrix),0)[2]", "name": "steps", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "rdmat": {"definition": "rdmat(rows)", "name": "rdmat", "description": "", "group": "Ungrouped variables", "templateType": "anything"}}, "functions": {"rref1": {"parameters": [["m", "list"]], "definition": "function finish() {\n  for(var i=0;i<rows;i++) {\n    var leader = null;\n    for(var j=0;j<columns;j++) {\n      if(leader!==null) {\n        m[i][j] /= leader;\n      } else if(m[i][j]!=0) {\n        leader = m[i][j];\n        m[i][j] = 1;\n      }\n    }\n  }\n  return [m,echelon,rank];\n}\n\n//This question used the function rref(m) defined in Functions & Rulesets to find the row reduced echelon form of a matrix m.\n\n//rref(m)[0] is the row reduced matrix, rref(m)[1] the echelon form.\n\n    var lead = 0;\n    var rank=0;\n    var echelon=[];\n    var rows = m.length;\n    for(var i=0;i<rows;i++){echelon.push(0);}\n    var columns=m[0].length;\n    for (var r = 0; r < rows; r++) {\n        if (columns <= lead) {\n            return finish();\n        }\n        var i = r;\n        while (m[i][lead] == 0) {\n            i++;\n            if (rows == i) {\n              \n                i = r;\n                lead++;\n                if (columns == lead) {\n                    return finish();\n                }\n            }\n        }\n \n        var tmp = m[i];\n        m[i] = m[r];\n        m[r] = tmp;\n        echelon[r]=lead+1;\n        rank+=1;\n        //var val = m[r][lead];\n        //var inval;\n        //nval=1/val;\n\n        // divide this row by the leading value\n      /*\n        for (var j = 0; j < columns; j++) {\n            m[r][j] = m[r][j]*inval;\n        }\n      */\n \n      // for each row, find a linear combination of m[r] and m[i] so that m[i][lead] becomes 0\n       for (var i = 0; i < rows; i++) {\n         if (i == r) continue;\n         var v_i = m[i][lead];\n         if(v_i!=0) {\n           var v_r = m[r][lead];\n           var lcm = Numbas.math.lcm(Math.abs(v_i),Math.abs(v_r));\n           var a = lcm/Math.abs(v_i);\n           var b = -Numbas.math.sign(v_i*v_r)*lcm/Math.abs(v_r);\n           for (var j = 0; j < columns; j++) {\n             m[i][j] = a*m[i][j] + b*m[r][j];\n           }\n         }\n        }\n        lead++;\n    }\n    return finish();\n", "type": "list", "language": "javascript"}, "testing": {"parameters": [["m", "list"]], "definition": "\nvar check=0;\nhere:\nfor(var i=0;i<m.length;i++){\n  for(var j=0;j<m[0].length;j++)\n  {if(Math.round(m[i][j])!=m[i][j]) {check= 1;break here;}\n  }\n}\nreturn check;\n      ", "type": "number", "language": "javascript"}, "solution": {"parameters": [["matrix_list", "list"], ["message_list", "list"]], "definition": "function st(o){\n  return '['+o.toString()+']';\n}\nfunction display_format(mat){\nvar s= st(mat.map(st));\n  return '$\\\\simplify[fractionNumbers]{{matrix('+s+')}}$';\n}\nvar table = $('<table>');\nvar n=matrix_list.length;\nvar m='';\nvar mess='';\nfor(var i=0;i<n;i++){\nm=display_format(matrix_list[i]);\nmess= message_list[i];\ntable.append('<tr><td style=\"text-align:left;\">'+mess+'</td></tr>');\ntable.append('<tr><td style=\"text-align:left;\"></td></tr>');\ntable.append('<tr><td style=\"text-align:left;\">'+m+'</td></tr>');  \n}\nreturn table;\n", "type": "html", "language": "javascript"}, "echelon_form_matrix": {"parameters": [["n", "number"], ["r", "number"], ["m", "number"], ["max", "number"]], "definition": "\nfunction getRandomInt(min, max) {\n  return Math.floor(Math.random() * (max - min + 1)) + min;\n};\n//generate a random echelon form\nfunction random_echelon_form(columns,rank){\nvar l=[];\nvar ech_form=[], ech_form1=[],b=[];\nfor(var i=1;i<columns;i++) {l.push(i);}\nech_form1=Numbas.math.shuffle(l);\nech_form1.splice(0,0,0);\nech_form=ech_form1.slice(0,rank);\nb=ech_form.sort(function(x,y){return x-y;});\nreturn b;\n}\n\n\n\nvar c=random_echelon_form(m,r);\n\nvar ind;\nvar temp;\nvar ech_matrix=[];\nfor(var i=0;i<r;i++){\n  ind=c[i];\n  temp=[];\n  for(var j=0;j<m;j++){\n    j>=ind?temp.push(getRandomInt(1,max)):temp.push(0);\n  }\n  for(var k=0;k<r;k++){temp.splice(c[k],1,0);}\n  temp.splice(ind,1,1);\n\n  ech_matrix.push(temp);\n}\nfor(var