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Merged
jkostiuk merged 4 commits into from
Feb 12, 2026
Merged

add solutions, clean up final task #928

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7dfe186
add solutions, clean up final task
StevenClontz Feb 11, 2026
9379ab9
typo
StevenClontz Feb 11, 2026
6fcfa59
clarify about choice C
StevenClontz Feb 11, 2026
4b55a0f
style guide fixes
StevenClontz Feb 11, 2026
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89 changes: 64 additions & 25 deletions source/linear-algebra/source/02-EV/04.ptx
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Original file line number Diff line number Diff line change
Expand Up @@ -148,7 +148,8 @@
</me>
</p>
</remark>
<activity estimated-time='10'>

<activity>
<introduction>
<p>Consider the following three vectors in <m>\IR^3</m>:
<me>\vec v_1=\left[\begin{array}{c}-2 \\ 0 \\ 0\end{array}\right],
Expand All @@ -159,6 +160,7 @@
</p>
</introduction>
<task>
<statement>
<p>
Let <m> \vec w = 3\vec v_1 - \vec v_2 - 5 \vec v_3 = \left[\begin{array}{c}\unknown \\ \unknown \\ \unknown\end{array}\right]</m>.
The set <m>\{\vec v_1,\vec v_2,\vec v_3,\vec w\}</m> is...
Expand All @@ -167,8 +169,16 @@
<li><p>linearly independent: no vector is a linear combination of others</p></li>
</ol>
</p>
</statement>
<answer>
<p>A.</p>
<p>
<m>\vec w</m> is a linear combination of the others, so the set is dependent.
</p>
</answer>
</task>
<task>
<statement>
<p>
Find <me>\RREF \left[\begin{array}{ccc|c}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec w \\
Expand Down Expand Up @@ -200,8 +210,14 @@
</li>
</ol>
</p>
</task>
</statement>
<answer>
<p>B.</p>
<p>Each variable is set equal to a specific value.</p>
</answer>
</task>
<task>
<statement>
<p>
Find <me>\RREF \left[\begin{array}{cccc|c}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec w &amp; \vec 0\\
Expand Down Expand Up @@ -233,52 +249,75 @@
</li>
</ol>
</p>
</statement>
<answer>
<p>C.</p>
<p><m>\vec w</m>'s column is not a pivot, revealing a free variable and infinitely-many solutions.</p>
</answer>
</task>
<task>
<task>
<statement>
<p>
Which of the following is the best conclusion obtained when we solved
<m>x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3 + x_4\vec w=\vec{0}</m>?
It follows that <m>\{\vec v_1,\vec v_2,\vec v_3,\vec w\}</m>
is linearly dependent because <m>x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3 + x_4\vec w=\vec{0}</m>
had the number of solutions you found in the previous task.
Which feature of which RREF matrix best reveals this?
<ol marker="A.">
<li>
<p>
A pivot column in the <em>augmented</em> matrix <m>\RREF \left[\begin{array}{cccc|c}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec w &amp; \vec 0 \\
\end{array}\right]</m> guarantees the linear independence
of <m>\{\vec v_1,\vec v_2,\vec v_3,\vec w\}</m>
by preventing contradictions.
The <em>pivot</em> column in the <em>augmented</em> matrix
<m>\RREF \left[\begin{array}{ccc|c}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec w
\end{array}\right]</m>.
</p>
</li>
<li>
<p>
A pivot column in the <em>coefficient</em> matrix <m>\RREF \left[\begin{array}{cccc}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec w \\
\end{array}\right]</m> guarantees the linear independence
of <m>\{\vec v_1,\vec v_2,\vec v_3,\vec w\}</m>
by preventing contradictions.
The <em>pivot</em> column in the <em>coefficient</em> matrix
<m>\RREF \left[\begin{array}{cccc}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec w
\end{array}\right]</m>.
</p>
</li>
<li>
<p>
A non-pivot column in the <em>augmented</em> matrix <m>\RREF \left[\begin{array}{cccc|c}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec w &amp; \vec 0 \\
\end{array}\right]</m> guarantees the linear dependence
of <m>\{\vec v_1,\vec v_2,\vec v_3,\vec w\}</m>
by describing a linear combination of one vector in terms of the others.
The <em>non-pivot</em> column in the <em>augmented</em> matrix
<m>\RREF \left[\begin{array}{ccc|c}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec w
\end{array}\right]</m>.
</p>
</li>
<li>
<p>
A non-pivot column in the <em>coefficient</em> matrix <m>\RREF \left[\begin{array}{cccc}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec w \\
\end{array}\right]</m> guarantees the linear dependence
of <m>\{\vec v_1,\vec v_2,\vec v_3,\vec w\}</m>
by describing a linear combination of one vector in terms of the others.
The <em>non-pivot</em> column in the <em>coefficient</em> matrix
<m>\RREF \left[\begin{array}{cccc}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec w
\end{array}\right]</m>.
</p>
</li>
</ol>
</p>
</statement>
<answer>
<p>D.</p>
<p>
<m>\RREF \left[\begin{array}{cccc}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec w
\end{array}\right]</m> (or <m>\RREF \left[\begin{array}{cccc|c}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec w &amp; \vec 0
\end{array}\right]</m>) can be used to solve
<m>x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3 + x_4\vec w=\vec{0}</m>,
and the non-pivot column reveals it has infinitely-many solutions.
</p>
<p>
Note that (C) technically represents the equation
<m>x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3 =\vec w</m>
which isn't what was asked about.
</p>
</answer>
</task>
</activity>

<sage language="octave">
</sage>

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