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Alter presentation of sample exercises #994

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b53975d
Tweak sample exercises
siwelwerd May 8, 2026
bb82521
Apply suggestion from @siwelwerd
siwelwerd May 8, 2026
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7 changes: 4 additions & 3 deletions source/linear-algebra/source/01-LE/samples/01.ptx
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<?xml version='1.0' encoding='UTF-8'?>
<example xml:id="sample-LE1">
<title>LE1</title>
<exercise xml:id="sample-LE1">
<title component="html">LE1</title>
<title component="print"><xref ref="LE1" text="global"/></title>
<introduction>
<p> Consider the vector equation <me>x_{1} \left[\begin{array}{c} 4 \\ -3 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ -3 \\ 3 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ 1 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} 18 \\ -7 \\ 15 \end{array}\right] = \left[\begin{array}{c} -11 \\ 5 \\ -9 \end{array}\right]</me></p>
</introduction>
Expand Down Expand Up @@ -34,4 +35,4 @@
</p>
</solution>
</task>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/01-LE/samples/02.ptx
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<?xml version='1.0' encoding='UTF-8'?>

<example xml:id="sample-LE2"><title>LE2</title>
<exercise xml:id="sample-LE2">
<title component="html">LE2</title>
<title component="print"><xref ref="LE2" text="global"/></title>
<task>
<introduction>
<p> For each of the following matrices, explain why it is not in reduced row echelon form. </p>
Expand Down Expand Up @@ -107,4 +109,4 @@
</p>
</solution>
</task>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/01-LE/samples/03.ptx
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<?xml version='1.0' encoding='UTF-8'?>
<example xml:id="sample-LE3"><title>LE3</title>
<exercise xml:id="sample-LE3">
<title component="html">LE3</title>
<title component="print"><xref ref="LE3" text="global"/></title>
<introduction>
<p> Consider each of the following systems of linear equations or vector equations. </p>
</introduction>
Expand Down Expand Up @@ -130,4 +132,4 @@
</solution>
</task>
</task>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/01-LE/samples/04.ptx
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<?xml version='1.0' encoding='UTF-8'?>
<example xml:id="sample-LE4"><title>LE4</title>
<exercise xml:id="sample-LE4">
<title component="html">LE4</title>
<title component="print"><xref ref="LE4" text="global"/></title>
<introduction>
<p> Consider the following vector equation. <me>x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 0 \\ -2 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} -5 \\ 1 \\ -5 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} 13 \\ -2 \\ 13 \\ 7 \end{array}\right] + x_{5} \left[\begin{array}{c} -14 \\ 3 \\ -14 \\ -5 \end{array}\right] = \left[\begin{array}{c} 18 \\ -3 \\ 18 \\ 9 \end{array}\right]</me> </p>
</introduction>
Expand Down Expand Up @@ -47,4 +49,4 @@
<p>Therefore, the solution set is <m>\left\{ \left[\begin{array}{c} 2 \, a - 3 \, b - c + 3 \\ a \\ 2 \, b - 3 \, c - 3 \\ b \\ c \end{array}\right] \,\middle|\, a,b,c \in\mathbb R \right\}</m>.</p>
</solution>
</task>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/02-EV/samples/01.ptx
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<?xml version='1.0' encoding='UTF-8'?>
<example xml:id="sample-EV1"><title>EV1</title>
<exercise xml:id="sample-EV1">
<title component="html">EV1</title>
<title component="print"><xref ref="EV1" text="global"/></title>
<introduction>
<p>Consider each of these claims about a vector equation.</p>
</introduction>
Expand Down Expand Up @@ -97,4 +99,4 @@ x_4\left[\begin{array}{c} -5 \\ 1 \\ -5 \end{array}\right] =
</solution>
</task>
</task>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/02-EV/samples/02.ptx
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<?xml version='1.0' encoding='UTF-8'?>
<example xml:id="sample-EV2"><title>EV2</title>
<exercise xml:id="sample-EV2">
<title component="html">EV2</title>
<title component="print"><xref ref="EV2" text="global"/></title>
<task>
<introduction>
<p> Consider the set of vectors <me>\left\{ \left[\begin{array}{c} -1 \\ 1 \\ 0 \\ 2 \end{array}\right] ,
Expand Down Expand Up @@ -137,4 +139,4 @@
</solution>
</task>
</task>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/02-EV/samples/03.ptx
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<?xml version='1.0' encoding='UTF-8'?>

<example xml:id="sample-EV3"><title>EV3</title>
<exercise xml:id="sample-EV3">
<title component="html">EV3</title>
<title component="print"><xref ref="EV3" text="global"/></title>
<introduction>
<p> Answer the following questions about Euclidean subspaces. </p>
</introduction>
Expand Down Expand Up @@ -95,4 +97,4 @@
</p>
</solution>
</task>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/02-EV/samples/04.ptx
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<?xml version='1.0' encoding='UTF-8'?>

