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jkostiuk merged 1 commit into from
Apr 23, 2026
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Add p tags so no bare text is in statements #965

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6 changes: 3 additions & 3 deletions source/linear-algebra/source/04-MX/01.ptx
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Expand Up @@ -348,7 +348,7 @@ in terms of matrix multiplication.</p>
<title>Mathematical Writing Explorations</title>
<exploration>
<statement>
Construct 3 matrices, <m>A,B,\mbox{ and } C</m>, such that
<p>Construct 3 matrices, <m>A,B,\mbox{ and } C</m>, such that</p>
<ul>
<li><m>AB:\mathbb{R}^4\rightarrow\mathbb{R}^2</m></li>
<li><m>BC:\mathbb{R}^2\rightarrow\mathbb{R}^3</m></li>
Expand All @@ -358,7 +358,7 @@ in terms of matrix multiplication.</p>
</statement></exploration>
<exploration>
<statement>
Construct 3 examples of matrix multiplication, with all matrix dimensions at least 2.
<p>Construct 3 examples of matrix multiplication, with all matrix dimensions at least 2.</p>
<ul>
<li>Where <m>A</m> and <m>B</m> are not square, but <m>AB</m> is square.</li>
<li>Where <m>AB = BA</m>.</li>
Expand All @@ -368,7 +368,7 @@ in terms of matrix multiplication.</p>
</exploration>
<exploration>
<statement>
Use the included map in this problem.
<p>Use the included map in this problem.</p>
<figure xml:id="MX1-adjacency">
<caption>Adjacency map, showing roads between 5 cities</caption>
<image xml:id="MX1-image-adjacency" width="50%">
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34 changes: 19 additions & 15 deletions source/linear-algebra/source/04-MX/02.ptx
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Expand Up @@ -496,7 +496,7 @@ Is the matrix <m>\left[\begin{array}{ccc} 2 &amp; 3 &amp; 1 \\ -1 &amp; -4 &amp;
<title>Mathematical Writing Explorations</title>
<exploration>
<statement>
Assume <m>A</m> is an <m>n \times n</m> matrix. Prove the following are equivalent. Some of these results you have proven previously.
<p>Assume <m>A</m> is an <m>n \times n</m> matrix. Prove the following are equivalent. Some of these results you have proven previously.</p>
<ul>
<li> <m>A</m> row reduces to the identity matrix.</li>
<li> For any choice of <m>\vec{b} \in \mathbb{R}^n</m>, the system of equations represented by the augmented matrix <m>[A|\vec{b}]</m> has a unique solution.</li>
Expand All @@ -517,31 +517,35 @@ Is the matrix <m>\left[\begin{array}{ccc} 2 &amp; 3 &amp; 1 \\ -1 &amp; -4 &amp;
</ul>
</statement>
</exploration>
<exploration>
<exploration>
<statement>
Use row reduction to find the inverse of the following general matrix. Give conditions on which this inverse exists.
<p>Use row reduction to find the inverse of the following general matrix. Give conditions on which this inverse exists.</p>
<me>\left[\begin{array}{ccc}1 &amp; b &amp; c \\ d &amp; e &amp; f \\ g &amp; h &amp; i \end{array}\right]</me>
</statement>
</exploration>

<exploration>
<statement>Assume that <m>H</m> is invertible, and that <m>HG</m> is the zero matrix. Prove that <m>G</m> must be the zero matrix. Would this still be true if <m>H</m> were not invertible?
</statement>
<exploration>
<statement>
<p>Assume that <m>H</m> is invertible, and that <m>HG</m> is the zero matrix. Prove that <m>G</m> must be the zero matrix. Would this still be true if <m>H</m> were not invertible?</p>
</statement>
</exploration>

<exploration>
<statement>If <m>H</m> is invertible and <m>r \in \mathbb{R}</m>, what is the inverse of <m>rH</m>?
</statement>
<exploration>
<statement>
<p>If <m>H</m> is invertible and <m>r \in \mathbb{R}</m>, what is the inverse of <m>rH</m>?</p>
</statement>
</exploration>

<exploration>
<statement>If <m>H</m> and <m>G</m> are invertible, is <m>H^{-1} + G^{-1} = (H+G)^{-1}</m>?
</statement>
<exploration>
<statement>
<p>If <m>H</m> and <m>G</m> are invertible, is <m>H^{-1} + G^{-1} = (H+G)^{-1}</m>?</p>
</statement>
</exploration>

<exploration>
<statement>Prove that if <m>A</m>, <m>P</m>, and <m>Q</m> are invertible with <m>PAQ = I</m>, then <m>A^{-1} = QP</m>.
</statement>
<exploration>
<statement>
<p>Prove that if <m>A</m>, <m>P</m>, and <m>Q</m> are invertible with <m>PAQ = I</m>, then <m>A^{-1} = QP</m>.</p>
</statement>
</exploration>

