// Numbas version: exam_results_page_options
{"name": "Durchschnitt von Kernen", "extensions": ["linearalgebra2"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Durchschnitt von Kernen", "tags": [], "metadata": {"description": "<p>Find a basis for the intersection of two kernels of linear maps $\\mathbb Q^3\\to \\mathbb Q^3$ and $\\mathbb Q^3\\to \\mathbb Q$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Wir betrachten die Abbildungen $f = \\mathbf f_A\\colon \\mathbb Q^3 \\to \\mathbb Q^3$ und $g = \\mathbf f_B\\colon \\mathbb Q^3\\to \\mathbb Q$ mit</p>\n<p>\\[ A = \\var{F},\\qquad B = \\var{G} \\]</p>\n<p></p>", "advice": "<p><strong>a)</strong></p>\n<p>Wegen der Dimensionsformel addieren sich in beiden F&auml;llen die Dimensionen von Kern und Bild zu $3$.</p>\n<p>Die Dimension des Kerns ist die Anzahl der frei w&auml;hlbaren Variablen in der L&ouml;sungsmenge des zugeh&ouml;rigen homogenen linearen Gleichungssystems. Die Dimension des Bildes ist die Dimension des von den Spaltenvektoren erzeugten Untervektorraums. Beide Dimensionen kann man in diesem Beispiel \"durch genaues Hinschauen\" erkennen (bei $g$ ist die einzige Frage, ob alle Eintr&auml;ge der Matrix $0$ sind, oder nicht; die Aufgabe ist so eingerichtet, dass $f$ Rang $1$ oder $2$ hat).</p>\n<p><strong>b)</strong></p>\n<p>Der Kern von $f$ ist die L&ouml;sungsmenge des Gleichungssystems $Ax=0$, analog f&uuml;r $g$ und $B$. Der Durchschnitt der Kerne ist also die L&ouml;sungsmenge des homogenen Gleichungssystems mit Koeffizientenmatrix $\\begin{pmatrix} A\\\\ B\\end{pmatrix}$ (als Blockmatrix geschrieben), wir nehmen also die durch $A$ und die durch $B$ gegebenen Gleichungen zusammen. Eine Basis der L&ouml;sungsmenge kann man dann nach dem &uuml;blichen Verfahren bestimmen.</p>\n<p>Die reduzierte Zeilenstufenform von $\\begin{pmatrix} A\\\\ B\\end{pmatrix}$ ist $\\var{reduced_row_echelon_form(matrix(list(F)+list(G)))}$.</p>", "rulesets": {}, "extensions": ["linearalgebra2"], "variables": {"rt": {"name": "rt", "group": "Ungrouped variables", "definition": "random_trafo_rng(3, [1, 2,3,1,1, 3, 2, 1, 1, 1,2, 1], [-3,-2,-1,1,2,3])", "description": "", "templateType": "anything"}, "F": {"name": "F", "group": "Ungrouped variables", "definition": "if(rnd < 3,\n  matrix(repeat(list(random(-2..2 except [0,1])*G)[0], 3)),  // go for two-diml intersection\n  modp(rt * random_ref(num_rows, num_columns, 1..random(1,2)), 11)\n)", "description": "", "templateType": "anything"}, "num_rows": {"name": "num_rows", "group": "Ungrouped variables", "definition": "3", "description": "", "templateType": "anything"}, "num_columns": {"name": "num_columns", "group": "Ungrouped variables", "definition": "3", "description": "", "templateType": "anything"}, "G": {"name": "G", "group": "Ungrouped variables", "definition": "matrix(repeat(random(-2..2), 3))", "description": "", "templateType": "anything"}, "basis": {"name": "basis", "group": "Ungrouped variables", "definition": "// [rank(F), rank(G), rank(matrix(list(F) + list(G)))] // reduced_row_echelon_form(matrix(list(F) + list(G)))]\nbasis_solution_set(matrix(list(F) + list(G)))", "description": "", "templateType": "anything"}, "rnd": {"name": "rnd", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "len(columns(basis)) > 0", "maxRuns": 100}, "ungrouped_variables": ["rt", "num_rows", "num_columns", "F", "G", "basis", "rnd"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "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>Geben Sie die folgenden Dimensionen an:</p>\n<p>$\\dim {\\rm Ker}(f) = $ [[0]]</p>\n<p>$\\dim {\\rm Im}(f) = $ [[1]]</p>\n<p>$\\dim {\\rm Ker}(g) = $ [[2]]</p>\n<p>$\\dim {\\rm Im}(g) = $ [[3]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{len(columns(F))-rank(F)}", "maxValue": "{len(columns(F))-rank(F)}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{rank(F)}", "maxValue": "{rank(F)}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{len(columns(G))-rank(G)}", "maxValue": "{len(columns(G))-rank(G)}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{rank(G)}", "maxValue": "{rank(G)}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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>Bestimmen Sie eine Basis von ${\\rm Ker}(f)\\cap {\\rm Ker}(g)$. Geben Sie Ihre Antwort als eine Matrix an, deren Spalten eine solche Basis bilden.</p>\n<p>[[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "\nmark:\n    apply(any_empty);\n    apply(any_invalid);\n    correctif(\n      rank(interpreted_answer) = len(columns(basis)) and\n      len(columns(interpreted_answer)) = len(columns(basis)) and\n      all_entries_zero(matrix(list(F)+list(G))*interpreted_answer)\n)", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "{basis}", "correctAnswerFractions": true, "numRows": "3", "numColumns": 1, "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ulrich G\u00f6rtz", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7603/"}]}]}], "contributors": [{"name": "Ulrich G\u00f6rtz", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7603/"}]}