<example xml:id="sample-EV4"><title>EV4</title>
<exercise xml:id="sample-EV4">
<title component="html">EV4</title>
<title component="print"><xref ref="EV4" text="global"/></title>
<task>
<introduction>
<p>
Expand Down Expand Up @@ -116,4 +118,4 @@
</solution>
</task>
</task>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/02-EV/samples/05.ptx
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<?xml version='1.0' encoding='UTF-8'?>
<example xml:id="sample-EV5"><title>EV5</title>
<exercise xml:id="sample-EV5">
<title component="html">EV5</title>
<title component="print"><xref ref="EV5" text="global"/></title>
<task>
<introduction>
<p> Consider the set of vectors <me>\left\{ \left[\begin{array}{c} 1 \\ -2 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ -6 \\ 0 \\ 3 \end{array}\right] , \left[\begin{array}{c} 5 \\ -5 \\ -2 \\ 4 \end{array}\right] \right\}</me> </p>
Expand Down Expand Up @@ -79,4 +81,4 @@
</solution>
</task>
</task>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/02-EV/samples/06.ptx
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<?xml version='1.0' encoding='UTF-8'?>

<example xml:id="sample-EV6"><title>EV6</title>
<exercise xml:id="sample-EV6">
<title component="html">EV6</title>
<title component="print"><xref ref="EV6" text="global"/></title>
<introduction>
<p>
Consider the subspace
Expand Down Expand Up @@ -63,4 +65,4 @@ Since this (and thus every other) basis has three vectors in it, the dimension o
</p>
</solution>
</task>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/02-EV/samples/07.ptx
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<?xml version='1.0' encoding='UTF-8'?>

<example xml:id="sample-EV7"><title>EV7</title>
<exercise xml:id="sample-EV7">
<title component="html">EV7</title>
<title component="print"><xref ref="EV7" text="global"/></title>
<introduction>
<p>
Consider the homogeneous system of equations
Expand Down Expand Up @@ -86,4 +88,4 @@ a basis for the solution space is
</p>
</solution>
</task>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/03-AT/samples/01.ptx
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<?xml version='1.0' encoding='UTF-8'?>

<example xml:id="sample-AT1"><title>AT1</title>
<exercise xml:id="sample-AT1">
<title component="html">AT1</title>
<title component="print"><xref ref="AT1" text="global"/></title>
<statement>
<p> Answer the following questions about transformations.
<ol>
Expand Down Expand Up @@ -124,4 +126,4 @@
</li>
</ol></p>
</solution>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/03-AT/samples/02.ptx
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<?xml version='1.0' encoding='UTF-8'?>
<example xml:id="sample-AT2"><title>AT2</title>
<exercise xml:id="sample-AT2">
<title component="html">AT2</title>
<title component="print"><xref ref="AT2" text="global"/></title>
<statement>
<p>
<ol>
Expand Down Expand Up @@ -52,4 +54,4 @@ S\left(\left[\begin{array}{c} -2 \\ 1 \\ 3 \\ 2 \end{array}\right] \right) = -2S
</ol>
</p>
</solution>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/03-AT/samples/03.ptx
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<?xml version='1.0' encoding='UTF-8'?>

<example xml:id="sample-AT3"><title>AT3</title>
<exercise xml:id="sample-AT3">
<title component="html">AT3</title>
<title component="print"><xref ref="AT3" text="global"/></title>
<statement>
<p>
Let <m>T: \IR^4 \rightarrow \IR^3</m> be the linear transformation given by
Expand Down Expand Up @@ -99,4 +101,4 @@ of the rank and nullity of <m>T</m> is the dimension of the domain of <m>T</m>.
</ol>
</p>
</solution>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/03-AT/samples/04.ptx
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<?xml version='1.0' encoding='UTF-8'?>

<example xml:id="sample-AT4"><title>AT4</title>
<exercise xml:id="sample-AT4">
<title component="html">AT4</title>
<title component="print"><xref ref="AT4" text="global"/></title>
<statement>
<p>
Let <m>T: \IR^4 \rightarrow \IR^3</m> be the linear transformation given by the standard matrix <m>\left[\begin{array}{cccc} 1 &amp; 3 &amp; 2 &amp; -3 \\ 2 &amp; 4 &amp; 6 &amp; -10 \\ 1 &amp; 6 &amp; -1 &amp; 3 \end{array}\right]</m>.
Expand Down Expand Up @@ -35,4 +37,4 @@ Since there are only two pivots, the image (i.e. the span of the columns) is a 2
</ol>
</p>
</solution>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/03-AT/samples/05.ptx
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<?xml version='1.0' encoding='UTF-8'?>


