0005.ptx

<exercise checkit-seed="0005" checkit-slug="V2" checkit-title="Linear combinations">
  <statement>
    <p>Consider the following statement.</p>
    <ul>
      <li>
            The vector <m> \left[\begin{array}{c}
-15 \\
8 \\
-4 \\
2
\end{array}\right] </m>is
                a linear combination of the vectors <m> \left[\begin{array}{c}
3 \\
-2 \\
-1 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-2 \\
1 \\
-3 \\
-1
\end{array}\right] , \text{ and } \left[\begin{array}{c}
-2 \\
0 \\
-1 \\
1
\end{array}\right] </m>.
            </li>
    </ul>
    <ol>
      <li> Write an equivalent statement using a vector equation.</li>
      <li> Explain why your statement is true or false.</li>
    </ol>
  </statement>
  <answer>
    <me>\operatorname{RREF}  \left[\begin{array}{ccc|c}
3 &amp; -2 &amp; -2 &amp; -15 \\
-2 &amp; 1 &amp; 0 &amp; 8 \\
-1 &amp; -3 &amp; -1 &amp; -4 \\
-1 &amp; -1 &amp; 1 &amp; 2
\end{array}\right] = \left[\begin{array}{ccc|c}
1 &amp; 0 &amp; 0 &amp; -3 \\
0 &amp; 1 &amp; 0 &amp; 2 \\
0 &amp; 0 &amp; 1 &amp; 1 \\
0 &amp; 0 &amp; 0 &amp; 0
\end{array}\right] </me>
    <ol>
      <li> 
                The vector equation <m> x_{1} \left[\begin{array}{c}
3 \\
-2 \\
-1 \\
-1
\end{array}\right] + x_{2} \left[\begin{array}{c}
-2 \\
1 \\
-3 \\
-1
\end{array}\right] + x_{3} \left[\begin{array}{c}
-2 \\
0 \\
-1 \\
1
\end{array}\right] = \left[\begin{array}{c}
-15 \\
8 \\
-4 \\
2
\end{array}\right] </m>has a solution.</li>
      <li>
        <p><m> \left[\begin{array}{c}
-15 \\
8 \\
-4 \\
2
\end{array}\right] </m> is 
                 a linear combination of the vectors <m> \left[\begin{array}{c}
3 \\
-2 \\
-1 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-2 \\
1 \\
-3 \\
-1
\end{array}\right] , \text{ and } \left[\begin{array}{c}
-2 \\
0 \\
-1 \\
1
\end{array}\right] </m>.
            </p>
      </li>
    </ol>
  </answer>
</exercise>