From 8edede9251bfae0f5ebefa31af989269adc81143 Mon Sep 17 00:00:00 2001 From: Drew Lewis Date: Wed, 22 Apr 2026 18:16:10 +0000 Subject: [PATCH] Add p tags so no bare text is in statements --- source/linear-algebra/source/04-MX/01.ptx | 6 ++-- source/linear-algebra/source/04-MX/02.ptx | 34 +++++++++++++---------- source/linear-algebra/source/05-GT/02.ptx | 7 +++-- source/linear-algebra/source/05-GT/03.ptx | 2 +- source/linear-algebra/source/05-GT/04.ptx | 2 +- source/precalculus/source/03-LF/03.ptx | 4 +-- source/precalculus/source/04-PR/04.ptx | 2 +- 7 files changed, 31 insertions(+), 26 deletions(-) diff --git a/source/linear-algebra/source/04-MX/01.ptx b/source/linear-algebra/source/04-MX/01.ptx index 1a410101e..3914ffbc2 100644 --- a/source/linear-algebra/source/04-MX/01.ptx +++ b/source/linear-algebra/source/04-MX/01.ptx @@ -348,7 +348,7 @@ in terms of matrix multiplication.

Mathematical Writing Explorations - Construct 3 matrices, A,B,\mbox{ and } C, such that +

Construct 3 matrices, A,B,\mbox{ and } C, such that

  • AB:\mathbb{R}^4\rightarrow\mathbb{R}^2
  • BC:\mathbb{R}^2\rightarrow\mathbb{R}^3
    • @@ -358,7 +358,7 @@ in terms of matrix multiplication.

    - Construct 3 examples of matrix multiplication, with all matrix dimensions at least 2. +

    Construct 3 examples of matrix multiplication, with all matrix dimensions at least 2.

    • Where A and B are not square, but AB is square.
    • Where AB = BA.
      • @@ -368,7 +368,7 @@ in terms of matrix multiplication.

      -Use the included map in this problem. +

      Use the included map in this problem.

      Adjacency map, showing roads between 5 cities diff --git a/source/linear-algebra/source/04-MX/02.ptx b/source/linear-algebra/source/04-MX/02.ptx index b7a08c652..5d7c2c141 100644 --- a/source/linear-algebra/source/04-MX/02.ptx +++ b/source/linear-algebra/source/04-MX/02.ptx @@ -496,7 +496,7 @@ Is the matrix \left[\begin{array}{ccc} 2 & 3 & 1 \\ -1 & -4 & Mathematical Writing Explorations - Assume A is an n \times n matrix. Prove the following are equivalent. Some of these results you have proven previously. +

      Assume A is an n \times n matrix. Prove the following are equivalent. Some of these results you have proven previously.

      • A row reduces to the identity matrix.
      • For any choice of \vec{b} \in \mathbb{R}^n, the system of equations represented by the augmented matrix [A|\vec{b}] has a unique solution.
        • @@ -517,31 +517,35 @@ Is the matrix \left[\begin{array}{ccc} 2 & 3 & 1 \\ -1 & -4 &
      - + - Use row reduction to find the inverse of the following general matrix. Give conditions on which this inverse exists. +

      Use row reduction to find the inverse of the following general matrix. Give conditions on which this inverse exists.

      \left[\begin{array}{ccc}1 & b & c \\ d & e & f \\ g & h & i \end{array}\right]       - - Assume that H is invertible, and that HG is the zero matrix. Prove that G must be the zero matrix. Would this still be true if H were not invertible? - + + +

      Assume that H is invertible, and that HG is the zero matrix. Prove that G must be the zero matrix. Would this still be true if H were not invertible?

      +       - - If H is invertible and r \in \mathbb{R}, what is the inverse of rH? - + + +

      If H is invertible and r \in \mathbb{R}, what is the inverse of rH?

      +       - - If H and G are invertible, is H^{-1} + G^{-1} = (H+G)^{-1}? - + + +

      If H and G are invertible, is H^{-1} + G^{-1} = (H+G)^{-1}?

      +       - - Prove that if A, P, and Q are invertible with PAQ = I, then A^{-1} = QP. - + + +

      Prove that if A, P, and Q are invertible with PAQ = I, then A^{-1} = QP.

