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 $\\mathbb{R^4}$
", "functions": {}, "advice": "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.
\n{message4}
\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?)
\n[[0]]
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"templateType": "anything", "name": "nt5"}, "ga": {"definition": "if(al*be=0, random(1,-1),0)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "ga"}, "v": {"definition": "[-b+c,a-c,-a+b,a-b+c] ", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "v"}, "tmm": {"definition": "list(transpose(matrix(mm)))", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "tmm"}, "ont2": {"definition": "if(mm2[1]=0.5,\"\", \"not\")", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "ont2"}, "a": {"definition": "random(-3..3 except 0)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "a"}, "p1": {"definition": "if(u<4,list(al*vector(x)),if(u<7,q,if(u<9,r,t)))", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "p1"}, "v3": {"definition": "if(u>6,z,if(u>3,p1,y))", "description": "", 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"mm2"}, "v4": {"definition": "if(u=7 or u=8,p1,if(u=9,v,if(u=4 or u=1,p2,z)))", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "v4"}, "ont4": {"definition": "if(mm4[1]=1,\"\", \"not\")", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "ont4"}, "v6": {"definition": "if(u>7 or u=3 or u=6, p2, v)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "v6"}, "b": {"definition": "random(-3..3 except 0)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "b"}, "r": {"definition": "list(al*vector(x)+be*vector(y)+ga*vector(z))", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "r"}, "mm4": {"definition": "if(u=2 or u=3 or u=5 or u=6 or u=9,[1,0], [0,1])", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "mm4"}, "v1": {"definition": "x", "description": "", "group": 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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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