i1=r;i1<n;i1++){\n var tempz=[];\n\n for(j=0;j<m;j++){tempz.push(0);} \n ech_matrix.push(tempz);\n}\n\n\nreturn [ech_matrix,c];\n\n\n\n\n\n", "type": "list", "language": "javascript"}, "rref": {"parameters": [["a", "list"], ["p", "number"]], "definition": "\nfunction findinv(a,q){\n  var s=0;\n  for(var j=1;j<q;j++){\n    if(modulo(a*j,q)==1){s=j;}\n  }\n  return s;\n}\nfunction copy_matrix(m){\n  return Numbas.util.copyarray(m,true);\n}\nfunction modulo(n,r){\n  if(r==0){return n;} else {return (r+n%r)%r;}\n}\n    var m=copy_matrix(a);\n    var lead = 0;\n    var echelon=[];\n    var deter=1;\n    var rows = m.length;\n    for(var i=0;i<rows;i++){echelon.push(0);}\n    var rank=rows;\n    var steps = 0;\n    var record_ops_matrix=[a];\n    var record_ops_message=['Starting with the matrix.'];\n    var columns=m[0].length;\n    for (var r = 0; r < rows; r++) {\n        if (columns <= lead) {\n          \n          record_ops_matrix.push(copy_matrix(m));\n          record_ops_message.push('Finished!');\n          return [record_ops_matrix, record_ops_message,steps,deter];\n        }\n        var i = r;\n        while (modulo(m[i][lead],p) == 0) {\n            i++;\n            if (rows == i) {\n                deter=0;\n                i = r;\n                rank-=1;\n                lead++;\n                if (columns == lead) {\n                    steps+=1;\n                    record_ops_matrix.push(copy_matrix(m));\n                    record_ops_message.push('Finished!');\n               return [record_ops_matrix, record_ops_message,steps,deter];\n                }\n            }\n        }\n \n        var tmp = m[i];\n        m[i] = m[r];\n        m[r] = tmp;\n        echelon[r]=lead+1;\n        var val = modulo(m[r][lead],p);\n        var inval;\n      if(p==0){inval=1/val;} else {inval=findinv(val,p);}\n      //if we have switched the ith and rth row i.e. a transposition, we have to multiply by -1\n      deter=(i==r?1:-1)*val*deter;  \n      for (var j = 0; j < columns; j++) {\n            m[r][j] = modulo(m[r][j]*inval,p);\n        }\n                    steps+=1;\n      var sw='';\n      var tw='';\n      if(val!=1){tw='We have divided throughout row '+(r+1)+' by $\\\\simplify[fractionNumbers]{{'+val+'}}$ so that the leading entry is 1.';}\n      if(i!=r)\n      {sw='We switch rows '+(r+1)+' and '+(i+1)+'.';\n      }\n      if(val!=1||i!=r){record_ops_matrix.push(copy_matrix(m));\n                 record_ops_message.push(sw+tw);}\n      \n        for (var i = 0; i < rows; i++) {\n            if (i == r) continue;\n            val = modulo(m[i][lead],p);\n            for (var j = 0; j < columns; j++) {\n                m[i][j] = modulo(m[i][j]-val * m[r][j],p);\n            }\n        }\n                    steps+=1;\n\n                    record_ops_matrix.push(copy_matrix(m));\n                    record_ops_message.push('We have reduced every other entry in column '+ (lead+1)+' to zero by adding suitable multiples of row '+(r+1)+ ' to the other rows.');\n      \n        lead++;\n    }\n          steps+=1;          \n          record_ops_matrix.push(copy_matrix(m));\n          record_ops_message.push('Finished!');\n          return [record_ops_matrix, record_ops_message,steps,deter];\n", "type": "list", "language": "javascript"}, "rdmat": {"parameters": [["n", "number"]], "definition": "function getRandomInt(min, max) {\nreturn Math.floor(Math.random() * (max - min + 1)) + min;\n};\nvar m=[];\nvar temp;\nfor(var i=0;i<n;i++){\n  temp=[];\n  for(var j=0;j<n;j++)\n  {temp.push(getRandomInt(-3,3));\n  }\n  m.push(temp);\n}\nm[0][0]=1;\nreturn m;", "type": "list", "language": "javascript"}}, "rulesets": {}, "extensions": ["stats"], "advice": "<p><strong>(a)</strong></p>\n<p>The following shows how $A$ is reduced to row-echelon form.</p>\n<p></p>\n<p>{solution(record_ops_matrix,record_ops_message)}</p>\n<p><strong></strong></p>\n<p><strong>(b)&nbsp;</strong></p>\n<p>Once a matrix is in reduced row echelon form, we can find the rank simply by counting the number of nonzero rows. This is also the same as the number of leading 1s.</p>\n<p>Thus in our example&nbsp;$Rank(A)=\\var{rank}$.