Clearly $\\textbf{v}_1$ is always in the required basis as it is non-zero.
\n$\\textbf{v}_2$ is {nt2} in the required basis as it is {ont2} a multiple of $\\textbf{v}_1$.
\n$\\textbf{v}_3$ is {nt3} in the required basis as it is {ont3} a linear combination of $\\textbf{v}_1$ and $\\textbf{v}_2$.
\n$\\textbf{v}_4$ is {nt4} in the required basis as it is {ont4} a linear combination of previous vectors.
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\n$\\textbf{v}_5$ is {nt5} in the required basis as it is {ont5} a linear combination of previous vectors.
\n{message5}
\n$\\textbf{v}_6$ is {nt6} in the required basis as it is {ont6} a linear combination of previous vectors.
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Your task is to find a basis for $\\mathbb{R^4}$ by finding a linearly independent subset of these vectors.
\nStart from $\\textbf{v}_1$ and work through each vector in turn.
\nDetermine if a vector is a linear combination of the previous vectors in the list.
\nIf it is not such a linear combination then include it in the basis by choosing Yes, otherwise choose No.
\nNote 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 $b$.
\nWhen you get to $\\textbf{v}_6$ it will be obvious if it is in the spanning set or not. (why?)
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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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can answer no to the next two.\",\" \"))", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "message4"}, "v4": {"definition": "if(u=7 or u=8,p1,if(u=9,v,if(u=4 or u=1,p2,z)))", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "v4"}, "mm5": {"definition": "if(u=2 or u=5 or u=7 or u=9,[0,1], [1,0])", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "mm5"}, "ont2": {"definition": "if(mm2[1]=0.5,\"\", \"not\")", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "ont2"}, "v2": {"definition": "if(u>3,y,p1)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "v2"}, "mm6": {"definition": "if(u=3 or u=6 or u=8 or u=9,[0,1], [1,0])", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "mm6"}, "mm2": {"definition": "switch(u>3,[0.5,0],[0,0.5])", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "mm2"}, "ga": {"definition": "if(al*be=0, random(1,-1),0)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "ga"}, "al": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "al"}, "al1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "al1"}, "v5": {"definition": "if(u=1 or u=4,z,if(u =2 or u=5 or u=7,p2, if(u=9,p1,v)))", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "v5"}}, "name": "John's copy of Find a basis given a spanning set of vectors", "statement": "Consider the following set of $6$ vectors in $\\mathbb{R^4}$ . .
\n\\[\\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}\\]
\nYou are given that this set of vectors is a spanning set for $\\mathbb{R^4}$
", "type": "question", "contributors": [{"name": "Mohamed Mehbali", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1764/"}, {"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}]}]}], "contributors": [{"name": "Mohamed Mehbali", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1764/"}, {"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}]}