0008.qti

<item ident="A3-0008" title="A3 | Image and kernel | ver. 0008">
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        <fieldentry>essay_question</fieldentry>
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  <presentation>
    <material>
      <mattextxml>
        <div class="exercise-statement"><p><strong>A3.</strong></p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T:\mathbb{R}^ 3 \to \mathbb{R}^ 4" alt="T:\mathbb{R}^ 3 \to \mathbb{R}^ 4" title="T:\mathbb{R}^ 3 \to \mathbb{R}^ 4" data-latex="T:\mathbb{R}^ 3 \to \mathbb{R}^ 4"/> be the linear transformation given by <p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T\left( \left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right) = \left[\begin{array}{c} -2 \, x_{1} + 4 \, x_{2} + 2 \, x_{3} \\ -x_{1} - 3 \, x_{2} - 4 \, x_{3} \\ -x_{1} + x_{2} \\ x_{1} + x_{3} \end{array}\right] ." alt="T\left( \left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right) = \left[\begin{array}{c} -2 \, x_{1} + 4 \, x_{2} + 2 \, x_{3} \\ -x_{1} - 3 \, x_{2} - 4 \, x_{3} \\ -x_{1} + x_{2} \\ x_{1} + x_{3} \end{array}\right] ." title="T\left( \left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right) = \left[\begin{array}{c} -2 \, x_{1} + 4 \, x_{2} + 2 \, x_{3} \\ -x_{1} - 3 \, x_{2} - 4 \, x_{3} \\ -x_{1} + x_{2} \\ x_{1} + x_{3} \end{array}\right] ." data-latex="T\left( \left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right) = \left[\begin{array}{c} -2 \, x_{1} + 4 \, x_{2} + 2 \, x_{3} \\ -x_{1} - 3 \, x_{2} - 4 \, x_{3} \\ -x_{1} + x_{2} \\ x_{1} + x_{3} \end{array}\right] ."/></p><ol type="a"><li>Explain how to find the image of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"/> and the kernel of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"/>.</li><li>Explain how to find a basis of the image of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"/> and a basis of the kernel of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"/>.</li><li>Explain how to find the rank and nullity of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"/>, and why the rank-nullity theorem holds for <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"/>.</li></ol></div>
      </mattextxml>
      <mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
&lt;p&gt;&lt;strong&gt;A3.&lt;/strong&gt;&lt;/p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T:%5Cmathbb%7BR%7D%5E%203%20%5Cto%20%5Cmathbb%7BR%7D%5E%204" alt="T:\mathbb{R}^ 3 \to \mathbb{R}^ 4" title="T:\mathbb{R}^ 3 \to \mathbb{R}^ 4" data-latex="T:\mathbb{R}^ 3 \to \mathbb{R}^ 4"&gt; be the linear transformation given by &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?T%5Cleft(%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%20x_%7B1%7D%20%5C%5C%20x_%7B2%7D%20%5C%5C%20x_%7B3%7D%20%5Cend%7Barray%7D%5Cright%5D%20%5Cright)%20=%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%20-2%20%5C,%20x_%7B1%7D%20+%204%20%5C,%20x_%7B2%7D%20+%202%20%5C,%20x_%7B3%7D%20%5C%5C%20-x_%7B1%7D%20-%203%20%5C,%20x_%7B2%7D%20-%204%20%5C,%20x_%7B3%7D%20%5C%5C%20-x_%7B1%7D%20+%20x_%7B2%7D%20%5C%5C%20x_%7B1%7D%20+%20x_%7B3%7D%20%5Cend%7Barray%7D%5Cright%5D%20." alt="T\left( \left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right) = \left[\begin{array}{c} -2 \, x_{1} + 4 \, x_{2} + 2 \, x_{3} \\ -x_{1} - 3 \, x_{2} - 4 \, x_{3} \\ -x_{1} + x_{2} \\ x_{1} + x_{3} \end{array}\right] ." title="T\left( \left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right) = \left[\begin{array}{c} -2 \, x_{1} + 4 \, x_{2} + 2 \, x_{3} \\ -x_{1} - 3 \, x_{2} - 4 \, x_{3} \\ -x_{1} + x_{2} \\ x_{1} + x_{3} \end{array}\right] ." data-latex="T\left( \left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right) = \left[\begin{array}{c} -2 \, x_{1} + 4 \, x_{2} + 2 \, x_{3} \\ -x_{1} - 3 \, x_{2} - 4 \, x_{3} \\ -x_{1} + x_{2} \\ x_{1} + x_{3} \end{array}\right] ."&gt;&lt;/p&gt;
&lt;ol type="a"&gt;
&lt;li&gt;Explain how to find the image of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"&gt; and the kernel of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"&gt;.&lt;/li&gt;
&lt;li&gt;Explain how to find a basis of the image of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"&gt; and a basis of the kernel of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"&gt;.&lt;/li&gt;
&lt;li&gt;Explain how to find the rank and nullity of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"&gt;, and why the rank-nullity theorem holds for &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"&gt;.&lt;/li&gt;
&lt;/ol&gt;
&lt;/div&gt;

</mattext>
    </material>
    <response_str ident="response1" rcardinality="Single">
      <render_fib>
        <response_label ident="answer1" rshuffle="No"/>
      </render_fib>
    </response_str>
  </presentation>
  <itemfeedback ident="general_fb">
    <flow_mat>
      <material>
        <mattextxml>
