// Numbas version: exam_results_page_options {"name": "Simon's copy of Do three given vectors form a spanning set and are they linearly independent?", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [{"variables": ["v1", "v2", "v3", "tt"], "name": "Shown to the student"}, {"variables": ["cannot", "only", "independent"], "name": "Strings"}, {"variables": ["is_independent", "t", "rs", "r1", "r2", "r3", "w_x", "w_y"], "name": "Setup"}, {"variables": ["marking_matrix_independent"], "name": "Marking"}, {"variables": ["w", "x", "y", "z"], "name": "Base vectors"}, {"variables": ["p", "q", "coeff_v1", "coeff_v2", "coeff_v3"], "name": "Solution"}], "tags": [], "parts": [{"unitTests": [], "marks": 0, "prompt": "

1. Do these vectors form a spanning set for $\\mathbb{R^4}$? [[0]]

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2. Is $\\{\\mathbf{v}_1,\\;\\mathbf{v}_2,\\;\\mathbf{v}_3\\}$ a linearly independent set of vectors?[[1]]

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Do there exist integers $p$ and $q$ such that the equation $\\mathbf{v}_{\\var{tt[0]}}=p\\mathbf{v}_{\\var{tt[1]}}+q\\mathbf{v}_{\\var{tt[2]}}$ holds?

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If you cannot find such an expression, input 0 for both $p$ and $q$.

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$p=$ [[0]]

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$q=$ [[1]]

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You are given the following three vectors in $\\mathbb{R}^4$:

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\$\\mathbf{v}_1=\\var{v1}, \\quad \\mathbf{v}_2=\\var{v2}, \\quad \\mathbf{v}_3=\\var{v3}\$

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Multiple of y in w

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Linearly independent to y and z

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Marking matrix for the \"are these vectors linearly independent?\" question

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Linearly independent to x and z

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Amount of y to use to make up -w (the variable q asked for in part b)

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Index of vector to write as a sum of multiples of the others. 4 if the vectors are linearly independent (in which case, the student will be asked to write v2 as a sum of multiples of v1 and v3)

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Multiple of z-y in w

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Should v1,v2,v3 be linearly independent?

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Linearly independent to x and y

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The student is asked to write v_tt[0] as a sum of multiples of v_tt[1] and v_tt[2].

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tt[0] is 2 if t=4, otherwise it's t. tt[1] and tt[2] are the two remaining indices, in increasing order.

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Amount of x to use to make up -w

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A linear combination of x and y

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Three distinct random numbers

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Coefficient of v1 in the equation a*v1 + b*v2 + c*v3 = 0

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Coefficient of v3 in the equation a*v1 + b*v2 + c*v3 = 0

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Coefficient of v2 in the equation a*v1 + b*v2 + c*v3 = 0

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#### (a)

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1. These vectors cannot form a spanning set as you need at least 4 vectors to form a spanning set in $\\mathbb{R^4}$.

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2. $\\mathbf{v}_1, \\; \\mathbf{v}_2 , \\; \\mathbf{v}_3$ is a linearly independent set if the only solution for \$a \\mathbf{v}_1 + b \\mathbf{v}_2 + c \\mathbf{v}_3 = \\var{vector(0,0,0,0)} \\qquad \\textbf{(1)}\$ is $a=0$, $b=0$, $c=0$.

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If there is any other solution then the set is linearly dependent. Note that if there is one other solution then there will be an infinite number of such solutions.

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On writing out the components given by equation (1), we see that we get the following four equations:

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\\\begin{align} \\simplify[std]{{v1[0]} * a + {v2[0]} * b + {v3[0]} * c }&= 0\\\\ \\simplify[std]{{v1[1]} * a + {v2[1]} * b + {v3[1]} * c }&= 0\\\\ \\simplify[std]{{v1[2]} *a+ {v2[2]} * b + {v3[2]} * c }&= 0\\\\ \\simplify[std]{{v1[3]} *a+ {v2[3]} * b + {v3[3]} * c }&= 0\\end{align}\

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We see that on solving these equations that there is {only} the solution:

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\$a = \\var{coeff_v1}, \\; b=\\var{coeff_v2}, \\; c=\\var{coeff_v3}\$

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It follows that $\\{\\mathbf{v}_1, \\; \\mathbf{v}_2 , \\; \\mathbf{v}_3\\}$ is a linearly {independent} set of vectors.

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#### (b)

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Because the vectors $\\mathbf{v}_1$, $\\mathbf{v}_2$ and $\\mathbf{v}_3$ are linearly independent, input 0 for both $p$ and $q$.Rearrange equation (1) to give \$\\var{[a,b,c][tt[0]-1]}\\mathbf{v}_{\\var{tt[0]}} = -\\var{[a,b,c][tt[1]-1]}\\mathbf{v}_{\\var{tt[1]}} - \\var{[a,b,c][tt[2]-1]}\\mathbf{v}_{\\var{tt[2]}}\$ So, using the solution found above, $p=-\\var{[a,b,c][tt[1]-1]} = \\var{p}$ and $q =-\\var{[a,b,c][tt[2]-1]} = \\var{q}$.

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Given the following three vectors $\\textbf{v}_1,\\;\\textbf{v}_2,\\;\\textbf{v}_3$ Find out whether they are a linearly independent set are not. Also if linearly dependent find the relationship $\\textbf{v}_{r}=p\\textbf{v}_{s}+q\\textbf{v}_{t}$ for suitable $r,\\;s,\\;t$ and integers $p,\\;q$.