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286 changes: 176 additions & 110 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 @@ -150,134 +150,203 @@
<activity estimated-time='10'>
<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],
<me>\vec v_1=\left[\begin{array}{c}-2 \\ 0 \\ 3\end{array}\right],
\vec v_2=\left[\begin{array}{c}1 \\ 3 \\ 0\end{array}\right],
\text{ and }
\vec v_3=\left[\begin{array}{c}-2 \\ 5 \\ 4\end{array}\right]
\vec v_3=\left[\begin{array}{c}-2 \\ 6 \\ 6\end{array}\right]
</me>.
</p>
</introduction>
<task>
<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...
<ol marker="A.">
<li><p>linearly dependent: at least one vector is a linear combination of others</p></li>
<li><p>linearly independent: no vector is a linear combination of others</p></li>
</ol>
</p>
<statement>
<p>
Let <m> \vec v_4 = 3\vec v_1 - \vec v_2 - 2 \vec v_3 =
\left[\begin{array}{c}-3 \\ -15 \\ -3\end{array}\right]</m>.
The set <m>\{\vec v_1,\vec v_2,\vec v_3,\vec v_4\}</m> is...
<ol marker="A.">
<li><p>linearly dependent: at least one vector is a linear combination of others</p></li>
<li><p>linearly independent: no vector is a linear combination of others</p></li>
</ol>
</p>
</statement>
<answer>
<p>
A. We explicitly constructed <m>\vec v_4</m> as a linear combination of the others.
(It turns out <m>\vec v_3</m> is a linear combo as well.)
</p>
</answer>
</task>
<task>
<p>
Find <me>\RREF \left[\begin{array}{ccc|c}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec w \\
\end{array}\right]=
\RREF \left[\begin{array}{ccc|c}
-2 &amp; 1 &amp;-2 &amp; \unknown \\
0 &amp; 3 &amp; 5 &amp; \unknown \\
0 &amp;0 &amp;4 &amp; \unknown
\end{array}\right]= \unknown .</me>
</p>
<p>
What does this tell you about solution set for the vector equation
<m>x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3 =\vec w</m>?
<ol marker="A.">
<li>
<p>
It is inconsistent.
</p>
</li>
<li>
<p>
It is consistent with one solution.
</p>
</li>
<li>
<p>
It is consistent with infinitely many solutions.
</p>
</li>
</ol>
</p>
</task>
<statement>
<p>
Find <me>\RREF \left[\begin{array}{ccc|c}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec v_4 \\
\end{array}\right]=
\RREF \left[\begin{array}{ccc|c}
-2 &amp; 1 &amp;-2 &amp; -3 \\
0 &amp; 3 &amp; 6 &amp; -15 \\
3 &amp;0 &amp;6 &amp; -3
\end{array}\right]= \unknown .</me>
</p>
<p>
What does this guarantee about the solution set for the vector equation
<m>x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3 =\vec v_4</m>?
<ol marker="A.">
<li>
<p>
It is inconsistent.
</p>
</li>
<li>
<p>
It is consistent with one solution.
</p>
</li>
<li>
<p>
It is consistent with infinitely many solutions.
</p>
</li>
<li>
<p>
None of these.
</p>
</li>
</ol>
</p>
</statement>
<answer>
<p>
C. Since <me>
\RREF \left[\begin{array}{ccc|c}
-2 &amp; 1 &amp;-2 &amp; -3 \\
0 &amp; 3 &amp; 6 &amp; -15 \\
3 &amp;0 &amp;6 &amp; -3
\end{array}\right] =
\RREF \left[\begin{array}{ccc|c}
1 &amp; 0 &amp;2 &amp; -1 \\
0 &amp; 1 &amp; 2 &amp; -5 \\
0 &amp;0 &amp;0 &amp; 0
\end{array}\right]</me>
has no contradiction and a free variable, it has infinitely-many solutions.
</p>
</answer>
</task>
<task>
<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\\
\end{array}\right]=
\RREF \left[\begin{array}{cccc|c}
-2 &amp; 1 &amp;-2 &amp; \unknown &amp; 0\\
0 &amp; 3 &amp; 5 &amp; \unknown &amp; 0 \\
0 &amp;0 &amp;4 &amp; \unknown &amp; 0
\end{array}\right]= \unknown .</me>
</p>
<p>
What does this tell you about solution set for the vector equation
<m>x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3 + x_4\vec w=\vec{0}</m>?
<ol marker="A.">
<li>
<p>
It is inconsistent.
</p>
</li>
<li>
<p>
It is consistent with one solution.
</p>
</li>
<li>
<p>
It is consistent with infinitely many solutions.
