CoCalc -- 0010.tex

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\begin{exercise}{V2}{Linear combinations}{0010}
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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 \( \left[\begin{array}{c}
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-8 \\
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1 \\
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0 \\
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-8
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\end{array}\right] \)is a linear combination of the vectors \( \left[\begin{array}{c}
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1 \\
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-1 \\
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2 \\
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0
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\end{array}\right] , \left[\begin{array}{c}
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-4 \\
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-3 \\
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2 \\
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-2
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\end{array}\right] , \left[\begin{array}{c}
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2 \\
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-2 \\
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2 \\
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2
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\end{array}\right] , \left[\begin{array}{c}
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-3 \\
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0 \\
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-5 \\
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2
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\end{array}\right] , \text{ and } \left[\begin{array}{c}
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0 \\
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0 \\
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0 \\
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2
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\end{array}\right] \). 
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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}\[\operatorname{RREF}  \left[\begin{array}{ccccc|c}
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1 & -4 & 2 & -3 & 0 & -8 \\
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-1 & -3 & -2 & 0 & 0 & 1 \\
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2 & 2 & 2 & -5 & 0 & 0 \\
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0 & -2 & 2 & 2 & 2 & -8
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\end{array}\right] = \left[\begin{array}{ccccc|c}
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1 & 0 & 0 & 0 & -\frac{64}{3} & 2 \\
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0 & 1 & 0 & 0 & 2 & 1 \\
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0 & 0 & 1 & 0 & \frac{23}{3} & -3 \\
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0 & 0 & 0 & 1 & -\frac{14}{3} & 0
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\end{array}\right] \]
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\begin{enumerate}[(a)]
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\item  The vector equation \( x_{1} \left[\begin{array}{c}
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1 \\
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-1 \\
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2 \\
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0
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\end{array}\right] + x_{2} \left[\begin{array}{c}
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-4 \\
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-3 \\
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2 \\
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-2
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\end{array}\right] + x_{3} \left[\begin{array}{c}
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2 \\
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-2 \\
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2 \\
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2
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\end{array}\right] + x_{4} \left[\begin{array}{c}
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-3 \\
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0 \\
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-5 \\
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2
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\end{array}\right] + x_{5} \left[\begin{array}{c}
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0 \\
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0 \\
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0 \\
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2
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\end{array}\right] = \left[\begin{array}{c}
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-8 \\
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1 \\
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0 \\
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-8
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\end{array}\right] \)has a solution.
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\item 
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\( \left[\begin{array}{c}
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-8 \\
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1 \\
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0 \\
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-8
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\end{array}\right] \) is a linear combination of the vectors \( \left[\begin{array}{c}
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1 \\
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-1 \\
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2 \\
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0
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\end{array}\right] , \left[\begin{array}{c}
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-4 \\
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-3 \\
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2 \\
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-2
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\end{array}\right] , \left[\begin{array}{c}
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2 \\
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-2 \\
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2 \\
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2
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\end{array}\right] , \left[\begin{array}{c}
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-3 \\
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0 \\
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-5 \\
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2
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\end{array}\right] , \text{ and } \left[\begin{array}{c}
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0 \\
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0 \\
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0 \\
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2
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\end{array}\right] \). 
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\end{enumerate}
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    \end{exerciseAnswer}
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\end{exercise}
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