0009.qti

<item ident="G4-0009" title="G4 | Eigenvectors | ver. 0009">
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  <presentation>
    <material>
      <mattextxml>
        <div class="exercise-statement">
          <p>
            <strong>G4.</strong>
          </p>
          <p>Explain how to find a basis for the eigenspace associated to the eigenvalue <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3" alt="3" title="3" data-latex="3"/> in the matrix <p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left[\begin{array}{cccc} 1 &amp; -4 &amp; 4 &amp; -8 \\ -2 &amp; -1 &amp; 4 &amp; -8 \\ 0 &amp; 0 &amp; 3 &amp; 0 \\ 1 &amp; 2 &amp; -2 &amp; 7 \end{array}\right]" alt="\left[\begin{array}{cccc} 1 &amp; -4 &amp; 4 &amp; -8 \\ -2 &amp; -1 &amp; 4 &amp; -8 \\ 0 &amp; 0 &amp; 3 &amp; 0 \\ 1 &amp; 2 &amp; -2 &amp; 7 \end{array}\right]" title="\left[\begin{array}{cccc} 1 &amp; -4 &amp; 4 &amp; -8 \\ -2 &amp; -1 &amp; 4 &amp; -8 \\ 0 &amp; 0 &amp; 3 &amp; 0 \\ 1 &amp; 2 &amp; -2 &amp; 7 \end{array}\right]" data-latex="\left[\begin{array}{cccc} 1 &amp; -4 &amp; 4 &amp; -8 \\ -2 &amp; -1 &amp; 4 &amp; -8 \\ 0 &amp; 0 &amp; 3 &amp; 0 \\ 1 &amp; 2 &amp; -2 &amp; 7 \end{array}\right]"/></p></p>
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      <mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;G4.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find a basis for the eigenspace associated to the eigenvalue &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3" alt="3" title="3" data-latex="3"&gt; in the matrix &lt;/p&gt;
&lt;p style="text-align:center;"&gt;&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D%201%20&amp;amp;%20-4%20&amp;amp;%204%20&amp;amp;%20-8%20%5C%5C%20-2%20&amp;amp;%20-1%20&amp;amp;%204%20&amp;amp;%20-8%20%5C%5C%200%20&amp;amp;%200%20&amp;amp;%203%20&amp;amp;%200%20%5C%5C%201%20&amp;amp;%202%20&amp;amp;%20-2%20&amp;amp;%207%20%5Cend%7Barray%7D%5Cright%5D" alt="\left[\begin{array}{cccc} 1 &amp;amp; -4 &amp;amp; 4 &amp;amp; -8 \\ -2 &amp;amp; -1 &amp;amp; 4 &amp;amp; -8 \\ 0 &amp;amp; 0 &amp;amp; 3 &amp;amp; 0 \\ 1 &amp;amp; 2 &amp;amp; -2 &amp;amp; 7 \end{array}\right]" title="\left[\begin{array}{cccc} 1 &amp;amp; -4 &amp;amp; 4 &amp;amp; -8 \\ -2 &amp;amp; -1 &amp;amp; 4 &amp;amp; -8 \\ 0 &amp;amp; 0 &amp;amp; 3 &amp;amp; 0 \\ 1 &amp;amp; 2 &amp;amp; -2 &amp;amp; 7 \end{array}\right]" data-latex="\left[\begin{array}{cccc} 1 &amp;amp; -4 &amp;amp; 4 &amp;amp; -8 \\ -2 &amp;amp; -1 &amp;amp; 4 &amp;amp; -8 \\ 0 &amp;amp; 0 &amp;amp; 3 &amp;amp; 0 \\ 1 &amp;amp; 2 &amp;amp; -2 &amp;amp; 7 \end{array}\right]"&gt;&lt;/p&gt;
&lt;/div&gt;

