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Issue 352-EV7-suggestion #353

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merged 10 commits into from
Dec 18, 2024
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Issue 352-EV7-suggestion #353

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50a047a
suggested edits
jkostiuk Oct 11, 2024
6e81096
adjusted MC options
jkostiuk Oct 11, 2024
85bee08
Update source/linear-algebra/source/02-EV/07.ptx
jkostiuk Oct 11, 2024
00c5823
accepted a suggestion and fixed some typos
jkostiuk Oct 11, 2024
fc75679
Update source/linear-algebra/source/02-EV/07.ptx
StevenClontz Oct 14, 2024
ac4a55d
implemented suggestions: added answers, textcolor
jkostiuk Nov 14, 2024
33473a9
Discard changes to .vscode/settings.json
siwelwerd Nov 14, 2024
c395136
Merge branch 'main' into 352-EV7-suggestion
StevenClontz Dec 17, 2024
17e749e
reorganize statement/answer
jkostiuk Dec 17, 2024
f5023c2
further reorganizing
jkostiuk Dec 17, 2024
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163 changes: 101 additions & 62 deletions source/linear-algebra/source/02-EV/07.ptx
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Original file line number Diff line number Diff line change
Expand Up @@ -127,15 +127,79 @@ Rewrite this solution space in the form <me>\setBuilder{ a \left[\begin{array}{c
</p>
</task>
<task>
<statement>
<p>
Rewrite this solution space in the form <me>\vspan\left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right]\right\}.</me>
Which of these choices best describes the set of two vectors
<m>\left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right]\right\}</m>
used in this solution space?
<ol marker="A.">
<li>
<p>
The set is linearly dependent.
</p>
</li>
<li>
<p>
The set is linearly independent.
</p>
</li>
<li>
<p>
The set spans the solution space.
</p>
</li>
<li>
<p>
The set is a basis of the solution space.
</p>
</li>
</ol>
</p>
</statement>
</task>
<answer>
<p>
D.
</p>
</answer>
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</activity>


<sage language="octave">
</sage>
<activity estimated-time='10'>
<introduction>
<p>
Consider the homogeneous system of equations
<md alignat-columns='5' alignment="alignat">
<mrow>
2x_1&amp;\,+\,&amp;4x_2&amp;\,+\,&amp;2x_3 &amp;\,-\,&amp;3 x_4 &amp;\,+\,&amp;31x_5&amp;\,+\,&amp;2x_6&amp;\,-\,&amp;16x_7&amp;=&amp; 0
</mrow>
<mrow>
-1x_1&amp;\,-\,&amp;2x_2&amp;\,+\,&amp;4x_3 &amp;\,-\,&amp;x_4 &amp;\,+\,&amp;2x_5&amp;\,+\,&amp;9x_6&amp;\,+\,&amp;3x_7&amp;=&amp; 0
</mrow>
<mrow>
x_1&amp;\,+\,&amp;2x_2&amp;\,+\,&amp;x_3 &amp;\,+\,&amp; x_4 &amp;\,+\,&amp;3x_5&amp;\,+\,&amp;6x_7&amp;\,+\,&amp;7x_7&amp;=&amp; 0
</mrow>
</md>
</p>
</introduction>
<task>
<p>
Find its solution set (a subspace of <m>\IR^7</m>).
</p>
</task>
<task>
<p>
Rewrite this solution space in the form <me>\setBuilder{ a \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right] + b \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right]+c \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right]+d \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right] }{a,b,c,d \in \IR}.</me>
</p>
</task>
<task>
<p>
Which of these choices best describes the set of two vectors
<m>\left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right]\right\}</m>
used in this span?
<m>\left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right],\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right],\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right]\right\}</m>
used in this solution space?
<ol marker="A.">
<li>
<p>
Expand All @@ -149,22 +213,34 @@ used in this span?
</li>
<li>
<p>
The set spans all of <m>\IR^4</m>.
The set spans the solution space.
</p>
</li>
<li>
<li>
<p>
The set fails to span the solution space.
The set is a basis for the solution space.
</p>
</li>
</ol>
</p>
</task>
</activity>
<answer>
<p>
D.
</p>
</answer>
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Similar here.


<sage language="octave">
<input>
row1=[]
row2=[]
row3=[]
rref([row1;row2;row3])
</input>
</sage>
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I think we usually just have a blank sage cell when we don't provide the matrix terms.