</p>\n<p>Remember that the Rank-Nullity Theorem says that:</p>\n<p>Number of Columns of $A = Rank(A) + Nullity(A)$</p>\n<p></p>\n<p>Hence $Nullity(A) = $ number of columns $-Rank(A)=5-\\var{rank}=\\var{nullity}$.</p>\n<p></p>\n<p>Also the dimensions of the row space and the column space are both equal to the rank.</p>\n<p>Finally, the dimension of the null space is the nullity.</p>\n<p></p>", "parts": [{"variableReplacements": [], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "gaps": [{"variableReplacements": [], "allowFractions": true, "type": "matrix", "extendBaseMarkingAlgorithm": true, "allowResize": false, "scripts": {}, "correctAnswer": "{echmatrix}", "showFeedbackIcon": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "tolerance": 0, "variableReplacementStrategy": "originalfirst", "markPerCell": false, "useCustomName": false, "customName": "", "numColumns": "5", "correctAnswerFractions": true, "numRows": "6", "marks": "4", "unitTests": []}], "prompt": "<p></p>\n<p>$R= $[[0]]</p>\n<p><strong>All entries in the matrix must be input as fractions or integers and not as decimals.</strong></p>\n<p></p>\n<p></p>", "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "useCustomName": false, "customName": "", "marks": 0, "unitTests": []}, {"variableReplacements": [], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "gaps": [{"variableReplacements": [], "maxValue": "rank", "type": "numberentry", "extendBaseMarkingAlgorithm": true, "scripts": {}, "allowFractions": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "correctAnswerFraction": false, "useCustomName": false, "customName": "", "mustBeReduced": false, "minValue": "rank", "mustBeReducedPC": 0, "marks": "0.2", "unitTests": []}, {"variableReplacements": [], "maxValue": "nullity", "type": "numberentry", "extendBaseMarkingAlgorithm": true, "scripts": {}, "allowFractions": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "correctAnswerFraction": false, "useCustomName": false, "customName": "", "mustBeReduced": false, "minValue": "nullity", "mustBeReducedPC": 0, "marks": "0.2", "unitTests": []}, {"variableReplacements": [], "maxValue": "rank", "type": "numberentry", "extendBaseMarkingAlgorithm": true, "scripts": {}, "allowFractions": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "correctAnswerFraction": false, "useCustomName": false, "customName": "", "mustBeReduced": false, "minValue": "rank", "mustBeReducedPC": 0, "marks": "0.2", "unitTests": []}, {"variableReplacements": [], "maxValue": "rank", "type": "numberentry", "extendBaseMarkingAlgorithm": true, "scripts": {}, "allowFractions": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "correctAnswerFraction": false, "useCustomName": false, "customName": "", "mustBeReduced": false, "minValue": "rank", "mustBeReducedPC": 0, "marks": "0.2", "unitTests": []}, {"variableReplacements": [], "maxValue": "nullity", "type": "numberentry", "extendBaseMarkingAlgorithm": true, "scripts": {}, "allowFractions": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "correctAnswerFraction": false, "useCustomName": false, "customName": "", "mustBeReduced": false, "minValue": "nullity", "mustBeReducedPC": 0, "marks": "0.2", "unitTests": []}], "prompt": "<p></p>\n<p>$\\operatorname{Rank}(A)=$[[0]]</p>\n<p>$\\operatorname{Nullity}(A)=$[[1]]</p>\n<p>$\\operatorname{dim}(\\operatorname{col}(A))=$[[2]]</p>\n<p>$\\operatorname{dim}(\\operatorname{row}(A))=$[[3]]</p>\n<p>$\\operatorname{dim}(\\operatorname{null}(A))=$[[4]]</p>\n<p></p>", "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "useCustomName": false, "customName": "", "marks": 0, "unitTests": []}], "name": "Simon's copy of Row reducing a matrix and finding its rank and nullity-MA2223", "statement": "<p>Find the row-reduced echelon form $R$ of the following matrix $A$.</p>\n<p>$A=\\var{testmatrix}$</p>\n<p></p>\n<p><span id=\"type\"></span></p>", "variable_groups": [], "ungrouped_variables": ["echelonform", "ech", "echform", "testmatrix", "record", "rref1", "record_ops_matrix", "record_ops_message", "rank", "maxint", "steps", "rdmat", "deter", "nullity", "echmatrix", "trtestmatrix", "rows", "columns", "column_basis", "sum", "null_basis", "ec", "chosen", "student_cols_matrix", "not_chosen_col"], "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}