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incorporating vector equations into AT3 #901

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125 changes: 97 additions & 28 deletions source/linear-algebra/source/03-AT/03.ptx
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Original file line number Diff line number Diff line change
Expand Up @@ -159,35 +159,77 @@ the set of all vectors that transform into <m>\vec 0</m>?
<activity estimated-time='10'>
<introduction>
<p>
Let <m>T: \IR^3 \rightarrow \IR^2</m> be the linear transformation given by the
standard matrix
<me>T\left( \left[\begin{array}{c} x \\ y \\ z \end{array}\right]\right) = \left[\begin{array}{c} 3x+4y-z \\ x+2y+z \end{array}\right]</me>
Let <m>T: \IR^3 \rightarrow \IR^2</m> be the linear transformation with the following standard matrix:
<me>A=\left[\begin{array}{ccc} 3 &amp; 4 &amp; -1 \\ 1 &amp; 2 &amp; 1 \end{array}\right]
=\left[\begin{array}{ccc} T(\vec e_1) &amp; T(\vec e_2) &amp; T(\vec e_3)\end{array}\right].</me>
</p>
</introduction>
<task>
<task>
<statement>
<p>
Which of the following vectors is an element of <m>\ker T</m>?
</p>
<ol marker="A.">
<li>
<p>
<m>\left[\begin{array}{c}1\\1\\1\end{array}\right]</m>
</p>
</li>
<li>
<p>
<m>\left[\begin{array}{c}3\\-2\\1\end{array}\right]</m>
</p>
</li>
<li>
<p>
<m>\left[\begin{array}{c}4\\-3\\1\\1\end{array}\right]</m>
</p>
</li>
</ol>
</statement>
</task>
<task>
<statement>
<p>
In general, <m>\ker T</m> is the set of solutions to the equation:
<me>
T\left(\vec{x}\right)=T\left(\left[\begin{array}{c}x_1\\x_2\\x_3\end{array}\right]\right)=\left[\begin{array}{c}0\\0\end{array}\right].
</me>
Write down an equivalent <em>vector equation</em>, solve it, and describe <m>\ker T</m>.
</p>
</statement>
</task>
</activity>
<sage language="octave">
</sage>

<observation xml:id="observation-kernel-homogeneous-solution">
<statement>
<p>
Set
<m>
T\left(\left[\begin{array}{c}x\\y\\z\end{array}\right]\right)
=
\left[\begin{array}{c}0\\0\end{array}\right]
</m> to find a linear system of equations whose solution set is the kernel.
The kernel of a transformation <m>T</m>
is exactly the solution space of
the homogeneous equation <m>T(\vec{x})=\vec{0}</m>.
If its standard matrix is <m>A</m>, then we may write
<m>A\vec x=\vec 0</m> and use <m>\RREF[A\,|\,\vec 0]</m> to
find this kernel.
</p>
</task>
<task>
<p>
Use <m>\RREF(A)</m> to solve this homogeneous system of equations and find a basis
for the kernel of <m>T</m>.
In particular, the kernel is a subspace of the transformation's
domain, and has a basis which may be found as in
<xref ref="fact-solution-space-basis"/>:
<me>
\ker T=\left\{\left[\begin{array}{c}3a\\-2a\\a\end{array}\right]\middle|
a\in\IR\right\} \hspace{2em}
\text{Basis for }\ker T=\left\{\left[\begin{array}{c}3\\-2\\1\end{array}\right]\right\}.
</me>
</p>
</task>
</activity>
<sage language="octave">
</sage>
</statement>
</observation>

<activity estimated-time='10'>
<statement>
<p>
Let <m>T: \IR^4 \rightarrow \IR^3</m> be the linear transformation given by
Let <m>T: \IR^4 \rightarrow \IR^3</m> be the linear transformation whose standard matrix is:
<me> T\left(\left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \right) =
\left[\begin{array}{c} 2x+4y+2z-4w \\ -2x-4y+z+w \\ 3x+6y-z-4w\end{array}\right].</me>
</p>
Expand Down Expand Up @@ -372,23 +414,50 @@ the set of all vectors that are the result of using <m>T</m> to transform
<task>
<statement>
<p>
Determine if <m>\left[\begin{array}{c} 12 \\ 3 \\ 3 \end{array}\right]</m> belongs to
<m>\Im T</m>.
Which of the following statements is most helpful in deciding if <m>\left[\begin{array}{c} 12 \\ 3 \\ 3 \end{array}\right]</m> is an element of <m>\Im T</m>?
</p>
<ol marker="A.">
<li>
<p>
The equation <m>T\left(\left[\begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array}\right]\right)=\left[\begin{array}{c} 12 \\ 3 \\ 3 \end{array}\right]</m> has infinitely many solutions.
</p>
</li>
<li>
<p>
The equation <m>T\left(\left[\begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array}\right]\right)=\left[\begin{array}{c} 12 \\ 3 \\ 3 \end{array}\right]</m> has at least one solution.
</p>
</li>
<li>
<p>
The equation <m>T\left(\left[\begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array}\right]\right)=\left[\begin{array}{c} 12 \\ 3 \\ 3 \end{array}\right]</m> has no solutions.
</p>
</li>
<li>
<p>
The equation <m>T\left(\left[\begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array}\right]\right)=\left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right]</m> has at least one solution.
</p>
</li>
</ol>
</statement>
</task>
<task>
<statement>
<p>
Determine if <m>\left[\begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right]</m> belongs to
<m>\Im T</m>.
</p>
</statement>
<statement>
<p>
Translate your choice into a statement about a specific vector equation and use it to determine whether or not the vector <m>\left[\begin{array}{c} 12 \\ 3 \\ 3 \end{array}\right]</m> is an element of <m>\Im T</m>.
</p>
</statement>
</task>
<task>
<statement>
<p>
Determine whether or not the vector <m>\left[\begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right]</m> is an element of <m>\Im T</m> by analyzing an appropriate vector equation.
</p>
</statement>
</task>
<task>
<statement>
<p>
An arbitrary vector <m>\left[\begin{array}{c}\unknown\\\unknown\\\unknown\end{array}\right]</m> belongs to
In general, an arbitrary vector <m>\left[\begin{array}{c}\unknown\\\unknown\\\unknown\end{array}\right]</m> belongs to
<m>\Im T</m> provided the equation
<me>x_1 T(\vec{e}_1)+x_2 T(\vec{e}_2)+x_3T(\vec{e}_3)+x_4T(\vec{e}_4)=\vec{w}</me> has...
<ol marker="A.">
Expand Down