      +       diff --git a/source/linear-algebra/source/05-GT/02.ptx b/source/linear-algebra/source/05-GT/02.ptx index 262436fe5..df5529dcb 100644 --- a/source/linear-algebra/source/05-GT/02.ptx +++ b/source/linear-algebra/source/05-GT/02.ptx @@ -475,20 +475,21 @@ Based on the previous activities, which technique is easier for computing determ Mathematical Writing Explorations - Prove that the equation of a line in the plane, through points (x_1,y_1), (x_2,y_2), when x_1 \neq x_2 is given by the equation +

      Prove that the equation of a line in the plane, through points (x_1,y_1), (x_2,y_2), when x_1 \neq x_2 is given by the equation \mbox{det}\left(\begin{array}{ccc}x&y&1\\x_1&y_1&1\\x_2&y_2&1\end{array}\right) = 0. +

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

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

             - What is the smallest number of zeros necessary to place in a 4 \times 4 matrix, and the placement of those zeros, such that the matrix has a zero determinant? +

      What is the smallest number of zeros necessary to place in a 4 \times 4 matrix, and the placement of those zeros, such that the matrix has a zero determinant?

             diff --git a/source/linear-algebra/source/05-GT/03.ptx b/source/linear-algebra/source/05-GT/03.ptx index c43db94a6..82948be05 100644 --- a/source/linear-algebra/source/05-GT/03.ptx +++ b/source/linear-algebra/source/05-GT/03.ptx @@ -455,7 +455,7 @@ which of these eigenvalues is associated to the eigenvector \left[\begin{arra Mathematical Writing Explorations - What are the maximum and minimum number of eigenvalues associated with an n \times n matrix? Write small examples to convince yourself you are correct, and then prove this in generality. +

      What are the maximum and minimum number of eigenvalues associated with an n \times n matrix? Write small examples to convince yourself you are correct, and then prove this in generality.

             diff --git a/source/linear-algebra/source/05-GT/04.ptx b/source/linear-algebra/source/05-GT/04.ptx index ed6310d7c..233f8295f 100644 --- a/source/linear-algebra/source/05-GT/04.ptx +++ b/source/linear-algebra/source/05-GT/04.ptx @@ -173,7 +173,7 @@ associated with the eigenvalue 2. Mathematical Writing Explorations - Given a matrix A, let \{\vec{v_1},\vec{v_2},\ldots,\vec{v_n}\} be the eigenvectors with associated distinct eigenvalues \{\lambda_1,\lambda_2,\ldots, \lambda_n\}. Prove the set of eigenvectors is linearly independent. +

      Given a matrix A, let \{\vec{v_1},\vec{v_2},\ldots,\vec{v_n}\} be the eigenvectors with associated distinct eigenvalues \{\lambda_1,\lambda_2,\ldots, \lambda_n\}. Prove the set of eigenvectors is linearly independent.

             diff --git a/source/precalculus/source/03-LF/03.ptx b/source/precalculus/source/03-LF/03.ptx index be09264a3..aada8a403 100644 --- a/source/precalculus/source/03-LF/03.ptx +++ b/source/precalculus/source/03-LF/03.ptx @@ -327,7 +327,7 @@ - After we plot the y-intercept, we can use the slope to find another point. Find another point and graph the resulting line. +

      After we plot the y-intercept, we can use the slope to find another point. Find another point and graph the resulting line.

      @@ -436,7 +436,7 @@ - After we plot this point, we can use the slope to find another point. Find another point and graph the resulting line. +

      After we plot this point, we can use the slope to find another point. Find another point and graph the resulting line.

      diff --git a/source/precalculus/source/04-PR/04.ptx b/source/precalculus/source/04-PR/04.ptx index 2f9f13d77..90129185d 100644 --- a/source/precalculus/source/04-PR/04.ptx +++ b/source/precalculus/source/04-PR/04.ptx @@ -30,7 +30,7 @@ - Label each of the following rational functions as either proper or improper. +

      Label each of the following rational functions as either proper or improper.

      1. \dfrac{x^3+x}{x^2+4}
      2. \dfrac{3}{x^2+3x+4}