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1. Is $\\{\\textbf{v}_1,\\;\\textbf{v}_2,\\;\\textbf{v}_3,\\;\\textbf{v}_4,\\;\\textbf{v}_5\\}$ a linearly independent set of vectors?  [[0]]

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2. Do the above vectors form a spanning set of $\\mathbb{R}^4$? [[1]]

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3. Does the set  $\\{\\textbf{v}_1,\\;\\textbf{v}_2,\\;\\textbf{v}_3,\\;\\textbf{v}_4,\\;\\textbf{v}_5\\}$ contain a linearly independent subset which forms a basis of  $\\mathbb{R}^4$?  [[2]]

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Consider the following $5$ 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)}\\end{align}\

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(a)

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Not linearly independent as any set of more than $4$ vectors in $\\mathbb{R^4}$ is linearly dependent.

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(b)

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Note that  \${v}_{\\var{t0}} =\\simplify{ {f1} * v_1+ {f2} * v_2 + {f3} * v_3 + {f4} * v_4}.\$

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Note also {test}

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Because 2 of our 5 vectors can be written as linear combinations of the other 3, there are not 4 linearly independent vectors in the list. Hence not all vectors in $\\mathbb{R^4}$ can be written as a linear combination of these vectors, so they are not spanning.

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(c)

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This set {contains} a linearly independent subset of $4$ vectors as it is {nt} spanning.

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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.

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