// Numbas version: finer_feedback_settings
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"extensions": [], "advice": "<p>Clearly $\\textbf{v}_1$ is always in the required basis as it is non-zero.</p>\n    <p>$\\textbf{v}_2$ is {nt2} in the required basis as it is {ont2} a multiple of $\\textbf{v}_1$.</p>\n    <p>$\\textbf{v}_3$ is {nt3} in the required basis as it is {ont3} a linear &nbsp;combination of $\\textbf{v}_1$ and $\\textbf{v}_2$.</p>\n    <p>$\\textbf{v}_4$ is {nt4} in the required basis as it is {ont4} a linear &nbsp;combination of previous vectors.</p>\n    <p>{message4}</p>\n    <p>$\\textbf{v}_5$ is {nt5} in the required basis as it is {ont5} a linear &nbsp;combination of previous vectors.</p>\n    <p>{message5}</p>\n    <p>$\\textbf{v}_6$ is {nt6} in the required basis as it is {ont6} a linear &nbsp;combination of previous vectors.</p>\n    <p>&nbsp;</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "<p>Given $6$ vectors in $\\mathbb{R^4}$ and given that they span $\\mathbb{R^4}$ find a &nbsp;basis.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "name": "Michael's copy of Find a basis given a spanning set of vectors", "ungrouped_variables": ["mm5", "mm4", "nt3", "mm6", "nt5", "al", "ga1", "mm2", "ga", "ep", "ont3", "ont2", "ont6", "ont5", "ont4", "message5", "message4", "be1", "tmm", "be", "nt2", "nt4", "v1", "v2", "v3", "v4", "mm3", "v6", "nt6", "b", "c5", "al1", "a", "c", "p1", "mm", "q", "p2", "r", "u", "t", "v", "y", "x", "z", "v5"], "statement": "<p>Consider the following $6$ vectors in $\\mathbb{R^4}$ . .&nbsp;</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)}\\\\ \\textbf{v}_6&amp;=\\var{rowvector(v6)}\\end{align}\\]&nbsp;</p>\n<p>You are given that this set of vectors is a spanning set for &nbsp;$\\mathbb{R^4}$</p>", "parts": [{"showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "marks": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "sortAnswers": false, "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "gaps": [{"showCorrectAnswer": true, "answers": ["Yes", "No"], "type": "m_n_x", "unitTests": [], "scripts": {}, "matrix": "mm", "warningType": "none", "extendBaseMarkingAlgorithm": true, "shuffleAnswers": false, "marks": 0, "minAnswers": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "minMarks": 0, "maxAnswers": 0, "displayType": "radiogroup", "layout": {"expression": "", "type": "all"}, "choices": ["$\\textbf{v}_1$", "$\\textbf{v}_2$", "$\\textbf{v}_3$", "$\\textbf{v}_4$", "$\\textbf{v}_5$", "$\\textbf{v}_6$"], "variableReplacementStrategy": "originalfirst", "shuffleChoices": false, "maxMarks": 0, "showFeedbackIcon": true}], "prompt": "<p>Your task is to find a basis for $\\mathbb{R^4}$ by finding a linearly independent subset of these vectors.</p>\n      <p>Start from $\\textbf{v}_1$ and work through each vector in turn.</p>\n      <p>Determine if a vector is a linear combination of the previous vectors in the list.</p>\n      <p>If it is not such a linear combination then include it in the basis by choosing Yes, otherwise choose No.</p>\n      <p>Note that if a vector $\\textbf{v}_i$ for $i=2,\\ldots 5$ is a linear combination of the previous vectors 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 $a$ can be $0,\\;1$ &nbsp;or $-1$ &nbsp;similarly for $b$.</p>\n      <p>When you get to $\\textbf{v}_6$ it will be obvious if it is in the spanning set or not. (why?)</p>\n      <p>[[0]] &nbsp;</p>\n      <p>&nbsp;</p>"}], "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Michael Hoffmann", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1625/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Michael Hoffmann", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1625/"}]}