</subsection>
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7 changes: 4 additions & 3 deletions source/linear-algebra/source/05-GT/02.ptx
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Expand Up @@ -475,20 +475,21 @@ Based on the previous activities, which technique is easier for computing determ
<title>Mathematical Writing Explorations</title>
<exploration>
<statement>
Prove that the equation of a line in the plane, through points <m>(x_1,y_1), (x_2,y_2)</m>, when <m>x_1 \neq x_2</m> is given by the equation
<p>Prove that the equation of a line in the plane, through points <m>(x_1,y_1), (x_2,y_2)</m>, when <m>x_1 \neq x_2</m> is given by the equation
<m>\mbox{det}\left(\begin{array}{ccc}x&amp;y&amp;1\\x_1&amp;y_1&amp;1\\x_2&amp;y_2&amp;1\end{array}\right) = 0.</m>
</p>
</statement>

</exploration>
<exploration>
<statement>
Show that, if an <m>n \times n</m> matrix <m>M</m> has a non-zero determinant, then any <m>\vec{v} \in \mathbb{R}^n</m> can be represented as a linear combination of the columns of <m>M</m>.
<p>Show that, if an <m>n \times n</m> matrix <m>M</m> has a non-zero determinant, then any <m>\vec{v} \in \mathbb{R}^n</m> can be represented as a linear combination of the columns of <m>M</m>.</p>
</statement>

</exploration>
<exploration>
<statement>
What is the smallest number of zeros necessary to place in a <m>4 \times 4</m> matrix, and the placement of those zeros, such that the matrix has a zero determinant?
<p>What is the smallest number of zeros necessary to place in a <m>4 \times 4</m> matrix, and the placement of those zeros, such that the matrix has a zero determinant?</p>
</statement>

</exploration>
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2 changes: 1 addition & 1 deletion source/linear-algebra/source/05-GT/03.ptx
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Expand Up @@ -455,7 +455,7 @@ which of these eigenvalues is associated to the eigenvector <m>\left[\begin{arra
<title>Mathematical Writing Explorations</title>
<exploration>
<statement>
What are the maximum and minimum number of eigenvalues associated with an <m>n \times n</m> matrix? Write small examples to convince yourself you are correct, and then prove this in generality.
<p>What are the maximum and minimum number of eigenvalues associated with an <m>n \times n</m> matrix? Write small examples to convince yourself you are correct, and then prove this in generality.</p>
</statement>
</exploration>
</subsection>
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2 changes: 1 addition & 1 deletion source/linear-algebra/source/05-GT/04.ptx
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Expand Up @@ -173,7 +173,7 @@ associated with the eigenvalue <m>2</m>.
<title>Mathematical Writing Explorations</title>
<exploration>
<statement>
Given a matrix <m>A</m>, let <m>\{\vec{v_1},\vec{v_2},\ldots,\vec{v_n}\}</m> be the eigenvectors with associated distinct eigenvalues <m>\{\lambda_1,\lambda_2,\ldots, \lambda_n\}</m>. Prove the set of eigenvectors is linearly independent.
<p>Given a matrix <m>A</m>, let <m>\{\vec{v_1},\vec{v_2},\ldots,\vec{v_n}\}</m> be the eigenvectors with associated distinct eigenvalues <m>\{\lambda_1,\lambda_2,\ldots, \lambda_n\}</m>. Prove the set of eigenvectors is linearly independent.</p>
</statement>
</exploration>
</subsection>
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4 changes: 2 additions & 2 deletions source/precalculus/source/03-LF/03.ptx
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Expand Up @@ -327,7 +327,7 @@
</task>
<task>
<statement>
After we plot the <m>y</m>-intercept, we can use the slope to find another point. Find another point and graph the resulting line.
<p>After we plot the <m>y</m>-intercept, we can use the slope to find another point. Find another point and graph the resulting line.</p>
</statement>
<answer>
<p>
Expand Down Expand Up @@ -436,7 +436,7 @@

<task>
<statement>
After we plot this point, we can use the slope to find another point. Find another point and graph the resulting line.
<p>After we plot this point, we can use the slope to find another point. Find another point and graph the resulting line.</p>
</statement>
<answer>
<p>
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2 changes: 1 addition & 1 deletion source/precalculus/source/04-PR/04.ptx
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Expand Up @@ -30,7 +30,7 @@

<activity xml:id="identify-improper-rational-functions">
<statement>
Label each of the following rational functions as either <term>proper</term> or <term>improper</term>.
<p>Label each of the following rational functions as either <term>proper</term> or <term>improper</term>.</p>
<ol marker="A.">
<li><m> \dfrac{x^3+x}{x^2+4}</m></li>
<li><m> \dfrac{3}{x^2+3x+4}</m></li>
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