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.