Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Jump to bottom

Issue 352-EV7-suggestion #353

Draft
wants to merge 7 commits into
base: main
Choose a base branch
from
Draft

Issue 352-EV7-suggestion #353

Changes from all commits
Commits
Show all changes
7 commits
Select commit Hold shift + click to select a range
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
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
161 changes: 98 additions & 63 deletions source/linear-algebra/source/02-EV/07.ptx
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -128,14 +128,74 @@ Rewrite this solution space in the form <me>\setBuilder{ a \left[\begin{array}{c
</task>
<task>
<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>
</task>
</activity>
<answer>
<p>
D.
</p>
</answer>

<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 +209,34 @@ 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>

<sage language="octave">
<input>
row1=[]
row2=[]
row3=[]
rref([row1;row2;row3])
</input>
</sage>


<fact xml:id="fact-solution-space-basis">
<statement>
<p>
Expand All @@ -175,73 +247,31 @@ 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 +310,11 @@ solution space?
</statement>
</task>
</activity>
<answer>
<p>
A.
</p>
</answer>
<sage language="octave">
</sage>

Expand All @@ -290,15 +325,15 @@ solution space?
To create a computer-animated film, an animator first models a scene
as a subset of <m>\mathbb R^3</m>. Then to transform this three-dimensional
visual data for display on a two-dimensional movie screen or television set,
the computer could apply a linear tranformation that maps visual information
the computer could apply a linear transformation that maps visual information
at the point <m>(x,y,z)\in\mathbb R^3</m> onto the pixel located at
<m>(x+y,y-z)\in\mathbb R^2</m>.
</p>
</introduction>
<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
Loading