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ubuntu2004
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\begin{exercise}{V2}{Linear combinations}{0008}

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\begin{exerciseStatement}

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Consider the following statement.

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\begin{itemize}

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\item  The vector \( \left[\begin{array}{c}

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7 \\

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2 \\

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-5 \\

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-3

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\end{array}\right] \)is a linear combination of the vectors \( \left[\begin{array}{c}

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1 \\

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-2 \\

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1 \\

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-1

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\end{array}\right] , \left[\begin{array}{c}

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2 \\

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1 \\

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-4 \\

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2

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\end{array}\right] , \left[\begin{array}{c}

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2 \\

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2 \\

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-5 \\

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-2

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\end{array}\right] , \left[\begin{array}{c}

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-12 \\

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-4 \\

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21 \\

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4

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\end{array}\right] , \text{ and } \left[\begin{array}{c}

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0 \\

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7 \\

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-8 \\

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-4

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\end{array}\right] \). 

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\end{itemize}

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\begin{enumerate}[(a)]

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\item  Write an equivalent statement using a vector equation.

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\item  Explain why your statement is true or false.

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\end{enumerate}

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    \end{exerciseStatement}

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\begin{exerciseAnswer}\[\operatorname{RREF}  \left[\begin{array}{ccccc|c}

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1 & 2 & 2 & -12 & 0 & 7 \\

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-2 & 1 & 2 & -4 & 7 & 2 \\

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1 & -4 & -5 & 21 & -8 & -5 \\

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-1 & 2 & -2 & 4 & -4 & -3

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\end{array}\right] = \left[\begin{array}{ccccc|c}

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1 & 0 & 0 & -2 & -2 & 0 \\

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0 & 1 & 0 & -2 & -1 & 0 \\

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0 & 0 & 1 & -3 & 2 & 0 \\

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0 & 0 & 0 & 0 & 0 & 1

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\end{array}\right] \]

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\begin{enumerate}[(a)]

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\item  The vector equation \( x_{1} \left[\begin{array}{c}

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1 \\

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-2 \\

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1 \\

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-1

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\end{array}\right] + x_{2} \left[\begin{array}{c}

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2 \\

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1 \\

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-4 \\

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2

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\end{array}\right] + x_{3} \left[\begin{array}{c}

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2 \\

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2 \\

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-5 \\

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-2

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\end{array}\right] + x_{4} \left[\begin{array}{c}

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-12 \\

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-4 \\

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21 \\

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4

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\end{array}\right] + x_{5} \left[\begin{array}{c}

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0 \\

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7 \\

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-8 \\

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-4

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\end{array}\right] = \left[\begin{array}{c}

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7 \\

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2 \\

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-5 \\

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-3

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\end{array}\right] \)has a solution.

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\item 

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\( \left[\begin{array}{c}

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7 \\

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2 \\

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-5 \\

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-3

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\end{array}\right] \) is not a linear combination of the vectors \( \left[\begin{array}{c}

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1 \\

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-2 \\

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1 \\

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-1

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\end{array}\right] , \left[\begin{array}{c}

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2 \\

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1 \\

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-4 \\

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2

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\end{array}\right] , \left[\begin{array}{c}

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2 \\

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2 \\

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-5 \\

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-2

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\end{array}\right] , \left[\begin{array}{c}

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-12 \\

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-4 \\

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21 \\

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4

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\end{array}\right] , \text{ and } \left[\begin{array}{c}

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0 \\

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7 \\

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-8 \\

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-4

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\end{array}\right] \). 

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\end{enumerate}

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    \end{exerciseAnswer}

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\end{exercise}

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