The answer is either 'True' or 'False'
", "help_url": "", "input_widget": "radios", "input_options": {"correctAnswer": "if(eval(settings[\"correct_answer_expr\"]), 0, 1)", "hint": {"static": true, "value": ""}, "choices": {"static": true, "value": ["True", "False"]}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\nif(studentanswer=correct_answer,\n correct(),\n incorrect()\n)\n\ninterpreted_answer:\nstudentAnswer=0\n\ncorrect_answer:\nif(eval(settings[\"correct_answer_expr\"]),0,1)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=correct_answer,\n correct(),\n incorrect()\n)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "studentAnswer=0"}, {"name": "correct_answer", "description": "", "definition": "if(eval(settings[\"correct_answer_expr\"]),0,1)"}], "settings": [{"name": "correct_answer_expr", "label": "Is the answer \"True\"", "help_url": "", "hint": "", "input_type": "mathematical_expression", "default_value": "true", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Direct sum and is sum whole space", "tags": ["direct sum", "Explore mode", "subspaces"], "metadata": {"description": "Student decides on four different examples whether two subspaces form a direct sum and whether the sum is the whole vectorspace.
\nNo randomisation, just the four examples. The question is set in explore mode, so that after deciding, students are asked to give reasons for their choices.
\n", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "For each of the following subspaces, determine whether the sum \\(U + W\\) is a direct sum, and whether the sum gives the whole vector space \\(V\\) (i.e. whether \\(U + W = V\\) ). The parts will talk you through some of the steps.
", "advice": "a) Let \\( U = \\left\\{ \\begin{pmatrix} x\\\\x \\end{pmatrix} \\middle| x \\in \\mathbb{R} \\right\\} \\) and \\( W = \\left\\{ \\begin{pmatrix} -y\\\\y \\end{pmatrix} \\middle| y \\in \\mathbb{R} \\right\\} \\) be subspaces of \\(\\mathbb{R}^2\\).
\nWe can see that if \\(v\\in U\\cap W\\), then we have to have \\(v=\\begin{pmatrix} x\\\\x \\end{pmatrix}=\\begin{pmatrix} -y\\\\y \\end{pmatrix}\\) for some \\(x\\) and \\(y\\). This means we need \\(x=-y\\) and \\(x=y\\), so the only option is \\(x=y=0\\). So \\(v=0\\) is the only vector in the intersection. So this is a direct sum.
\nWe can get any general vector \\(\\begin{pmatrix} a\\\\b\\end{pmatrix} \\in \\mathbb{R}^2\\) in this sum: \\(\\begin{pmatrix} a\\\\b\\end{pmatrix}=\\begin{pmatrix} \\frac{b+a}{2}\\\\\\frac{b+a}{2}\\end{pmatrix} +\\begin{pmatrix} -\\frac{b-a}{2}\\\\\\frac{b-a}{2}\\end{pmatrix}\\). I.e. \\(x=\\frac{b+a}{2}\\) and \\(y=\\frac{b-a}{2}\\). There is only one option here.
\nb) Let \\( U = \\left\\{ \\begin{pmatrix} x\\\\x\\\\-x \\end{pmatrix} \\middle| x \\in \\mathbb{R} \\right\\} \\) and \\( W = \\left\\{ \\begin{pmatrix} y\\\\z\\\\-z \\end{pmatrix} \\middle| y,z \\in \\mathbb{R} \\right\\} \\) be subspaces of \\( \\mathbb{R}^3 \\).
\nHere we can see that there is some overlap: for example, the vector \\(\\begin{pmatrix}1\\\\1\\\\-1\\end{pmatrix}\\) is in \\(U\\cap W\\). In fact, \\(U\\subseteq W\\): any vector in \\(U\\) is also in \\(W\\). For any given \\(x\\), we just have to choose \\(y=x\\) and \\(z=x\\). So this is not a direct sum.
\nIt also does not give the whole space, as the second and third entry are very linked. So we cannot get the vector \\(\\begin{pmatrix}0\\\\1\\\\0\\end{pmatrix}\\) or the vector \\(\\begin{pmatrix}0\\\\0\\\\1\\end{pmatrix}\\) in the sum. (Or any vector where the third entry is different to minus the second entry.) In fact, since \\(U\\subseteq W\\), we get \\(U+W=W\\).
\nc) Let \\( U = \\left\\{ \\begin{pmatrix} x\\\\y\\\\-y \\end{pmatrix} \\middle| x, y \\in \\mathbb{R} \\right\\} \\) and \\( W = \\left\\{ \\begin{pmatrix} 0\\\\z\\\\-z \\end{pmatrix} \\middle| z \\in \\mathbb{R} \\right\\} \\) be subspaces of \\( \\mathbb{R}^3 \\).