<example xml:id="sample-AT5"><title>AT5</title>
<exercise xml:id="sample-AT5">
<title component="html">AT5</title>
<title component="print"><xref ref="AT5" text="global"/></title>
<statement>
<p>
Let <m>V</m> be the set of all pairs of numbers <m>(x,y)</m> of real numbers together with the following operations:
Expand Down Expand Up @@ -128,4 +130,4 @@ Since these ordered pairs are different, the property fails to hold.
</ol>
</p>
</solution>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/03-AT/samples/06.ptx
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<?xml version='1.0' encoding='UTF-8'?>
<example xml:id="sample-AT6"><title>AT6</title>
<exercise xml:id="sample-AT6">
<title component="html">AT6</title>
<title component="print"><xref ref="AT6" text="global"/></title>
<statement>
<p>
<ol>
Expand Down Expand Up @@ -76,4 +78,4 @@ The system has (infintely many) nontrivial solutions, so we that the set of poly
is linearly <em>dependent</em>.
</p>
</solution>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/04-MX/samples/01.ptx
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<?xml version='1.0' encoding='UTF-8'?>

<example xml:id="sample-MX1"><title>MX1</title>
<exercise xml:id="sample-MX1">
<title component="html">MX1</title>
<title component="print"><xref ref="MX1" text="global"/></title>
<introduction>
<p>
Of the following three matrices, only two may be multiplied.
Expand Down Expand Up @@ -95,4 +97,4 @@ ans =
</p>
</solution>
</task>
</example>
</exercise>
7 changes: 4 additions & 3 deletions source/linear-algebra/source/04-MX/samples/02.ptx
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<?xml version='1.0' encoding='UTF-8'?>


<example xml:id="sample-MX2">
<title>MX2</title>
<exercise xml:id="sample-MX2">
<title component="html">MX2</title>
<title component="print"><xref ref="MX2" text="global"/></title>
<introduction>
<p> Consider each of the following matrices. </p>
</introduction>
Expand Down Expand Up @@ -90,4 +91,4 @@
</solution>
</task>
</task>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/04-MX/samples/03.ptx
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<?xml version='1.0' encoding='UTF-8'?>

<example xml:id="sample-MX3"><title>MX3</title>
<exercise xml:id="sample-MX3">
<title component="html">MX3</title>
<title component="print"><xref ref="MX3" text="global"/></title>
<introduction>
<p>
Let <m>\mathcal{B}=\setList{\begin{bmatrix}-2\\-2\\1\end{bmatrix},\begin{bmatrix}-1\\-2\\-1\end{bmatrix},\begin{bmatrix}1\\3\\2\end{bmatrix}}</m>, and <m>\vec{v}=\begin{bmatrix}1\\2\\3\end{bmatrix}</m>.
Expand Down Expand Up @@ -47,4 +49,4 @@
</p>
</solution>
</task>
</example>
</exercise>
7 changes: 4 additions & 3 deletions source/linear-algebra/source/04-MX/samples/04.ptx
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<?xml version='1.0' encoding='UTF-8'?>
<example xml:id="sample-MX4">
<title>MX4</title>
<exercise xml:id="sample-MX4">
<title component="html">MX4</title>
<title component="print"><xref ref="MX4" text="global"/></title>
<task>
<statement>
<p> Give a <m>3 \times 3</m> matrix <m>C</m> that may be used to perform the row operation <m>-5 R_1 \to R_1</m>. </p>
Expand Down Expand Up @@ -123,4 +124,4 @@ ans =
</sage>
</answer>
</task>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/05-GT/samples/01.ptx
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<?xml version='1.0' encoding='UTF-8'?>

<example xml:id="sample-GT1"><title>GT1</title>
<exercise xml:id="sample-GT1">
<title component="html">GT1</title>
<title component="print"><xref ref="GT1" text="global"/></title>
<introduction>
<p>
Let <m>A</m> be <em>any</em> <m>4 \times 4</m> matrix with determinant
Expand Down Expand Up @@ -93,4 +95,4 @@ so that <m>M=RA</m>. It follows that
</p>
</solution>
</task>
</example>
</exercise>
6 changes: 4 additions & 2 deletions source/linear-algebra/source/05-GT/samples/02.ptx
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<?xml version='1.0' encoding='UTF-8'?>


<example xml:id="sample-GT2"><title>GT2</title>
<exercise xml:id="sample-GT2">
<title component="html">GT2</title>
<title component="print"><xref ref="GT2" text="global"/></title>
<statement>
<p>
Show how to compute the determinant of the matrix
Expand Down Expand Up @@ -80,4 +82,4 @@ the determinant to a <m>3\times 3</m> matrix and then applying a formula:
</mrow>
</md>
</solution>
</example>
</exercise>
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