// Numbas version: exam_results_page_options
{"name": "Determine if vectors form a spanning set", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"contains": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(mm[0]=1, \"contains\", \"does not contain\")", "description": "", "name": "contains"}, "mm": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(u=3 or u=6 or u=7 or u=8 or u=9,[1,0],[0,1])", "description": "", "name": "mm"}, "t0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u<4,2,if(u<7,3,if(u<9,4,5)))", "description": "", "name": "t0"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "name": "b"}, "f1": {"templateType": "anything", 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variables", "definition": "if(mm[0]=1,\"is only\", \"are\")", "description": "", "name": "are"}, "al": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "al"}, "eg": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(mm[0]=1,'','This is one of the relations.')", "description": "", "name": "eg"}, "be1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1,-1)", "description": "", "name": "be1"}, "v4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=7 or u=8,p1,if(u=9,v,if(u=4 or u=1,p2,z)))", "description": "", "name": "v4"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "list(al*vector(x)+be*vector(y)+ga*vector(z))", "description": "", "name": "r"}, "ga": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(al*be=0, random(1,-1),0)", "description": "", "name": "ga"}, 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spanning set. \", \"There is one other simple relationship - you find this! So this is not a spanning set as it contains less than 4 linearly independent vectors.\")", "description": "", "name": "another"}, "v3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u>6,z,if(u>3,p1,y))", "description": "", "name": "v3"}}, "ungrouped_variables": ["f1", "f2", "f3", "f4", "another", "eg", "al", "ga1", "are", "ga", "ep", "es", "contains", "be1", "nt", "be", "v1", "v2", "v3", "v4", "v5", "b", "c5", "al1", "a", "c", "p1", "thismany", "mm", "t0", "q", "p2", "r", "u", "t", "v", "y", "x", "z"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Determine if vectors form a spanning set", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [0, 1], "marks": 0}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "mm", "marks": 0}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "mm", "marks": 0}], "type": "gapfill", "prompt": "<p>1. Is $\\{\\textbf{v}_1,\\;\\textbf{v}_2,\\;\\textbf{v}_3,\\;\\textbf{v}_4,\\;\\textbf{v}_5\\}$ a linearly independent set of vectors? &nbsp;[[0]]</p>\n      <p>&nbsp;</p>\n      <p>2. Do the above vectors form a spanning set of $\\mathbb{R}^4$? [[1]]</p>\n      <p>&nbsp;</p>\n      <p>3. Does the set&nbsp;&nbsp;$\\{\\textbf{v}_1,\\;\\textbf{v}_2,\\;\\textbf{v}_3,\\;\\textbf{v}_4,\\;\\textbf{v}_5\\}$&nbsp;contain a linearly independent subset which forms a basis of&nbsp;&nbsp;$\\mathbb{R}^4$? &nbsp;[[2]]</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Consider the following $5$ vectors in $\\mathbb{R^4}$ .</p>\n  <p>\\[\\begin{align} \\textbf{v}_1&amp;=\\var{rowvector(v1)}\\\\&nbsp;\\textbf{v}_2&amp;=\\var{rowvector(v2)}\\\\&nbsp;\\textbf{v}_3&amp;=\\var{rowvector(v3)}\\\\ \\textbf{v}_4&amp;=\\var{rowvector(v4)}\\\\&nbsp;\\textbf{v}_5&amp;=\\var{rowvector(v5)}\\end{align}\\]</p>\n  <p>&nbsp;Observe: In this set of vectors at least one and sometimes two of the vectors are a simple linear combination of one or two of the previous vectors in the list. There are no other linear relations between the vectors.</p>\n  <p>Here, if a vector is in the span of the vectors earlier in the list then it will satisfy a simple relation of the form &nbsp;$\\textbf{v}_i=a\\textbf{v}_j +b\\textbf{v}_k$ where &nbsp;$a$ can be $0,\\;1$ or $-2$, &nbsp;similarly for $b$ .</p>", "tags": ["basis", "checked2015", "euclidean space", "linear algebra", "linear combination", "linear dependence", "linear independence", "linear spaces", "MAS2223", "span", "spanning set", "vector spaces"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>12/02/2013:</strong></p>\n    <p>First draft finished.</p>\n    <p><strong>06/11/2013:</strong></p>\n    <p>Got rid of 2 cases which were incorrect (u=7,9). Tested and seems OK .</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given $5$ vectors in $\\mathbb{R^4}$ determine if a spanning set for $\\mathbb{R^4}$ or not by looking for any simple dependencies between the vectors.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>1. Not linearly independent as any set of more than $4$ vectors in $\\mathbb{R^4}$ is linearly dependent.</p>\n  <p>2. They are spanning if any vector in $\\mathbb{R^4}$ can be written as a linear combination of these vectors. This means that there must be $4$ linearly independent vectors in the list. If there are not then it is not spanning.</p>\n  <p>Given the hint in the question, if only one vector can be expressed as a linear combination of the others then there are $4$ linearly independent vectors.&nbsp;</p>\n  <p>However if two can be so expressed then there are only $3$ independent vectors and so cannot be spanning.</p>\n  <p>We note that for this example there {are} {thismany} vector{es} dependent on the others. {eg} \\[\\textbf{v}_{\\var{t0}} =\\simplify{ {f1} * v_1+ {f2} * v _2 + {f3} * v_3 + {f4} * v_4}.\\]</p>\n  <p>{another}</p>\n  <p>&nbsp;</p>\n  <p>3. This set {contains} a linearly independent subset of $4$ vectors as it is {nt} spanning.</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}