\nAgain we see some overlap: \\( \\begin{pmatrix} 0\\\\z\\\\-z \\end{pmatrix} \\in U\\cap W\\) for any \\(z\\): we just put \\(x=0\\) and \\(y=z\\). So \\(W\\subseteq U\\). So it is not a direct sum.
\nTherefore \\(U+W=U\\), which is not all of \\(\\mathbb{R}^3\\). As in b), we cannot get the vector \\(\\begin{pmatrix}0\\\\1\\\\0\\end{pmatrix}\\) or the vector \\(\\begin{pmatrix}0\\\\0\\\\1\\end{pmatrix}\\) in the sum.
\nd) Let \\( P_2 \\) be the vectorspace of polynomials of degree at most \\( 2 \\) and consider the subspaces \\( U = \\{ p=a_2 X^2 + a_1 X \\vert a_1,a_2 \\in \\mathbb{R}\\} \\) and \\( W = P_1= \\{ q=b_1 X + b_0 \\vert b_0,b_1 \\in \\mathbb{R}\\}\\).
\nThere is an overlap here: the polynomial \\(X\\) is in \\(U\\cap W\\), because \\(X=0\\cdot X^2+1\\cdot X=1\\cdot X+0\\). So of course any \\(dX\\in U\\cap W\\). So this is not a direct sum. But we do not have one of the two spaces contained in the other: no polynomials in \\(U\\) have any non-zero constant terms, and no polynomials in \\(W\\) have any non-zero \\(X^2\\) term.
\nHowever, we do get the whole space as the sum:
\n\\(c_2X^2+c_1X+c_0= (c_2X^2+c_1X) + (0X+c_0)\\), or \\(c_2X^2+c_1X+c_0= (c_2X^2+0X) + (c_1X+c_0)\\), or \\(c_2X^2+c_1X+c_0= (c_2X^2+0.4c_1X) + (0.6c_1X+c_0)\\), or \\(c_2X^2+c_1X+c_0= (c_2X^2+(c_1+14)X) + ((c_1-14)X+c_0)\\), etc. As long as \\(a_1+b_1=c_1\\), and \\(a_2=c_2\\), \\(b_0=c_0\\), we can make this sum. So you see there is some redundancy coming from the overlap.
\ne) Let \\( P_2 \\) be the vectorspace of polynomials of degree at most \\( 2 \\) and consider the subspaces \\( U = \\{ p=a_2 X^2 \\vert a_2 \\in \\mathbb{R}\\} \\) and \\( W = P_0=\\{ q=b_0 \\vert b_0 \\in \\mathbb{R}\\} \\).
\nWe see here that there is no overlap: if \\(p\\in U\\cap W\\), then \\(p=a_2X^2 + 0X+0=0X^2+0X+b_0\\), so \\(a_2=0\\) and \\(b_0=0\\), so \\(0\\) is the only element in the intersection. So this is a direct sum.
\nThe sum does not give all of \\(P_2\\): we cannot get any non-zero \\(X\\) term. So for example \\(X\\notin U+W\\), or any \\(c_2X^2+c_1X+c_0\\) with \\(c_1\\neq 0\\).
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\nIs \\(U + W \\) a direct sum?
\n[[0]]
\nIs \\(U + W = \\mathbb{R}^2\\) ?
\n[[1]]
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\nYou correctly said that \\(U + W \\) is a direct sum.
\nCheck the statements which, after established true, prove your claim.
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\nIn order to show that \\(U + W = \\mathbb{R}^2\\), choose \\(x,y\\) such that \\( \\begin{pmatrix} x\\\\x \\end{pmatrix} + \\begin{pmatrix} -y\\\\y \\end{pmatrix} = \\begin{pmatrix} a\\\\b \\end{pmatrix} \\), for a general vector \\(\\begin{pmatrix}a\\\\b\\end{pmatrix}\\) in \\(\\mathbb{R}^2\\).
\n\\(x = \\) [[0]]
\n\\(y = \\) [[1]]
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\nIs \\(U + W \\) a direct sum?
\n[[0]]
\nIs \\(U + W = \\mathbb{R}^3 \\)?
\n[[1]]
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\nIn order to show that \\(U + V \\) is not a direct sum, you need to show that \\(U \\cap V \\) contains at least one vector which is not zero.
