// Numbas version: finer_feedback_settings
{"name": "Ugur's copy of 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": [{"name": "Ugur's copy of Determine if vectors form a spanning set", "tags": ["basis", "checked2015", "euclidean space", "linear algebra", "linear combination", "linear dependence", "linear independence", "linear spaces", "span", "spanning set", "vector spaces"], "metadata": {"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>", "licence": "Creative Commons Attribution 4.0 International"}, "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;</p>", "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>3. This set {contains} a linearly independent subset of $4$ vectors as it is {nt} spanning.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"contains": {"name": "contains", "group": "Ungrouped variables", "definition": "if(mm[0]=1, \"contains\", \"does not contain\")", "description": "", "templateType": "anything", "can_override": false}, "mm": {"name": "mm", "group": "Ungrouped variables", "definition": "switch(u=3 or u=6 or u=7 or u=8 or u=9,[1,0],[0,1])", "description": "", "templateType": "anything", "can_override": false}, "t0": {"name": "t0", "group": "Ungrouped variables", "definition": "if(u<4,2,if(u<7,3,if(u<9,4,5)))", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "templateType": "anything", "can_override": false}, "f1": {"name": "f1", "group": 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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. 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