          <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}{ccc} -2 &amp; 4 &amp; 2 \\ -1 &amp; -3 &amp; -4 \\ -1 &amp; 1 &amp; 0 \\ 1 &amp; 0 &amp; 1 \end{array}\right] = \left[\begin{array}{ccc} 1 &amp; 0 &amp; 1 \\ 0 &amp; 1 &amp; 1 \\ 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 \end{array}\right]" alt="\operatorname{RREF} \left[\begin{array}{ccc} -2 &amp; 4 &amp; 2 \\ -1 &amp; -3 &amp; -4 \\ -1 &amp; 1 &amp; 0 \\ 1 &amp; 0 &amp; 1 \end{array}\right] = \left[\begin{array}{ccc} 1 &amp; 0 &amp; 1 \\ 0 &amp; 1 &amp; 1 \\ 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 \end{array}\right]" title="\operatorname{RREF} \left[\begin{array}{ccc} -2 &amp; 4 &amp; 2 \\ -1 &amp; -3 &amp; -4 \\ -1 &amp; 1 &amp; 0 \\ 1 &amp; 0 &amp; 1 \end{array}\right] = \left[\begin{array}{ccc} 1 &amp; 0 &amp; 1 \\ 0 &amp; 1 &amp; 1 \\ 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 \end{array}\right]" data-latex="\operatorname{RREF} \left[\begin{array}{ccc} -2 &amp; 4 &amp; 2 \\ -1 &amp; -3 &amp; -4 \\ -1 &amp; 1 &amp; 0 \\ 1 &amp; 0 &amp; 1 \end{array}\right] = \left[\begin{array}{ccc} 1 &amp; 0 &amp; 1 \\ 0 &amp; 1 &amp; 1 \\ 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 \end{array}\right]"/>
              </p>
            </p>
            <ol type="a">
              <li>
                <p style="text-align:center;">
                  <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\operatorname{Im}\ T = \operatorname{span}\ \left\{ \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 0 \end{array}\right] \right\}" alt="\operatorname{Im}\ T = \operatorname{span}\ \left\{ \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 0 \end{array}\right] \right\}" title="\operatorname{Im}\ T = \operatorname{span}\ \left\{ \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 0 \end{array}\right] \right\}" data-latex="\operatorname{Im}\ T = \operatorname{span}\ \left\{ \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 0 \end{array}\right] \right\}"/>
                </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{ker}\ T = \left\{ \left[\begin{array}{c} -a \\ -a \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}" alt="\operatorname{ker}\ T = \left\{ \left[\begin{array}{c} -a \\ -a \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}" title="\operatorname{ker}\ T = \left\{ \left[\begin{array}{c} -a \\ -a \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}" data-latex="\operatorname{ker}\ T = \left\{ \left[\begin{array}{c} -a \\ -a \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}"/>
                </p>
              </li>
              <li> A basis of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\operatorname{Im}\ T" alt="\operatorname{Im}\ T" title="\operatorname{Im}\ T" data-latex="\operatorname{Im}\ T"/> 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 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 0 \end{array}\right] \right\}" alt="\left\{ \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 0 \end{array}\right] \right\}" title="\left\{ \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 0 \end{array}\right] \right\}" data-latex="\left\{ \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 0 \end{array}\right] \right\}"/>. A basis of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\operatorname{ker}\ T" alt="\operatorname{ker}\ T" title="\operatorname{ker}\ T" data-latex="\operatorname{ker}\ T"/> is <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 1 \end{array}\right] \right\}" alt="\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 1 \end{array}\right] \right\}" title="\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 1 \end{array}\right] \right\}" data-latex="\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 1 \end{array}\right] \right\}"/></li>
              <li> The rank of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"/> is <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2" alt="2" title="2" data-latex="2"/>, the nullity of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"/> is <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?1" alt="1" title="1" data-latex="1"/>, and the dimension of the domain of <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"/> is <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"/>. The rank-nullity theorem asserts that <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 + 1 = 3" alt="2 + 1 = 3" title="2 + 1 = 3" data-latex="2 + 1 = 3"/>, which we see to be true. </li>
            </ol>
