// Numbas version: exam_results_page_options
{"name": "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": [{"name": "Row reducing a matrix and finding its rank and nullity-MA2223", "tags": ["bases", "basis", "column space", "dimension", "echelon form", "linear algebra", "linear independence", "linearly dependent", "matrices", "matrix type", "null space", "nullity", "rank", "row operations", "row reduced", "row space", "row-reduced"], "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"}, "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>", "advice": "<p>a) 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>b)&nbsp;</p>\n<p>The rank of A is the number of columns of R with pivots, i.e. $\\var{rank}$.</p>\n<p>The nullity of A is the number of columns of R without pivots, i.e. $5-\\var{rank}=\\var{5-rank}$.</p>\n<p>Remember that the Rank-Nullity Theorem says that the number of columns of A equals the rank of A plus the nullity of A, i.e. $5=\\var{rank}+\\var{nullity}$.</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>", "rulesets": {}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"column_basis": {"name": "column_basis", "group": "Ungrouped variables", "definition": "set(map(trtestmatrix[x],x,echelonform[1]))", "description": "", "templateType": "anything", "can_override": false}, "maxint": {"name": "maxint", "group": "Ungrouped variables", "definition": "4", "description": "", "templateType": "anything", "can_override": false}, "echmatrix": {"name": "echmatrix", "group": "Ungrouped variables", "definition": "matrix(ech)", "description": "", "templateType": "anything", "can_override": false}, "deter": {"name": "deter", "group": "Ungrouped variables", "definition": "rref(rdmat,0)[3]", "description": "", "templateType": "anything", "can_override": false}, "record": {"name": "record", "group": "Ungrouped variables", "definition": "rref(list(testmatrix),0)", "description": "", "templateType": "anything", "can_override": false}, "echelonform": {"name": "echelonform", "group": "Ungrouped variables", "definition": "echelon_form_matrix(rows,rank,columns,maxint)", "description": "", "templateType": "anything", "can_override": false}, "nullity": {"name": "nullity", "group": "Ungrouped variables", "definition": "5-rank", "description": "", "templateType": "anything", "can_override": false}, "ech": {"name": "ech", "group": "Ungrouped variables", "definition": "echelonform[0]", "description": "", "templateType": "anything", "can_override": false}, "null_basis": {"name": "null_basis", "group": "Ungrouped variables", "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]])", "description": "", "templateType": "anything", "can_override": false}, "trtestmatrix": {"name": "trtestmatrix", "group": "Ungrouped variables", "definition": "transpose(testmatrix)", "description": "", "templateType": "anything", "can_override": false}, "columns": {"name": "columns", "group": "Ungrouped variables", "definition": "5", "description": "", "templateType": "anything", "can_override": false}, "testmatrix": {"name": "testmatrix", "group": "Ungrouped variables", "definition": "matrix(rdmat)*matrix(ech)", "description": "", "templateType": "anything", "can_override": false}, "sum": {"name": "sum", "group": "Ungrouped variables", "definition": "sum(echform)", "description": "", "templateType": "anything", "can_override": false}, "record_ops_matrix": {"name": "record_ops_matrix", "group": "Ungrouped variables", "definition": "record[0]", "description": "", "templateType": "anything", "can_override": false}, "rows": {"name": "rows", "group": "Ungrouped variables", "definition": "6", "description": "", "templateType": "anything", "can_override": false}, "rank": {"name": "rank", "group": "Ungrouped variables", "definition": "random(2,3,4)\n", "description": "", "templateType": "anything", "can_override": false}, "rref1": {"name": "rref1", "group": "Ungrouped variables", "definition": "rref1(list(testmatrix))[0]", "description": "", "templateType": "anything", "can_override": false}, "chosen": {"name": "chosen", "group": "Ungrouped variables", "definition": "[true,true,false,true,false]", "description": "", "templateType": "anything", "can_override": false}, "numsteps": {"name": "numsteps", "group": "Ungrouped variables", "definition": "rref(list(testmatrix),0)[2]", "description": "", "templateType": "anything", "can_override": false}, "rdmat": {"name": "rdmat", "group": "Ungrouped variables", "definition": "rdmat(rows)", "description": "", "templateType": "anything", "can_override": false}, "echform": {"name": "echform", "group": "Ungrouped variables", "definition": "map(x+1,x,echelonform[1])", "description": "", "templateType": "anything", "can_override": false}, "student_cols_matrix": {"name": "student_cols_matrix", "group": "Ungrouped variables", "definition": "transpose(matrix(map(transpose(echmatrix)[n],n,filter(chosen[j],j,0..4))))", "description": "", "templateType": "anything", "can_override": false}, "record_ops_message": {"name": "record_ops_message", "group": "Ungrouped variables", "definition": "record[1]", "description": "", "templateType": "anything", "can_override": false}, "not_chosen_col": {"name": "not_chosen_col", "group": "Ungrouped variables", "definition": "transpose(matrix(map(transpose(echmatrix)[n],n,filter(!chosen[j],j,0..4)))[0])", "description": "", "templateType": "anything", "can_override": false}, "ec": {"name": "ec", "group": "Ungrouped variables", "definition": "echmatrix", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "deter<>0\nand\nmax(map(testing(record[0][x]),x,0..length(record[0])-1))=0", "maxRuns": "200"}, "ungrouped_variables": ["echelonform", "ech", "echform", "testmatrix", "record", "rref1", "record_ops_matrix", "record_ops_message", "rank", "maxint", "numsteps", "rdmat", "deter", "nullity", "echmatrix", "trtestmatrix", "rows", "columns", "column_basis", "sum", "null_basis", "ec", "chosen", "student_cols_matrix", "not_chosen_col"], "variable_groups": [], "functions": {"solution": {"parameters": [["matrix_list", "list"], ["message_list", "list"]], "type": "html", "language": "javascript", "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"}, "rref1": {"parameters": [["m", "list"]], "type": "list", "language": "javascript", "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"}, "echelon_form_matrix": {"parameters": [["n", "number"], ["r", "number"], ["m", "number"], ["max", "number"]], "type": "list", "language": "javascript", "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"}, "testing": {"parameters": [["m", "list"]], "type": "number", "language": "javascript", "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      "}, "rref": {"parameters": [["a", "list"], ["p", "number"]], "type": "list", "language": "javascript", "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"}, "rdmat": {"parameters": [["n", "number"]], "type": "list", "language": "javascript", "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;"}}, "preamble": {"js": 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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", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p></p>\n<p>$R= $[[0]]</p>\n<p><strong>All entries in the matrix must be 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"showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p></p>\n<p>$\\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>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "rank", "maxValue": "rank", "correctAnswerFraction": false, 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"objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}