// Numbas version: finer_feedback_settings
{"name": "Berechne Eigenraum zu gegebenem Eigenwert (WiWi)", "extensions": ["linearalgebra2"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Berechne Eigenraum zu gegebenem Eigenwert (WiWi)", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Wir betrachten die Matrix</p>\n<p>\\[ A=\\var[fractionNumbers]{A} \\in {\\rm Mat}_{\\var{n}\\times \\var{n}} \\]</p>", "advice": "<p>Der gesuchte Untervektorraum ist der Unterraum ${\\rm Ker}(A-c E_{\\var{n}}) = {\\rm Ker}\\left(\\var[fractionNumbers]{A} - \\var{lambda * unit_matrix(n)}\\right)$.</p>\n<p>Die Aufgabe besteht nun darin, eine Basis f&uuml;r die L&ouml;sungsmenge des durch die Matrix</p>\n<p>\\[ \\var[fractionNumbers]{A -lambda * unit_matrix(n)} \\]</p>\n<p>gegebenen homogenen linearen Gleichungssystems zu bestimmen. Das kann man wie &uuml;blich mit dem Gau&szlig;-Algorithmus machen.</p>", "rulesets": {}, "extensions": ["linearalgebra2"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"A": {"name": "A", "group": "Ungrouped variables", "definition": "S * (\n  matrix(diag(repeat(lambda, dimES) + repeat(random(-2..2 except lambda), n-dimES))) +\n  transpose(matrix(repeat(repeat(0,n), dimES) + map(repeat(random(-2, -1, 0, 0, 0, 1, 1, 2), x+dimES-1) + repeat(0, n-x-dimES+1), x, 1..n-dimES)))\n)\n* inverse_matrix(s)", "description": "", "templateType": "anything", "can_override": false}, "basis": {"name": "basis", "group": "Ungrouped variables", "definition": "S * transpose(matrix(columns(unit_matrix(n))[0..dimES]))", "description": "", "templateType": "anything", "can_override": false}, "S": {"name": "S", "group": "Ungrouped variables", "definition": "random_trafo(n, [1,2,1,2,1,2,2,1,2,1])", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(3..4)", "description": "", "templateType": "anything", "can_override": false}, "dimES": {"name": "dimES", "group": "Ungrouped variables", "definition": "random(1..n-1)", "description": "<p>dimension of eigenspace we are looking for</p>", "templateType": "anything", "can_override": false}, "lambda": {"name": "lambda", "group": "Ungrouped variables", "definition": "random(-3..3)/random(1,2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "basis", "S", "n", "dimES", "lambda"], "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>Die Zahl $c = \\var[fractionNumbers]{lambda}$ ist ein Eigenwert von $A$. Berechnen Sie eine Basis des Untervektorraums</p>\n<p>$U = \\{ v\\in\\mathbb R^{\\var{n}};\\ A\\cdot v = c\\cdot v \\}$</p>\n<p>und geben Sie die Vektoren der Basis als die Spalten einer Matrix an.</p>\n<p>[[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "mark:\n    apply(any_empty);\n    apply(any_invalid);\n    apply(wrong_size);\n    correctif(\n      rank(interpreted_answer) = dimES and\n      rank(matrix(columns(basis) + columns(interpreted_answer))) = dimES)", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "{basis}", "correctAnswerFractions": true, "numRows": "n", "numColumns": 1, "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Ulrich G\u00f6rtz", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7603/"}], "resources": []}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Ulrich G\u00f6rtz", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7603/"}]}