</p>
</li>
</ol>
</p>
</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>?
Let <m>\vec w=\left[\begin{array}{c}\unknown \\ \unknown \\ \unknown\end{array}\right]</m>
be some arbitrary vector in <m>\mathbb R^3</m>.
Consider <me>\RREF \left[\begin{array}{cccc}
\vec v_1 &amp; \vec v_2 &amp; \vec v_3 &amp; \vec v_4 \\
\end{array}\right]=
\RREF \left[\begin{array}{cccc}
-2 &amp; 1 &amp;-2 &amp; -3 \\
0 &amp; 3 &amp; 6 &amp; -15 \\
3 &amp;0 &amp;6 &amp; -3
\end{array}\right]= \unknown .</me>
</p>
<p>
What does this guarantee about the solution set for the vector equation
<m>x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3 + x_4\vec v_4=\vec w</m>?
<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.
It is inconsistent.
</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.
It is consistent with exactly one solution.
</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.
It is not consistent with exactly one solution.
</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.
It is consistent with infinitely many solutions.
</p>
</li>
</ol>
</p>
</task>
</activity>
</statement>
<answer>
<p>
D. Due to the row of zeros, it might be inconsistent. But if it is consistent,
the free variable column guarantees it will have infinitely-many solutions.
</p>
</answer>
</task>
<task>
<statement>
<p>
Which of the following describes any solution to the equation
<m>x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3 + x_4\vec v_4=\vec{w}</m>
for any <em>linearly dependent</em> set <m>\{\vec v_1,\vec v_2,\vec v_3,\vec v_4\}</m>
and some vector <m>\vec w\in \mathbb R^n</m>?
<ol marker="A.">
<li>
<p>
No such solution can exist.
</p>
</li>
<li>
<p>
It cannot be unique.
</p>
</li>
<li>
<p>
It must be unique.
</p>
</li>
</ol>
</p>
</statement>
<answer>
<p>
B. If we assume the solution exists, the previous answer shows us that it cannot be the only one,
due to the free variable column.
</p>
</answer>
</task>
<task>
<statement>
<p>
Which of the following describes any solution to the equation
<m>x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3 + x_4\vec v_4=\vec{w}</m>
for any <em>linearly independent</em> set <m>\{\vec v_1,\vec v_2,\vec v_3,\vec v_4\}</m>
and some vector <m>\vec w\in \mathbb R^n</m>?
<ol marker="A.">
<li>
<p>
No such solution can exist.
</p>
</li>
<li>
<p>
It cannot be unique.
</p>
</li>
<li>
<p>
It must be unique.
</p>
</li>
</ol>
</p>
</statement>
<answer>
<p>
C. If we assume the solution exists, it must be the only one,
due to the lack of a free variable column.
</p>
</answer>
</task>
</activity>
<sage language="octave">
</sage>

Expand All @@ -286,18 +355,15 @@ by describing a linear combination of one vector in terms of the others.
<p>
For any vector space,
the set <m>\{\vec v_1,\dots\vec v_n\}</m> is linearly dependent if and only
if the vector equation <m>x_1\vec v_1+ x_2 \vec v_2+\dots+x_n\vec v_n=\vec{0}</m> is consistent with
infinitely many solutions.
if the vector equation <m>x_1\vec v_1+ x_2 \vec v_2+\dots+x_n\vec v_n=\vec w</m> cannot have
a unique solution for any <m>\vec w</m>.
</p>
<p>
Likewise, the set of vectors
On the other hand, the set of vectors
<m>\{\vec v_1,\dots\vec v_n\}</m> is linearly independent
if and only the vector equation <me>x_1\vec v_1+ x_2 \vec v_2 + \cdots + x_n\vec v_n = \vec{0}</me>
has exactly one solution: <m>\left[\begin{array}{c}
x_1 \\ \vdots \\ x_n
\end{array}\right]=\left[\begin{array}{c}
0 \\ \vdots \\ 0
\end{array}\right]</m>.
if and only if any solution to the vector equation
<me>x_1\vec v_1+ x_2 \vec v_2 + \cdots + x_n\vec v_n = \vec w</me>
must be unique for any <m>\vec w</m>.
</p>
</statement>
</fact>
Expand All @@ -322,8 +388,8 @@ by describing a linear combination of one vector in terms of the others.
\left[\begin{array}{c}4\\3\\0\\1\end{array}\right]
\right\}
</me>
is linearly dependent (the part that shows its linear system has
infinitely many solutions).
is linearly dependent (the part that shows its linear system cannot have
a unique solution).
</p>
</statement>
</activity>
Expand Down