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    <response_str ident="response1" rcardinality="Single">
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        <response_label ident="answer1" rshuffle="No"/>
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  </presentation>
  <itemfeedback ident="general_fb">
    <flow_mat>
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          <div class="exercise-answer">
            <h4>Partial Answer:</h4>
            <p>
              <p style="text-align:center;">
                <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\operatorname{RREF} \left[\begin{array}{cccc} -2 &amp; -4 &amp; 4 &amp; -8 \\ -2 &amp; -4 &amp; 4 &amp; -8 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 1 &amp; 2 &amp; -2 &amp; 4 \end{array}\right] = \left[\begin{array}{cccc} 1 &amp; 2 &amp; -2 &amp; 4 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \end{array}\right]" alt="\operatorname{RREF} \left[\begin{array}{cccc} -2 &amp; -4 &amp; 4 &amp; -8 \\ -2 &amp; -4 &amp; 4 &amp; -8 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 1 &amp; 2 &amp; -2 &amp; 4 \end{array}\right] = \left[\begin{array}{cccc} 1 &amp; 2 &amp; -2 &amp; 4 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \end{array}\right]" title="\operatorname{RREF} \left[\begin{array}{cccc} -2 &amp; -4 &amp; 4 &amp; -8 \\ -2 &amp; -4 &amp; 4 &amp; -8 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 1 &amp; 2 &amp; -2 &amp; 4 \end{array}\right] = \left[\begin{array}{cccc} 1 &amp; 2 &amp; -2 &amp; 4 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \end{array}\right]" data-latex="\operatorname{RREF} \left[\begin{array}{cccc} -2 &amp; -4 &amp; 4 &amp; -8 \\ -2 &amp; -4 &amp; 4 &amp; -8 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 1 &amp; 2 &amp; -2 &amp; 4 \end{array}\right] = \left[\begin{array}{cccc} 1 &amp; 2 &amp; -2 &amp; 4 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \end{array}\right]"/>
              </p>
            </p>
            <p>A basis of the eigenspace is <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}" alt="\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}" title="\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}" data-latex="\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}"/>.</p>
          </div>
        </mattextxml>
        <mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt;
    &lt;/p&gt;
&lt;p style="text-align:center;"&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Coperatorname%7BRREF%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D%20-2%20&amp;amp;%20-4%20&amp;amp;%204%20&amp;amp;%20-8%20%5C%5C%20-2%20&amp;amp;%20-4%20&amp;amp;%204%20&amp;amp;%20-8%20%5C%5C%200%20&amp;amp;%200%20&amp;amp;%200%20&amp;amp;%200%20%5C%5C%201%20&amp;amp;%202%20&amp;amp;%20-2%20&amp;amp;%204%20%5Cend%7Barray%7D%5Cright%5D%20=%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D%201%20&amp;amp;%202%20&amp;amp;%20-2%20&amp;amp;%204%20%5C%5C%200%20&amp;amp;%200%20&amp;amp;%200%20&amp;amp;%200%20%5C%5C%200%20&amp;amp;%200%20&amp;amp;%200%20&amp;amp;%200%20%5C%5C%200%20&amp;amp;%200%20&amp;amp;%200%20&amp;amp;%200%20%5Cend%7Barray%7D%5Cright%5D" alt="\operatorname{RREF} \left[\begin{array}{cccc} -2 &amp;amp; -4 &amp;amp; 4 &amp;amp; -8 \\ -2 &amp;amp; -4 &amp;amp; 4 &amp;amp; -8 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \\ 1 &amp;amp; 2 &amp;amp; -2 &amp;amp; 4 \end{array}\right] = \left[\begin{array}{cccc} 1 &amp;amp; 2 &amp;amp; -2 &amp;amp; 4 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \end{array}\right]" title="\operatorname{RREF} \left[\begin{array}{cccc} -2 &amp;amp; -4 &amp;amp; 4 &amp;amp; -8 \\ -2 &amp;amp; -4 &amp;amp; 4 &amp;amp; -8 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \\ 1 &amp;amp; 2 &amp;amp; -2 &amp;amp; 4 \end{array}\right] = \left[\begin{array}{cccc} 1 &amp;amp; 2 &amp;amp; -2 &amp;amp; 4 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \end{array}\right]" data-latex="\operatorname{RREF} \left[\begin{array}{cccc} -2 &amp;amp; -4 &amp;amp; 4 &amp;amp; -8 \\ -2 &amp;amp; -4 &amp;amp; 4 &amp;amp; -8 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \\ 1 &amp;amp; 2 &amp;amp; -2 &amp;amp; 4 \end{array}\right] = \left[\begin{array}{cccc} 1 &amp;amp; 2 &amp;amp; -2 &amp;amp; 4 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 0 &amp;amp; 0 \end{array}\right]"&gt;
    &lt;/p&gt;
  
  &lt;p&gt;A basis of the eigenspace is &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft%5C%7B%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%20-2%20%5C%5C%201%20%5C%5C%200%20%5C%5C%200%20%5Cend%7Barray%7D%5Cright%5D%20,%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%202%20%5C%5C%200%20%5C%5C%201%20%5C%5C%200%20%5Cend%7Barray%7D%5Cright%5D%20,%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%20-4%20%5C%5C%200%20%5C%5C%200%20%5C%5C%201%20%5Cend%7Barray%7D%5Cright%5D%20%5Cright%5C%7D" alt="\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}" title="\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}" data-latex="\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}"&gt;.&lt;/p&gt;
&lt;/div&gt;

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