\nThus choose \\(x,y,z\\) such that \\( \\begin{pmatrix} x\\\\x\\\\-x \\end{pmatrix} = \\begin{pmatrix} y\\\\z\\\\-z \\end{pmatrix} \\neq \\begin{pmatrix} 0\\\\0\\\\0 \\end{pmatrix}.\\)
\n\\( x = \\) [[0]]
\n\\( y = \\) [[1]]
\n\\( z = \\) [[2]]
\nTrue or false: actually all vectors in \\(U\\) are also in \\(W\\), i.e. \\(U\\subseteq W\\).
\n[[3]]
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\nYou have seen that \\(U + W \\neq \\mathbb{R}^3\\).
\nGive an example of a vector in \\(\\mathbb{R}^3\\) which is not contained in \\(U + W,\\) i.e. choose \\( v \\in \\mathbb{R}^3 \\) such that for all \\(x,y,z \\in \\mathbb{R} \\) \\( v \\neq \\begin{pmatrix} x\\\\x\\\\-x \\end{pmatrix} + \\begin{pmatrix} y\\\\z\\\\-z \\end{pmatrix} \\)
\n\\(v = \\) [[0]]
\nTrue or false: in this case, \\(U+W\\) is one of the two original subspaces (\\(U+W=U\\) or \\(U+W=W\\)).
\n[[1]]
", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "v", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([0],[100],[0])", "correctAnswerFractions": false, "numRows": "3", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "true-false", "useCustomName": true, "customName": "true/false", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_answer_expr": "true"}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Try part c)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Give the reaons for \"direct sum\" answer", "rawLabel": "Give the reaons for \"direct sum\" answer", "otherPart": 7, "variableReplacements": [], "availabilityCondition": "answered", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": true}, {"label": "Give the reaons for \"whole space\" answer", "rawLabel": "Give the reaons for \"whole space\" answer", "otherPart": 5, "variableReplacements": [], "availabilityCondition": "answered", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": true}, {"label": "Try part d)", "rawLabel": "", "otherPart": 9, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": false}, {"label": "Try part e)", "rawLabel": "", "otherPart": 12, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Let \\( U = \\left\\{ \\begin{pmatrix} x\\\\y\\\\-y \\end{pmatrix} \\middle| x, y \\in \\mathbb{R} \\right\\} \\) and \\( W = \\left\\{ \\begin{pmatrix} 0\\\\z\\\\-z \\end{pmatrix} \\middle| z \\in \\mathbb{R} \\right\\} \\) be subspaces of \\( \\mathbb{R}^3 \\).
\nIs \\(U + W \\) a direct sum?
\n[[0]]
\nIs \\(U + W = \\mathbb{R}^3 \\)?
\n[[1]]
\n", "gaps": [{"type": "1_n_2", "useCustomName": true, "customName": "direct sum?", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Decide if direct sum", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["yes", "no"], "matrix": ["-1", "1"], "distractors": ["", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "whole space?", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Decide if sum gives whole space", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["yes", "no"], "matrix": ["-1", "1"], "distractors": ["", ""]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "c) counterexample direct sum", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "mark:\ncorrectif(answers[0]=0 && answers[1] = answers[2] && answers[2] <> 0)", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Give the reaons for \"whole space\" answer", "rawLabel": "Give the reaons for \"whole space\" answer", "otherPart": 8, "variableReplacements": [], "availabilityCondition": "answered", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Reasonings", "prompt": "Let \\( U = \\left\\{ \\begin{pmatrix} x\\\\y\\\\-y \\end{pmatrix} \\middle| x, y \\in \\mathbb{R} \\right\\} \\) and \\( W = \\left\\{ \\begin{pmatrix} 0\\\\z\\\\-z \\end{pmatrix} \\middle| z \\in \\mathbb{R} \\right\\} \\) be subspaces of \\( \\mathbb{R}^3\\).
\nIn order to show that \\(U + V \\) is not a direct sum you need to show that \\(U \\cap V \\) contains at least one vector which is not zero.
\nThus choose \\(x,y,z\\) such that \\( \\begin{pmatrix} x\\\\y\\\\-y \\end{pmatrix} = \\begin{pmatrix} 0\\\\z\\\\-z \\end{pmatrix} \\neq \\begin{pmatrix} 0\\\\0\\\\0 \\end{pmatrix}.\\)
\n\\( x = \\) [[0]]
\n\\( y = \\) [[1]]
\n\\( z = \\) [[2]]
\nTrue or false: actually all vectors in \\(W\\) are also in \\(U\\), i.e. \\(W\\subseteq U\\).
