0010.qti

<item ident="A1-0010" title="A1 | Linear maps | ver. 0010">
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  <presentation>
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        <div class="exercise-statement">
          <p>
            <strong>A1.</strong>
          </p>
          <p>Consider the following maps of polynomials <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?S:\mathcal{P}\rightarrow\mathcal{P}" alt="S:\mathcal{P}\rightarrow\mathcal{P}" title="S:\mathcal{P}\rightarrow\mathcal{P}" data-latex="S:\mathcal{P}\rightarrow\mathcal{P}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T:\mathcal{P}\rightarrow\mathcal{P}" alt="T:\mathcal{P}\rightarrow\mathcal{P}" title="T:\mathcal{P}\rightarrow\mathcal{P}" data-latex="T:\mathcal{P}\rightarrow\mathcal{P}"/> defined by <p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?S(h(x))= -h\left(x\right)^{2} - 2 \, h'\left(x\right) \hspace{1em} \text{and} \hspace{1em} T(h(x))= 3 \, h\left(4\right) + 4 \, h\left(x^{2}\right)" alt="S(h(x))= -h\left(x\right)^{2} - 2 \, h'\left(x\right) \hspace{1em} \text{and} \hspace{1em} T(h(x))= 3 \, h\left(4\right) + 4 \, h\left(x^{2}\right)" title="S(h(x))= -h\left(x\right)^{2} - 2 \, h'\left(x\right) \hspace{1em} \text{and} \hspace{1em} T(h(x))= 3 \, h\left(4\right) + 4 \, h\left(x^{2}\right)" data-latex="S(h(x))= -h\left(x\right)^{2} - 2 \, h'\left(x\right) \hspace{1em} \text{and} \hspace{1em} T(h(x))= 3 \, h\left(4\right) + 4 \, h\left(x^{2}\right)"/></p> Explain why one these maps is a linear transformation and why the other map is not. </p>
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      <mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;A1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Consider the following maps of polynomials &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?S:%5Cmathcal%7BP%7D%5Crightarrow%5Cmathcal%7BP%7D" alt="S:\mathcal{P}\rightarrow\mathcal{P}" title="S:\mathcal{P}\rightarrow\mathcal{P}" data-latex="S:\mathcal{P}\rightarrow\mathcal{P}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?T:%5Cmathcal%7BP%7D%5Crightarrow%5Cmathcal%7BP%7D" alt="T:\mathcal{P}\rightarrow\mathcal{P}" title="T:\mathcal{P}\rightarrow\mathcal{P}" data-latex="T:\mathcal{P}\rightarrow\mathcal{P}"&gt; defined by &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?S(h(x))=%20-h%5Cleft(x%5Cright)%5E%7B2%7D%20-%202%20%5C,%20h'%5Cleft(x%5Cright)%20%5Chspace%7B1em%7D%20%5Ctext%7Band%7D%20%5Chspace%7B1em%7D%20T(h(x))=%203%20%5C,%20h%5Cleft(4%5Cright)%20+%204%20%5C,%20h%5Cleft(x%5E%7B2%7D%5Cright)" alt="S(h(x))= -h\left(x\right)^{2} - 2 \, h'\left(x\right) \hspace{1em} \text{and} \hspace{1em} T(h(x))= 3 \, h\left(4\right) + 4 \, h\left(x^{2}\right)" title="S(h(x))= -h\left(x\right)^{2} - 2 \, h'\left(x\right) \hspace{1em} \text{and} \hspace{1em} T(h(x))= 3 \, h\left(4\right) + 4 \, h\left(x^{2}\right)" data-latex="S(h(x))= -h\left(x\right)^{2} - 2 \, h'\left(x\right) \hspace{1em} \text{and} \hspace{1em} T(h(x))= 3 \, h\left(4\right) + 4 \, h\left(x^{2}\right)"&gt;&lt;/p&gt; Explain why one these maps is a linear transformation and why the other map is not. 
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  <itemfeedback ident="general_fb">
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            <h4>Partial Answer:</h4>
            <p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?S" alt="S" title="S" data-latex="S"/> is not linear and <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 linear.</p>
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        <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?S" alt="S" title="S" data-latex="S"&gt; is not linear and &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 linear.&lt;/p&gt;
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