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<exercise checkit-seed="0006" checkit-slug="V2" checkit-title="Linear combinations">

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  <statement>

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    <p>Consider the following statement.</p>

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    <ul>

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      <li>

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            The vector <m> \left[\begin{array}{c}

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-1 \\

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-9 \\

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6 \\

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9

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\end{array}\right] </m>is not

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                a linear combination of the vectors <m> \left[\begin{array}{c}

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0 \\

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-1 \\

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1 \\

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0

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

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-2 \\

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-3 \\

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-1 \\

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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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-1 \\

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-3 \\

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2

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

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-4 \\

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-3 \\

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-5 \\

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4

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\end{array}\right] </m>.

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            </li>

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    </ul>

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    <ol>

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      <li> Write an equivalent statement using a vector equation.</li>

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      <li> Explain why your statement is true or false.</li>

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    </ol>

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  </statement>

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  <answer>

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    <me>\operatorname{RREF}  \left[\begin{array}{cccc|c}

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0 &amp; -2 &amp; -2 &amp; -4 &amp; -1 \\

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-1 &amp; -3 &amp; -1 &amp; -3 &amp; -9 \\

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1 &amp; -1 &amp; -3 &amp; -5 &amp; 6 \\

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0 &amp; 2 &amp; 2 &amp; 4 &amp; 9

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

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1 &amp; 0 &amp; -2 &amp; -3 &amp; 0 \\

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0 &amp; 1 &amp; 1 &amp; 2 &amp; 0 \\

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0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 \\

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0 &amp; 0 &amp; 0 &amp; 0 &amp; 0

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\end{array}\right] </me>

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    <ol>

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      <li> 

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                The vector equation <m> x_{1} \left[\begin{array}{c}

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0 \\

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-1 \\

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1 \\

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0

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

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-2 \\

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-3 \\

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-1 \\

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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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-1 \\

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-3 \\

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2

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

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-4 \\

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-3 \\

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-5 \\

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4

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

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-1 \\

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-9 \\

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6 \\

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9

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\end{array}\right] </m>has no solutions.</li>

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      <li>

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        <p><m> \left[\begin{array}{c}

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-1 \\

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-9 \\

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6 \\

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9

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\end{array}\right] </m> is not 

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                 a linear combination of the vectors <m> \left[\begin{array}{c}

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0 \\

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-1 \\

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1 \\

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0

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

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-2 \\

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-3 \\

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-1 \\

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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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-1 \\

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-3 \\

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2

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

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-4 \\

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-3 \\

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-5 \\

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4

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\end{array}\right] </m>.

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            </p>

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      </li>

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    </ol>

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  </answer>

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</exercise>

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