<fact xml:id="fact-solution-space-basis">
<statement>
<p>
Expand All @@ -175,73 +251,31 @@ used in this span?
Thus if
<me>
\setBuilder{
a \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0\end{array}\right] +
b \left[\begin{array}{c} -1 \\ 0 \\ -4 \\ 1 \end{array}\right]
a \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0\\0\\0\\0\end{array}\right] +
b \left[\begin{array}{c} -7 \\ 0 \\ -1 \\ 5\\1\\0\\0 \end{array}\right]+
c \left[\begin{array}{c} -1 \\ 0 \\ -3 \\ -2\\0\\1\\0 \end{array}\right]+
d \left[\begin{array}{c} 1 \\ 0 \\ -2 \\ -6\\0\\0\\1 \end{array}\right]
}{
a,b \in \IR
} = \vspan\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0\end{array}\right],
\left[\begin{array}{c} -1 \\ 0 \\ -4 \\ 1 \end{array}\right] \right\}
a,b,c,d \in \IR
} = \vspan\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0\\0\\0\\0\end{array}\right],
\left[\begin{array}{c} -7 \\ 0 \\ -1 \\ 5\\1\\0\\0 \end{array}\right],
\left[\begin{array}{c} -1 \\ 0 \\ -3 \\ -2\\0\\1\\0 \end{array}\right],
\left[\begin{array}{c} 1 \\ 0 \\ -2 \\ -6\\0\\0\\1 \end{array}\right] \right\}
</me>
is the solution space for a homogeneous system, then
<me>
\setList{
\left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0\end{array}\right],
\left[\begin{array}{c} -1 \\ 0 \\ -4 \\ 1 \end{array}\right]
\left[\begin{array}{c} -2 \\ \textcolor{blue}{1} \\ 0 \\ 0\\\textcolor{blue}{0}\\\textcolor{blue}{0}\\\textcolor{blue}{0}\end{array}\right],
\left[\begin{array}{c} -7 \\ \textcolor{blue}{0} \\ -1 \\ 5\\\textcolor{blue}{1}\\\textcolor{blue}{0}\\\textcolor{blue}{0} \end{array}\right],
\left[\begin{array}{c} -1 \\ \textcolor{blue}{0} \\ -3 \\ -2\\\textcolor{blue}{0}\\\textcolor{blue}{1}\\\textcolor{blue}{0} \end{array}\right],
\left[\begin{array}{c} 1 \\ \textcolor{blue}{0} \\ -2 \\ -6\\\textcolor{blue}{0}\\\textcolor{blue}{0}\\\textcolor{blue}{1} \end{array}\right]
}
</me>
is a basis for the solution space.
</p>
</statement>
</fact>



<activity estimated-time='10'>
<statement>
<p>
Consider the homogeneous system of equations
<md alignat-columns='5' alignment="alignat">
<mrow>
2x_1&amp;\,+\,&amp;4x_2&amp;\,+\,&amp; 2x_3&amp;\,-\,&amp;4x_4 &amp;=&amp; 0
</mrow>
<mrow>
-2x_1&amp;\,-\,&amp;4x_2&amp;\,+\,&amp;x_3 &amp;\,+\,&amp; x_4 &amp;=&amp; 0
</mrow>
<mrow>
3x_1&amp;\,+\,&amp;6x_2&amp;\,-\,&amp;x_3 &amp;\,-\,&amp;4 x_4 &amp;=&amp; 0
</mrow>
</md>
</p>
<p>
Find a basis for its solution space.
</p>
</statement>
</activity>
<sage language="octave">
</sage>


<activity estimated-time='10'>
<statement>
<p>
Consider the homogeneous vector equation
<me>
x_1 \left[\begin{array}{c} 2 \\ -2 \\ 3 \end{array}\right]+
x_2 \left[\begin{array}{c} 4 \\ -4 \\ 6 \end{array}\right]+
x_3 \left[\begin{array}{c} 2 \\ 1 \\ -1 \end{array}\right]+
x_4 \left[\begin{array}{c} -4 \\ 1 \\ -4 \end{array}\right]=
\left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right]
</me>
</p>
<p>
Find a basis for its solution space.
</p>
</statement>
</activity>
<sage language="octave">
</sage>


<activity estimated-time='5'>
<introduction>
<p>
Expand Down Expand Up @@ -280,6 +314,11 @@ solution space?
</statement>
</task>
</activity>
<answer>
<p>
A.
</p>
</answer>
<sage language="octave">
</sage>

Expand All @@ -298,7 +337,7 @@ at the point <m>(x,y,z)\in\mathbb R^3</m> onto the pixel located at
<task>
<statement>
<p>
What homoegeneous linear system describes the positions <m>(x,y,z)</m>
What homogeneous linear system describes the positions <m>(x,y,z)</m>
within the original scene that would be aligned with the
pixel <m>(0,0)</m> on the screen?
</p>
Expand Down
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