          </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%7Bccc%7D%20-2%20&amp;amp;%204%20&amp;amp;%202%20%5C%5C%20-1%20&amp;amp;%20-3%20&amp;amp;%20-4%20%5C%5C%20-1%20&amp;amp;%201%20&amp;amp;%200%20%5C%5C%201%20&amp;amp;%200%20&amp;amp;%201%20%5Cend%7Barray%7D%5Cright%5D%20=%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%201%20&amp;amp;%200%20&amp;amp;%201%20%5C%5C%200%20&amp;amp;%201%20&amp;amp;%201%20%5C%5C%200%20&amp;amp;%200%20&amp;amp;%200%20%5C%5C%200%20&amp;amp;%200%20&amp;amp;%200%20%5Cend%7Barray%7D%5Cright%5D" alt="\operatorname{RREF} \left[\begin{array}{ccc} -2 &amp;amp; 4 &amp;amp; 2 \\ -1 &amp;amp; -3 &amp;amp; -4 \\ -1 &amp;amp; 1 &amp;amp; 0 \\ 1 &amp;amp; 0 &amp;amp; 1 \end{array}\right] = \left[\begin{array}{ccc} 1 &amp;amp; 0 &amp;amp; 1 \\ 0 &amp;amp; 1 &amp;amp; 1 \\ 0 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 0 \end{array}\right]" title="\operatorname{RREF} \left[\begin{array}{ccc} -2 &amp;amp; 4 &amp;amp; 2 \\ -1 &amp;amp; -3 &amp;amp; -4 \\ -1 &amp;amp; 1 &amp;amp; 0 \\ 1 &amp;amp; 0 &amp;amp; 1 \end{array}\right] = \left[\begin{array}{ccc} 1 &amp;amp; 0 &amp;amp; 1 \\ 0 &amp;amp; 1 &amp;amp; 1 \\ 0 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 0 \end{array}\right]" data-latex="\operatorname{RREF} \left[\begin{array}{ccc} -2 &amp;amp; 4 &amp;amp; 2 \\ -1 &amp;amp; -3 &amp;amp; -4 \\ -1 &amp;amp; 1 &amp;amp; 0 \\ 1 &amp;amp; 0 &amp;amp; 1 \end{array}\right] = \left[\begin{array}{ccc} 1 &amp;amp; 0 &amp;amp; 1 \\ 0 &amp;amp; 1 &amp;amp; 1 \\ 0 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 0 \end{array}\right]"&gt;
    &lt;/p&gt;
  
  &lt;ol type="a"&gt;
    &lt;li&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%7BIm%7D%5C%20T%20=%20%5Coperatorname%7Bspan%7D%5C%20%5Cleft%5C%7B%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%20-2%20%5C%5C%20-1%20%5C%5C%20-1%20%5C%5C%201%20%5Cend%7Barray%7D%5Cright%5D%20,%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%204%20%5C%5C%20-3%20%5C%5C%201%20%5C%5C%200%20%5Cend%7Barray%7D%5Cright%5D%20%5Cright%5C%7D" alt="\operatorname{Im}\ T = \operatorname{span}\ \left\{ \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 0 \end{array}\right] \right\}" title="\operatorname{Im}\ T = \operatorname{span}\ \left\{ \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 0 \end{array}\right] \right\}" data-latex="\operatorname{Im}\ T = \operatorname{span}\ \left\{ \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 0 \end{array}\right] \right\}"&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%7Bker%7D%5C%20T%20=%20%5Cleft%5C%7B%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%20-a%20%5C%5C%20-a%20%5C%5C%20a%20%5Cend%7Barray%7D%5Cright%5D%20%5Cmiddle%7C%5C,a%5Cin%5Cmathbb%7BR%7D%5Cright%5C%7D" alt="\operatorname{ker}\ T = \left\{ \left[\begin{array}{c} -a \\ -a \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}" title="\operatorname{ker}\ T = \left\{ \left[\begin{array}{c} -a \\ -a \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}" data-latex="\operatorname{ker}\ T = \left\{ \left[\begin{array}{c} -a \\ -a \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt; A basis of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Coperatorname%7BIm%7D%5C%20T" alt="\operatorname{Im}\ T" title="\operatorname{Im}\ T" data-latex="\operatorname{Im}\ T"&gt; 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%20-1%20%5C%5C%20-1%20%5C%5C%201%20%5Cend%7Barray%7D%5Cright%5D%20,%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%204%20%5C%5C%20-3%20%5C%5C%201%20%5C%5C%200%20%5Cend%7Barray%7D%5Cright%5D%20%5Cright%5C%7D" alt="\left\{ \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 0 \end{array}\right] \right\}" title="\left\{ \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 0 \end{array}\right] \right\}" data-latex="\left\{ \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 0 \end{array}\right] \right\}"&gt;. A basis of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Coperatorname%7Bker%7D%5C%20T" alt="\operatorname{ker}\ T" title="\operatorname{ker}\ T" data-latex="\operatorname{ker}\ T"&gt; 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%20-1%20%5C%5C%20-1%20%5C%5C%201%20%5Cend%7Barray%7D%5Cright%5D%20%5Cright%5C%7D" alt="\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 1 \end{array}\right] \right\}" title="\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 1 \end{array}\right] \right\}" data-latex="\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 1 \end{array}\right] \right\}"&gt;
&lt;/li&gt;
    &lt;li&gt; The rank of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"&gt; is &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2" alt="2" title="2" data-latex="2"&gt;, the nullity of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"&gt; is &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?1" alt="1" title="1" data-latex="1"&gt;, and the dimension of the domain of &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T" alt="T" title="T" data-latex="T"&gt; is &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;. The rank-nullity theorem asserts that &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20+%201%20=%203" alt="2 + 1 = 3" title="2 + 1 = 3" data-latex="2 + 1 = 3"&gt;, which we see to be true. &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

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