\n[[3]]
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "x", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "123", "maxValue": "123", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "y", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "123", "maxValue": "123", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "z", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "123", "maxValue": "123", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "true-false", "useCustomName": true, "customName": "True/false", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_answer_expr": "true"}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "c) counterexample whole space", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "mark:\n assert(all_valid or not settings[\"sortAnswers\"], fail(translate(\"question.can not submit\")));\n apply(answers);\n apply(gap_feedback);\n correctif(rank(matrix([1,0,0],[0,1,-1],transpose(answers[0])[0])) = 3)", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Try part d)", "rawLabel": "", "otherPart": 9, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": false}, {"label": "Try part e)", "rawLabel": "", "otherPart": 12, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": false}, {"label": "Give the reaons for \"direct sum\" answer if you haven't done it yet", "rawLabel": "Give the reaons for \"direct sum\" answer if you haven't done it yet", "otherPart": 7, "variableReplacements": [], "availabilityCondition": "answered", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Reasonings", "prompt": "Let \\( U = \\left\\{ \\begin{pmatrix} x\\\\y\\\\-y \\end{pmatrix} \\middle| x, y \\in \\mathbb{R} \\right\\} \\) and \\( W = \\left\\{ \\begin{pmatrix} 0\\\\z\\\\-z \\end{pmatrix} \\middle| z \\in \\mathbb{R} \\right\\} \\) be subspaces of \\( \\mathbb{R}^3\\).
\nYou have seen that \\(U + W \\neq \\mathbb{R}^3\\).
\nGive an example of a vector in \\(\\mathbb{R}^3\\) which is not contained in \\(U + W,\\) i.e. choose \\( v \\in \\mathbb{R}^3 \\) such that for all \\(x,y,z \\in \\mathbb{R} \\) \\( v \\neq \\begin{pmatrix} x\\\\y\\\\-y \\end{pmatrix} + \\begin{pmatrix} 0\\\\z\\\\-z \\end{pmatrix} \\)
\n\\(v = \\) [[0]]
\nTrue or false: in this case, \\(U+W\\) is one of the two original subspaces (\\(U+W=U\\) or \\(U+W=W\\)).
\n[[1]]
", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "v", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([0],[100],[0])", "correctAnswerFractions": false, "numRows": "3", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "true-false", "useCustomName": true, "customName": "True/false", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_answer_expr": "true"}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Try part d)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Give the reaons for \"direct sum\" answer", "rawLabel": "Give the reaons for \"direct sum\" answer", "otherPart": 10, "variableReplacements": [], "availabilityCondition": "answered", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": true}, {"label": "Give the reaons for \"whole space\" answer", "rawLabel": "Give the reaons for \"whole space\" answer", "otherPart": 11, "variableReplacements": [], "availabilityCondition": "answered", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": true}, {"label": "Try part e)", "rawLabel": "", "otherPart": 12, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Let \\( P_2 \\) be the vectorspace of polynomials of degree at most \\( 2 \\) and consider the subspaces \\( U = \\{ p=a_2 X^2 + a_1 X \\vert a_1,a_2 \\in \\mathbb{R}\\} \\) and \\( W = P_1= \\{ q=b_1 X + b_0 \\vert b_0,b_1 \\in \\mathbb{R}\\}\\).
\nIs \\(U+W\\) a direct sum?
\n[[0]]
\nIs \\( U+W=P_2\\)?
\n[[1]]
", "gaps": [{"type": "1_n_2", "useCustomName": true, "customName": "direct sum?", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Decide if direct sum", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["yes", "no"], "matrix": [0, "1"], "distractors": ["", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "whole space?", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Decide if sum gives whole space", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["yes", "no"], "matrix": ["1", "0"], "distractors": ["", ""]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "d) counterexample direct sum", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "mark:\n assert(all_valid or not settings[\"sortAnswers\"], fail(translate(\"question.can not submit\")));\n apply(answers);\n correctif(answers[1] = answers[2] && answers[1] <> 0 && answers[0] = answers[3] = 0)", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Give the reaons for \"whole space\" answer", "rawLabel": "Give the reaons for \"whole space\" answer", "otherPart": 11, "variableReplacements": [], "availabilityCondition": "answered", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Reasonings", "prompt": "Let \\( P_2 \\) be the vectorspace of polynomials of degree at most \\( 2 \\) and consider the subspaces \\( U = \\{ p=a_2 X^2 + a_1 X \\vert a_1,a_2 \\in \\mathbb{R}\\} \\) and \\( W = P_1= \\{ q=b_1 X + b_0 \\vert b_0,b_1 \\in \\mathbb{R}\\}\\).
\nYou have seen that \\(U + W\\) is not a direct sum.
\nChoose \\(a_2,a_1,b_1,b_0\\) such that \\( a_2 X^2 + a_1 X = b_1 X + b_0 \\neq 0 \\).
