CoCalc -- 0009.tex

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\begin{exercise}{V2}{Linear combinations}{0009}
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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 \( \left[\begin{array}{c}
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1 \\
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3 \\
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6 \\
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-10
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\end{array}\right] \)is not a linear combination of the vectors \( \left[\begin{array}{c}
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-1 \\
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-1 \\
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4 \\
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2
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\end{array}\right] , \left[\begin{array}{c}
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0 \\
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-2 \\
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2 \\
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-1
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\end{array}\right] , \left[\begin{array}{c}
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3 \\
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2 \\
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4 \\
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2
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\end{array}\right] , \left[\begin{array}{c}
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-7 \\
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-9 \\
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0 \\
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-4
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\end{array}\right] , \text{ and } \left[\begin{array}{c}
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-6 \\
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2 \\
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-14 \\
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-1
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\end{array}\right] \). 
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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}\[\operatorname{RREF}  \left[\begin{array}{ccccc|c}
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-1 & 0 & 3 & -7 & -6 & 1 \\
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-1 & -2 & 2 & -9 & 2 & 3 \\
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4 & 2 & 4 & 0 & -14 & 6 \\
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2 & -1 & 2 & -4 & -1 & -10
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\end{array}\right] = \left[\begin{array}{ccccc|c}
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1 & 0 & 0 & 1 & 0 & 0 \\
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0 & 1 & 0 & 2 & -3 & 0 \\
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0 & 0 & 1 & -2 & -2 & 0 \\
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0 & 0 & 0 & 0 & 0 & 1
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\end{array}\right] \]
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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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-1 \\
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4 \\
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2
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\end{array}\right] + x_{2} \left[\begin{array}{c}
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0 \\
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-2 \\
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2 \\
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-1
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\end{array}\right] + x_{3} \left[\begin{array}{c}
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3 \\
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2 \\
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4 \\
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2
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\end{array}\right] + x_{4} \left[\begin{array}{c}
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-7 \\
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-9 \\
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0 \\
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-4
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\end{array}\right] + x_{5} \left[\begin{array}{c}
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-6 \\
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2 \\
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-14 \\
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-1
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\end{array}\right] = \left[\begin{array}{c}
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1 \\
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3 \\
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6 \\
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-10
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\end{array}\right] \)has no solutions.
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\item 
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\( \left[\begin{array}{c}
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1 \\
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3 \\
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6 \\
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-10
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\end{array}\right] \) is not a linear combination of the vectors \( \left[\begin{array}{c}
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-1 \\
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-1 \\
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4 \\
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2
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\end{array}\right] , \left[\begin{array}{c}
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0 \\
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-2 \\
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2 \\
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-1
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\end{array}\right] , \left[\begin{array}{c}
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3 \\
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2 \\
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4 \\
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2
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\end{array}\right] , \left[\begin{array}{c}
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-7 \\
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-9 \\
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0 \\
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-4
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\end{array}\right] , \text{ and } \left[\begin{array}{c}
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-6 \\
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2 \\
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-14 \\
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-1
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\end{array}\right] \). 
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\end{enumerate}
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    \end{exerciseAnswer}
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\end{exercise}
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