0007.qti

<item ident="V1-0007" title="V1 | Vector spaces | ver. 0007">
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  <presentation>
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        <div class="exercise-statement">
          <p>
            <strong>V1.</strong>
          </p>
          <p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?V" alt="V" title="V" data-latex="V"/> be the set of all pairs <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(x,y)" alt="(x,y)" title="(x,y)" data-latex="(x,y)"/> of real numbers together with the following operations: </p>
          <p style="text-align:center;">
            <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2} + 4,\,\sqrt{y_{1}^{2} + y_{2}^{2}}\right)" alt="(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2} + 4,\,\sqrt{y_{1}^{2} + y_{2}^{2}}\right)" title="(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2} + 4,\,\sqrt{y_{1}^{2} + y_{2}^{2}}\right)" data-latex="(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2} + 4,\,\sqrt{y_{1}^{2} + y_{2}^{2}}\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?c \odot (x,y) = \left(c x,\,c y\right) ." alt="c \odot (x,y) = \left(c x,\,c y\right) ." title="c \odot (x,y) = \left(c x,\,c y\right) ." data-latex="c \odot (x,y) = \left(c x,\,c y\right) ."/>
          </p>
          <p> (a) Show that vector addition is associative, that is: </p>
          <p style="text-align:center;">
            <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right)." alt="\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right)." title="\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right)." data-latex="\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right)."/>
          </p>
          <p> (b) Explain why <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?V" alt="V" title="V" data-latex="V"/> nonetheless is not a vector space. </p>
        </div>
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      <mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;V1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Let &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?V" alt="V" title="V" data-latex="V"&gt; be the set of all pairs &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(x,y)" alt="(x,y)" title="(x,y)" data-latex="(x,y)"&gt; of real numbers together with the following operations: &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?(x_1,y_1)%5Coplus%20(x_2,y_2)=%20%5Cleft(x_%7B1%7D%20+%20x_%7B2%7D%20+%204,%5C,%5Csqrt%7By_%7B1%7D%5E%7B2%7D%20+%20y_%7B2%7D%5E%7B2%7D%7D%5Cright)" alt="(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2} + 4,\,\sqrt{y_{1}^{2} + y_{2}^{2}}\right)" title="(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2} + 4,\,\sqrt{y_{1}^{2} + y_{2}^{2}}\right)" data-latex="(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2} + 4,\,\sqrt{y_{1}^{2} + y_{2}^{2}}\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?c%20%5Codot%20(x,y)%20=%20%5Cleft(c%20x,%5C,c%20y%5Cright)%20." alt="c \odot (x,y) = \left(c x,\,c y\right) ." title="c \odot (x,y) = \left(c x,\,c y\right) ." data-latex="c \odot (x,y) = \left(c x,\,c y\right) ."&gt;
  &lt;/p&gt;
  &lt;p&gt; (a) Show that vector addition is associative, that is: &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((x_1,y_1)%5Coplus(x_2,y_2)%5Cright)%5Coplus(x_3,y_3)=(x_1,y_1)%5Coplus%5Cleft((x_2,y_2)%5Coplus(x_3,y_3)%5Cright)." alt="\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right)." title="\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right)." data-latex="\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right)."&gt;
  &lt;/p&gt;
  &lt;p&gt; (b) Explain why &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?V" alt="V" title="V" data-latex="V"&gt; nonetheless is not a vector space. &lt;/p&gt;
&lt;/div&gt;

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        <response_label ident="answer1" rshuffle="No"/>
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  <itemfeedback ident="general_fb">
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          <div class="exercise-answer">
            <h4>Partial Answer:</h4>
            <p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?V" alt="V" title="V" data-latex="V"/> is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold: </p>
            <ul>
              <li>there is no additive identity element</li>
              <li>scalar multiplication does not distribute over vector addition</li>
              <li>scalar multiplication does not distribute over scalar addition</li>
            </ul>
          </div>
        </mattextxml>
        <mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt;&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?V" alt="V" title="V" data-latex="V"&gt; is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold: &lt;/p&gt;
  &lt;ul&gt;
    &lt;li&gt;there is no additive identity element&lt;/li&gt;
    &lt;li&gt;scalar multiplication does not distribute over vector addition&lt;/li&gt;
    &lt;li&gt;scalar multiplication does not distribute over scalar addition&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

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