\n\\(a_2 = \\) [[0]]
\n\\(a_1 = \\) [[1]]
\n\\(b_1 = \\) [[2]]
\n\\(b_0 = \\) [[3]]
\nTrue or false: actually all vectors in \\(U\\) are also in \\(W\\), i.e. \\(U\\subseteq W\\).
\n[[4]]
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "a_2", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "a_1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1", "maxValue": "1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "b_1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1", "maxValue": "1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "b_0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "true-false", "useCustomName": true, "customName": "True/false", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_answer_expr": "false"}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "d) proof whole space", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "sum_gap1_gap2 (Sum of gaps 1 and gaps 2, to check for correctness):\n simplify(expression(answergap1+\"+\"+answergap2),\"all\")\n\nanswergap1:\n string(answers[1])\n\nanswergap2:\n string(answers[2])\n\nchecksum (Check the sum of the two middle gaps):\n if(sum_gap1_gap2=expression(\"c_1\"),2,0)\n\ngap_credits:\n map(result[\"credit\"],result,gap_feedback)\n\ngap_credits_corrected:\n if(checksum=2,[gap_credits[0],1,1,gap_credits[3]],[gap_credits[0],0,0,gap_credits[3]])\n\ncorrected_mark:\n set_credit(sum(gap_credits_corrected)/4,\"\")\n\n\nmark:\n assert(all_valid or not settings[\"sortAnswers\"], fail(translate(\"question.can not submit\")));\n apply(answers);\n apply(gap_feedback);\n apply(corrected_mark)", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Try part e)", "rawLabel": "", "otherPart": 12, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": false}, {"label": "Give the reaons for \"direct sum\" answer if you haven't done it yet", "rawLabel": "Give the reaons for \"direct sum\" answer if you haven't done it yet", "otherPart": 10, "variableReplacements": [], "availabilityCondition": "answered", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Reasonings", "prompt": "Let \\( P_2 \\) be the vectorspace of polynomials of degree at most \\( 2 \\) and consider the subspaces \\( U = \\{ p=a_2 X^2 + a_1 X \\vert a_1,a_2 \\in \\mathbb{R}\\} \\) and \\( W = P_1= \\{ q=b_1 X + b_0 \\vert b_0,b_1 \\in \\mathbb{R}\\}\\).
\nIn order to show that \\(U + W = P_2,\\) choose \\(a_2,a_1,b_1,b_0\\) such that \\( a_2 X^2 + a_1 X + b_1 X + b_0 = c_2 X^2 + c_1 X + c_0 \\). I.e. show that any general polynomial of degree at most two can be written as the sum of a poly in \\(U\\) and a poly in \\(W\\).
\nEnter subscripts as c_2 or c_1 etc. Please use decimals instead of fractions: the marking algorithm will not understand fractions in this question.
\\(a_2 = \\) [[0]]
\n\\(a_1 = \\) [[1]]
\n\\(b_1 = \\) [[2]]
\n\\(b_0 = \\) [[3]]
", "gaps": [{"type": "jme", "useCustomName": true, "customName": "a_2", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "c_2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c_2", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "a_1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "c_1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c_1", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "b_1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "0", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": true, "customName": "b_0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "c_0", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c_0", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Try part e)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Give the reaons for \"whole space\" answer", "rawLabel": "Give the reaons for \"whole space\" answer", "otherPart": 13, "variableReplacements": [], "availabilityCondition": "answered", "penalty": "", "penaltyAmount": 0, "lockAfterLeaving": true}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Let \\( P_2 \\) be the vectorspace of polynomials of degree at most \\( 2 \\) and consider the subspaces \\( U = \\{ p=a_2 X^2 \\vert a_2 \\in \\mathbb{R}\\} \\) and \\( W = P_0=\\{ q=b_0 \\vert b_0 \\in \\mathbb{R}\\} \\).
\nIs \\(U+W\\) a direct sum?
\n[[0]]
\nIs \\( U+W=P_2\\)?
\n[[1]]
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\nYou have seen that \\(U + W \\neq P_2\\).
\nGive an example of a Polynomial in \\(P_2,\\) which is not contained in \\(U + W,\\) i.e. choose \\(c_0,c_1,c_2\\) such that \\( c_2 X^2 + c_1 X + c_0 \\notin U + W. \\)
\n\\(c_0 = \\) [[0]]
\n\\(c_1 = \\) [[1]]
\n\\(c_2 = \\) [[2]]
\nTrue or false: in this case, \\(U+W\\) is one of the two original subspaces (\\(U+W=U\\) or \\(U+W=W\\)).
\n[[3]]
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