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AT2 - add scaffolded flunecy builder similar to checkit bank #363

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Oct 18, 2024
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AT2 - add scaffolded flunecy builder similar to checkit bank #363

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80 changes: 44 additions & 36 deletions source/linear-algebra/source/03-AT/02.ptx
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Expand Up @@ -377,47 +377,55 @@ of a vector <m>\vec x</m>, we will often write
</statement>
</fact>

<activity estimated-time='5'>
<introduction>
<activity estimated-time='3'>
<statement>
<p>
Let <m>T: \IR^3 \rightarrow \IR^3</m> be the linear transformation given by the standard matrix
Let <m>T: \IR^4 \rightarrow \IR^3</m> be the linear transformation given by
<me>
\left[\begin{array}{ccc} 3 &amp; -2 &amp; -1 \\ 4 &amp; 5 &amp; 2 \\ 0 &amp; -2 &amp; 1 \end{array}\right]
.</me>
T\left(\vec e_1 \right)
=
\left[\begin{array}{c} 0 \\ 3 \\ -2\end{array}\right]
\hspace{2em}
T\left(\vec e_2 \right)
=
\left[\begin{array}{c} -3 \\ 0 \\ 1\end{array}\right]
\hspace{2em}
T\left(\vec e_3 \right)
=
\left[\begin{array}{c} 4 \\ -2 \\ 1\end{array}\right]
\hspace{2em}
T\left(\vec e_4 \right)
=
\left[\begin{array}{c} 2 \\ 0 \\ 0\end{array}\right]
</me>
Write the standard matrix <m>[T(\vec e_1) \,\cdots\, T(\vec e_n)]</m> for <m>T</m>.
</p>
</introduction>
<task>
<p>
Compute <m>T\left(\left[\begin{array}{c} 1\\ 2 \\ 3 \end{array}\right] \right) </m>.
</p>
</task>
<task>
<p>
Compute <m>T\left(\left[\begin{array}{c} x\\ y \\ z \end{array}\right] \right) </m>.
</p>
</task>
</statement>
</activity>


<activity estimated-time='15'>
<introduction> <p>Compute the following linear transformations of vectors given their
standard matrices.</p></introduction>

<task><p> <me>
T_1\left(\left[\begin{array}{c}1\\2\end{array}\right]\right)
\text{ for the standard matrix }
A_1=\left[\begin{array}{cc}4&amp;3\\0&amp;-1\\1&amp;1\\3&amp;0\end{array}\right]
</me></p></task>
<task><p><me>
T_2\left(\left[\begin{array}{c}1\\1\\0\\-3\end{array}\right]\right)
\text{ for the standard matrix }
A_2=\left[\begin{array}{cccc}4&amp;3&amp;0&amp;-1\\1&amp;1&amp;3&amp;0\end{array}\right]
</me></p></task>
<task><p><me>
T_3\left(\left[\begin{array}{c}0\\-2\\0\end{array}\right]\right)
\text{ for the standard matrix }
A_3=\left[\begin{array}{ccc}4&amp;3&amp;0\\0&amp;-1&amp;3\\5&amp;1&amp;1\\3&amp;0&amp;0\end{array}\right]
</me></p></task>
<activity>
<task>
<statement>
<p>Explain and demonstrate how to compute the standard matrix for the linear transformation <m>S:\mathbb{R}^2 \to \mathbb{R}^4</m> given by <me>S\left( \left[\begin{array}{c} x_{1} \\ x_{2} \end{array}\right] \right) = \left[\begin{array}{c} 9 \, x_{1} - 2 \, x_{2} \\ -3 \, x_{1} \\ 5 \, x_{1} - x_{2} \\ -6 \, x_{2} \end{array}\right]</me> by computing transformations of the standard basic vectors:</p>
<p><me>S(\vec e_1)=\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right]\hspace{1em}S(\vec e_2)=\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right]\hspace{1em}\rightarrow\hspace{1em}\left[\begin{array}{cc} \unknown &amp; \unknown \\ \unknown &amp; \unknown \\ \unknown &amp; \unknown \\\unknown &amp; \unknown \end{array}\right]</me></p>
</statement>
<answer>
<p>
<me>\left[\begin{array}{cc} 9 &amp; -2 \\ -3 &amp; 0 \\ 5 &amp; -1 \\ 0 &amp; -6 \end{array}\right]</me>
</p>
</answer>
</task>
<task>
<statement>
<p>Let <m>T:\mathbb{R}^4 \to \mathbb{R}^3</m> be the linear transformation given by the standard matrix <me>\left[\begin{array}{cccc} -2 &amp; -4 &amp; 2 &amp; -2 \\ -4 &amp; 3 &amp; -3 &amp; 2 \\ 5 &amp; 0 &amp; 2 &amp; -6 \end{array}\right].</me> Explain and demonstrate how to compute <m>T\left(\left[\begin{array}{c} -5 \\ 0 \\ -3 \\ -2 \end{array}\right]\right)</m> by using the values of transformed standard basic vectors:</p>
<p><me>T\left(\left[\begin{array}{c} -5 \\ 0 \\ -3 \\ -2 \end{array}\right]\right)=\unknown T(\vec e_1)+\unknown T(\vec e_2)+\unknown T(\vec e_3)+\unknown T(\vec e_4)</me></p>
</statement>
<answer>
<p>
<me>T\left(\left[\begin{array}{c} -5 \\ 0 \\ -3 \\ -2 \end{array}\right]\right)=\left[\begin{array}{c} 8 \\ 25 \\ -19 \end{array}\right]</me>
</p>
</answer>
</task>
</activity>

</subsection>
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