// Numbas version: exam_results_page_options {"name": "Combine linearly dependent vectors to get zero", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"x": {"templateType": "anything", "group": "Ungrouped variables", "definition": "rowvector(-a+b-c,-b+c,a-c,-a+b)", "description": "", "name": "x"}, "w": {"templateType": "anything", "group": "Ungrouped variables", "definition": "rowvector(-al*a+(del-2*ga)*b-del*c,(del-al)*a-(del-ga)*b+del*c,(-del+2*al)*a+ga*b-del*c,(del-2*al)*a+(del-2*ga)*b)", "description": "", "name": "w"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "name": "c"}, "z": {"templateType": "anything", "group": "Ungrouped variables", "definition": "rowvector(-b-c,a+c,-a+b-c,a-b)", "description": "", "name": "z"}, "al": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-2,-1,1,2)", "description": "", "name": "al"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "t"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=2,1/(1-del),if(t=1,-al/(1-del),-(del-al-ga)/(1-del)))", "description": "", "name": "c2"}, "v4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=4,w,z)", "description": "", "name": "v4"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,1/(1-del),-al/(1-del))", "description": "", "name": "c1"}, "ga": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-2,-1,1,2)", "description": "", "name": "ga"}, "y": {"templateType": "anything", "group": "Ungrouped variables", "definition": "rowvector(b-c,a-b+c,-a-c,a+b)", "description": "", "name": "y"}, "v1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,w, x)", "description": "", "name": "v1"}, "c4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=4,1/(1-del),-ga/(1-del))", "description": "", "name": "c4"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=3,1/(1-del),if(t=4,-ga/(1-del),-(del-al-ga)/(1-del)))", "description": "", "name": "c3"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "name": "a"}, "del": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,-1,-3,-4)", "description": "", "name": "del"}, "v2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,x,t=2,w,y)", "description": "", "name": "v2"}, "v3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=3,w,t=4,z,y)", "description": "", "name": "v3"}}, "ungrouped_variables": ["a", "x", "c", "b", "ga", "al", "y", "v1", "v2", "v3", "v4", "t", "w", "c2", "c3", "del", "c1", "z", "c4"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Combine linearly dependent vectors to get zero", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "c1", "minValue": "c1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "c2", "minValue": "c2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "c3", "minValue": "c3", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "c4", "minValue": "c4", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "

Find $a,\\;b,\\;c$ and $d$

$a=$ [[0]]

$b=$ [[1]]

$c=$ [[2]]

$d=$ [[3]]

Input all values as exact decimals.

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You are given the following four vectors in $\\mathbb{R}^4$: \\[\\begin{align} \\textbf{v}_1&=\\var{v1}\\\\ \\textbf{v}_2&=\\var{v2}\\\\ \\textbf{v}_3&=\\var{v3}\\\\ \\textbf{v}_4&=\\var{v4}\\end{align}\\]

You are asked to show that the vectors $\\textbf{v}_1,\\;\\textbf{v}_2,\\;\\textbf{v}_3,\\;\\textbf{v}_4$ are linearly dependent.

Find real numbers $a,\\;b,\\;c$ and $d$ such that \\[a\\textbf{v}_1+b\\textbf{v}_2+c\\textbf{v}_3+d\\textbf{v}_4=\\textbf{0},\\;\\;\\; a+b+c+d=1\\]where $\\textbf{0}=\\var{rowvector(0,0,0,0)}$ is the zero vector.

", "tags": ["checked2015", "dependent vectors", "linear algebra", "linear combination of vectors", "linear dependence", "linear equations", "linearly dependent vectors", "MAS2223", "solving linear equations", "vectors"], "rulesets": {"std": ["all", "!collectNumbers", "!noleadingminus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

10/02/2013:

Finished first draft. Need to resolve the display of row vectors etc.

Display of linear equations difficult to format e.g. variables under one another as got to use \\simplify where there are randomised coefficients.

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Real numbers $a,\\;b,\\;c$ and $d$ are such that $a+b+c+d=1$ and for the given vectors $\\textbf{v}_1,\\;\\textbf{v}_2,\\;\\textbf{v}_3,\\;\\textbf{v}_4$ $a\\textbf{v}_1+b\\textbf{v}_2+c\\textbf{v}_3+d\\textbf{v}_4=\\textbf{0}$. Find $a,\\;b,\\;c,\\;d$.

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On putting $a=1-b-c-d$ we have

\\[(1-b-c-d)\\textbf{v}_1+b\\textbf{v}_2+c\\textbf{v}_3+d\\textbf{v}_4=0\\]

Hence on combining the vectors and equating the four components each to $0$ we have the four equations:

\\[\\begin{align} \\simplify[std]{{v1[0][0]} * (1 -b -c -d) + {v2[0][0]} * b + {v3[0][0]} * c + {v4[0][0]} * d }&= 0\\\\ \\simplify[std]{{v1[0][1]} * (1 -b -c -d) + {v2[0][1]} * b + {v3[0][1]} * c + {v4[0][1]} * d }&= 0\\\\ \\simplify[std]{{v1[0][2]} * (1 -b -c -d) + {v2[0][2]} * b + {v3[0][2]} * c + {v4[0][2]} * d }&= 0\\\\ \\simplify[std]{{v1[0][3]} * (1 -b -c -d) + {v2[0][3]} * b + {v3[0][3]} * c + {v4[0][3]} * d }&= 0\\end{align}\\]

On rearranging these equations in the unknowns $b,\\;c,\\;d$ we get:

\\[\\begin{align} \\simplify[std]{{v2[0][0] -v1[0][0]} * b + {v3[0][0] -v1[0][0]} * c + {v4[0][0] -v1[0][0]} * d} &= \\var{-v1[0][0]}\\\\ \\simplify[std]{{v2[0][1] -v1[0][1]} * b + {v3[0][1] -v1[0][1]} * c + {v4[0][1] -v1[0][1]} * d }&= \\var{-v1[0][1]}\\\\ \\simplify[std]{{v2[0][2] -v1[0][2]} * b + {v3[0][2] -v1[0][2]} * c + {v4[0][2] -v1[0][2]} * d }&= \\var{-v1[0][2]}\\\\ \\simplify[std]{{v2[0][3] -v1[0][3]} * b + {v3[0][3] -v1[0][3]} * c + {v4[0][3] -v1[0][3]} * d}&= \\var{-v1[0][3]}\\end{align}\\]

On solving these equations we see that although there are more equations than unknowns, the equations are consistent and they have the solution:

$b=\\var{c2}$, $c=\\var{c3}$, $d=\\var{c4}$ and hence $a=\\simplify[std]{1- {c2}-{c3}-{c4}= {c1}}$.

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