CoCalc -- 0010.ptx

banks / tbil-la / __build__ / V2 / 0010.ptx
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ubuntu2004
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<exercise checkit-seed="0010" 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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-8 \\
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1 \\
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0 \\
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-8
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\end{array}\right] </m>is
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                a linear combination of the vectors <m> \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] </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}{ccccc|c}
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1 &amp; -4 &amp; 2 &amp; -3 &amp; 0 &amp; -8 \\
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-1 &amp; -3 &amp; -2 &amp; 0 &amp; 0 &amp; 1 \\
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2 &amp; 2 &amp; 2 &amp; -5 &amp; 0 &amp; 0 \\
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0 &amp; -2 &amp; 2 &amp; 2 &amp; 2 &amp; -8
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\end{array}\right] = \left[\begin{array}{ccccc|c}
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1 &amp; 0 &amp; 0 &amp; 0 &amp; -\frac{64}{3} &amp; 2 \\
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0 &amp; 1 &amp; 0 &amp; 0 &amp; 2 &amp; 1 \\
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0 &amp; 0 &amp; 1 &amp; 0 &amp; \frac{23}{3} &amp; -3 \\
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0 &amp; 0 &amp; 0 &amp; 1 &amp; -\frac{14}{3} &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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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] </m>has a solution.</li>
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      <li>
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        <p><m> \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] </m> is 
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                 a linear combination of the vectors <m> \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] </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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