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variables", "name": "mm5"}}, "functions": {}, "extensions": [], "advice": "

Clearly $\\textbf{v}_1$ is always in the required basis as it is non-zero.

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$\\textbf{v}_2$ is {nt2} in the required basis as it is {ont2} a multiple of $\\textbf{v}_1$.

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$\\textbf{v}_3$ is {nt3} in the required basis as it is {ont3} a linear  combination of $\\textbf{v}_1$ and $\\textbf{v}_2$.

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$\\textbf{v}_4$ is {nt4} in the required basis as it is {ont4} a linear  combination of previous vectors.

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{message4}

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$\\textbf{v}_5$ is {nt5} in the required basis as it is {ont5} a linear  combination of previous vectors.

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{message5}

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$\\textbf{v}_6$ is {nt6} in the required basis as it is {ont6} a linear  combination of previous vectors.

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Given $6$ vectors in $\\mathbb{R^4}$ and given that they span $\\mathbb{R^4}$ find a  basis.

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Consider the following $6$ vectors in $\\mathbb{R^4}$ . . 

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\\[\\begin{align} \\textbf{v}_1&=\\var{rowvector(v1)}\\\\ \\textbf{v}_2&=\\var{rowvector(v2)}\\\\ \\textbf{v}_3&=\\var{rowvector(v3)}\\\\ \\textbf{v}_4&=\\var{rowvector(v4)}\\\\ \\textbf{v}_5&=\\var{rowvector(v5)}\\\\ \\textbf{v}_6&=\\var{rowvector(v6)}\\end{align}\\] 

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You are given that this set of vectors is a spanning set for  $\\mathbb{R^4}$

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Your task is to find a basis for $\\mathbb{R^4}$ by finding a linearly independent subset of these vectors.

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Start from $\\textbf{v}_1$ and work through each vector in turn.

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Determine if a vector is a linear combination of the previous vectors in the list.

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If it is not such a linear combination then include it in the basis by choosing Yes, otherwise choose No.

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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  $ \\textbf{v}_i=a\\textbf{v}_j +b\\textbf{v}_k$ where $a$ can be $0,\\;1$  or $-1$  similarly for $b$.

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When you get to $\\textbf{v}_6$ it will be obvious if it is in the spanning set or not. (why?)

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[[0]]  

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"}], "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/"}]}