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jkostiukStevenClontz
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StevenClontz
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incorporating vector equations into AT3 (#901)
* incorporating vector equations into AT3 * Update source/linear-algebra/source/03-AT/03.ptx Co-authored-by: Steven Clontz <steven.clontz@gmail.com> * Update source/linear-algebra/source/03-AT/03.ptx Co-authored-by: Steven Clontz <steven.clontz@gmail.com> * Update source/linear-algebra/source/03-AT/03.ptx Co-authored-by: Steven Clontz <steven.clontz@gmail.com> --------- Co-authored-by: Steven Clontz <steven.clontz@gmail.com>
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‎source/linear-algebra/source/03-AT/03.ptx‎

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@@ -165,35 +165,79 @@ the set of all vectors that transform into <m>\vec 0</m>?
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<activity estimated-time='10'>
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<introduction>
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<p>
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Let <m>T: \IR^3 \rightarrow \IR^2</m> be the linear transformation given by the
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standard matrix
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<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>
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Let <m>T: \IR^3 \rightarrow \IR^2</m> be the linear transformation with the following standard matrix:
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<me>A=\left[\begin{array}{ccc} 3 &amp; 4 &amp; -1 \\ 1 &amp; 2 &amp; 1 \end{array}\right]
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=\left[\begin{array}{ccc} T(\vec e_1) &amp; T(\vec e_2) &amp; T(\vec e_3)\end{array}\right].</me>
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</p>
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</introduction>
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<task>
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<task>
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<statement>
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<p>
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Which of the following vectors is an element of <m>\ker T</m>?
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</p>
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<ol marker="A." cols="3">
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<li>
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<p>
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<m>\left[\begin{array}{c}1\\1\\1\end{array}\right]</m>
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</p>
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</li>
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<li>
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<p>
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<m>\left[\begin{array}{c}3\\-2\\1\end{array}\right]</m>
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</p>
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</li>
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<li>
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<p>
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<m>\left[\begin{array}{c}4\\-3\\1\\1\end{array}\right]</m>
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</p>
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</li>
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</ol>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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In general, <m>\ker T</m> is the set of solutions to the equation:
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<me>
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T\left(\vec{x}\right)=T\left(\left[\begin{array}{c}x_1\\x_2\\x_3\end{array}\right]\right)
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=x_1\left[\begin{array}{c}\unknown\\\unknown\end{array}\right]
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+x_2\left[\begin{array}{c}\unknown\\\unknown\end{array}\right]+x_3\left[\begin{array}{c}\unknown\\\unknown\end{array}\right]=\left[\begin{array}{c}0\\0\end{array}\right].
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</me>
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Complete and solve this vector equation to find <m>\ker T</m>.
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</p>
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</statement>
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</task>
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</activity>
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<sage language="octave">
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</sage>
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<observation xml:id="observation-kernel-homogeneous-solution">
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<statement>
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<p>
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Set
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<m>
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T\left(\left[\begin{array}{c}x\\y\\z\end{array}\right]\right)
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=
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\left[\begin{array}{c}0\\0\end{array}\right]
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</m> to find a linear system of equations whose solution set is the kernel.
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The kernel of a transformation <m>T</m>
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is exactly the solution space of
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the homogeneous equation <m>T(\vec{x})=\vec{0}</m>.
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If its standard matrix is <m>A</m>, then we may write
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<m>A\vec x=\vec 0</m> and use <m>\RREF[A\,|\,\vec 0]</m> to
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find this kernel.
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</p>
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</task>
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<task>
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<p>
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Use <m>\RREF(A)</m> to solve this homogeneous system of equations and find a basis
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for the kernel of <m>T</m>.
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In particular, the kernel is a subspace of the transformation's
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domain, and has a basis which may be found as in
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<xref ref="fact-solution-space-basis"/>:
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<me>
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\ker T=\left\{\left[\begin{array}{c}3a\\-2a\\a\end{array}\right]\middle|
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a\in\IR\right\} \hspace{2em}
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\text{Basis for }\ker T=\left\{\left[\begin{array}{c}3\\-2\\1\end{array}\right]\right\}.
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</me>
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</p>
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</task>
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</activity>
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<sage language="octave">
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</sage>
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</statement>
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</observation>
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<activity estimated-time='10'>
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<statement>
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<p>
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Let <m>T: \IR^4 \rightarrow \IR^3</m> be the linear transformation given by
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Let <m>T: \IR^4 \rightarrow \IR^3</m> be the linear transformation whose standard matrix is:
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<me> T\left(\left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \right) =
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\left[\begin{array}{c} 2x+4y+2z-4w \\ -2x-4y+z+w \\ 3x+6y-z-4w\end{array}\right].</me>
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</p>
@@ -389,25 +433,52 @@ the set of all vectors that are the result of using <m>T</m> to transform
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<task>
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<statement>
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<p>
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Determine if <m>\left[\begin{array}{c} 12 \\ 3 \\ 3 \end{array}\right]</m> belongs to
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<m>\Im T</m>.
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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>?
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</p>
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<ol marker="A.">
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<li>
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<p>
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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.
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</p>
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</li>
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<li>
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<p>
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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.
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</p>
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</li>
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<li>
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<p>
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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.
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</p>
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</li>
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<li>
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<p>
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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.
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</p>
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</li>
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</ol>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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Determine if <m>\left[\begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right]</m> belongs to
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<m>\Im T</m>.
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</p>
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</statement>
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<statement>
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<p>
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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>.
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</p>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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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.
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</p>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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An arbitrary vector <m>\left[\begin{array}{c}\unknown\\\unknown\\\unknown\end{array}\right]</m> belongs to
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In general, an arbitrary vector <m>\left[\begin{array}{c}\unknown\\\unknown\\\unknown\end{array}\right]</m> belongs to
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<m>\Im T</m> provided the equation
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<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...
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<me>x_1 T(\vec{e}_1)+x_2 T(\vec{e}_2)+x_3T(\vec{e}_3)+x_4T(\vec{e}_4)=\left[\begin{array}{c}\unknown\\\unknown\\\unknown\end{array}\right]</me> has...
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<ol marker="A.">
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<li>no solutions.</li>
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<li>exactly one solution.</li>

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