From f4df65cc2bb8f927d7f815a52a3bf2db5f0eda54 Mon Sep 17 00:00:00 2001
From: Longye Tian <longye.tian@outlook.com>
Date: Tue, 23 Jul 2024 17:15:04 +1000
Subject: [PATCH 1/6] update the example environment

---
 lectures/linear_equations.md | 24 ++++++++++++++----------
 1 file changed, 14 insertions(+), 10 deletions(-)

diff --git a/lectures/linear_equations.md b/lectures/linear_equations.md
index dc4469b6e..fb209ead1 100644
--- a/lectures/linear_equations.md
+++ b/lectures/linear_equations.md
@@ -141,11 +141,10 @@ column vectors.
 
 The set of all $n$-vectors is denoted by $\mathbb R^n$.
 
-For example, 
-
+```{prf:example}
 * $\mathbb R^2$ is the plane --- the set of pairs $(x_1, x_2)$.
 * $\mathbb R^3$ is 3 dimensional space --- the set of vectors $(x_1, x_2, x_3)$.
-
+```
 Often vectors are represented visually as arrows from the origin to the point.
 
 Here's a visualization.
@@ -185,8 +184,7 @@ multiplication, which we now describe.
 
 When we add two vectors, we add them element-by-element.
 
-For example,
-
+```{prf:example}
 $$
 \begin{bmatrix}
     4 \\
@@ -208,6 +206,7 @@ $$
     1
 \end{bmatrix}.
 $$
+```
 
 In general,
 
@@ -273,8 +272,7 @@ plt.show()
 
 Scalar multiplication is an operation that multiplies a vector $x$ with a scalar elementwise.
 
-For example,
-
+```{prf: example}
 $$
 -2
 \begin{bmatrix}
@@ -292,6 +290,7 @@ $$
     14
 \end{bmatrix}.
 $$
+```
 
 More generally, it takes a number $\gamma$ and a vector $x$ and produces
 
@@ -429,8 +428,7 @@ matrices.
 
 Scalar multiplication and addition are generalizations of the vector case:
 
-Here is an example of scalar multiplication
-
+```{prf: example}
 $$
 3
 \begin{bmatrix}
@@ -443,6 +441,7 @@ $$
     0 & 15
 \end{bmatrix}.
 $$
+```
 
 In general for a number $\gamma$ and any matrix $A$,
 
@@ -461,6 +460,7 @@ $$
 \end{bmatrix}.
 $$
 
+```{prf: example}
 Consider this example of matrix addition,
 
 $$
@@ -479,6 +479,7 @@ $$
     7 & 12
 \end{bmatrix}.
 $$
+```
 
 In general,
 
@@ -518,6 +519,7 @@ $j$-th column of $B$.
 If $A$ is $n \times k$ and $B$ is $j \times m$, then to multiply $A$ and $B$
 we require $k = j$, and the resulting matrix $A B$ is $n \times m$.
 
+```{prf: example}
 Here's an example of a $2 \times 2$ matrix multiplied by a $2 \times 1$ vector.
 
 $$
@@ -536,6 +538,7 @@ Ax =
     a_{21}x_1 + a_{22}x_2
 \end{bmatrix}
 $$
+```
 
 As an important special case, consider multiplying $n \times k$
 matrix $A$ and $k \times 1$ column vector $x$.
@@ -839,6 +842,7 @@ In matrix form, the system {eq}`la_se` becomes
     \end{bmatrix}.
 ```
 
+```{prf:example}
 For example, {eq}`n_eq_sys_la` has this form with 
 
 $$ 
@@ -848,7 +852,7 @@ $$
     \quad \text{and} \quad
     x = p.
 $$
-
+```
 
 When considering problems such as {eq}`la_gf`, we need to ask at least some of
 the following questions

From f924ed9cbb38e54c4623e5de5a48f9001e805152 Mon Sep 17 00:00:00 2001
From: Longye Tian <longye.tian@outlook.com>
Date: Tue, 23 Jul 2024 18:04:03 +1000
Subject: [PATCH 2/6] update example environment

---
 lectures/linear_equations.md | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/lectures/linear_equations.md b/lectures/linear_equations.md
index fb209ead1..72e0741f1 100644
--- a/lectures/linear_equations.md
+++ b/lectures/linear_equations.md
@@ -272,7 +272,7 @@ plt.show()
 
 Scalar multiplication is an operation that multiplies a vector $x$ with a scalar elementwise.
 
-```{prf: example}
+```{prf:example}
 $$
 -2
 \begin{bmatrix}
@@ -428,7 +428,7 @@ matrices.
 
 Scalar multiplication and addition are generalizations of the vector case:
 
-```{prf: example}
+```{prf:example}
 $$
 3
 \begin{bmatrix}
@@ -460,7 +460,7 @@ $$
 \end{bmatrix}.
 $$
 
-```{prf: example}
+```{prf:example}
 Consider this example of matrix addition,
 
 $$
@@ -519,7 +519,7 @@ $j$-th column of $B$.
 If $A$ is $n \times k$ and $B$ is $j \times m$, then to multiply $A$ and $B$
 we require $k = j$, and the resulting matrix $A B$ is $n \times m$.
 
-```{prf: example}
+```{prf:example}
 Here's an example of a $2 \times 2$ matrix multiplied by a $2 \times 1$ vector.
 
 $$

From 82b403b5816af2d628471ae7fcabaeeeced39268 Mon Sep 17 00:00:00 2001
From: Longye Tian <longye.tian@outlook.com>
Date: Tue, 23 Jul 2024 18:30:20 +1000
Subject: [PATCH 3/6] update example environment

---
 lectures/linear_equations.md | 13 +++++++++++++
 1 file changed, 13 insertions(+)

diff --git a/lectures/linear_equations.md b/lectures/linear_equations.md
index 72e0741f1..c862d16f3 100644
--- a/lectures/linear_equations.md
+++ b/lectures/linear_equations.md
@@ -142,6 +142,8 @@ column vectors.
 The set of all $n$-vectors is denoted by $\mathbb R^n$.
 
 ```{prf:example}
+:label:ex_dim
+
 * $\mathbb R^2$ is the plane --- the set of pairs $(x_1, x_2)$.
 * $\mathbb R^3$ is 3 dimensional space --- the set of vectors $(x_1, x_2, x_3)$.
 ```
@@ -185,6 +187,8 @@ multiplication, which we now describe.
 When we add two vectors, we add them element-by-element.
 
 ```{prf:example}
+:label:ex_add
+
 $$
 \begin{bmatrix}
     4 \\
@@ -273,6 +277,8 @@ plt.show()
 Scalar multiplication is an operation that multiplies a vector $x$ with a scalar elementwise.
 
 ```{prf:example}
+:label:ex_mul
+
 $$
 -2
 \begin{bmatrix}
@@ -429,6 +435,8 @@ matrices.
 Scalar multiplication and addition are generalizations of the vector case:
 
 ```{prf:example}
+:label:ex_asm
+
 $$
 3
 \begin{bmatrix}
@@ -461,6 +469,8 @@ $$
 $$
 
 ```{prf:example}
+:label:ex_ma
+
 Consider this example of matrix addition,
 
 $$
@@ -520,6 +530,8 @@ If $A$ is $n \times k$ and $B$ is $j \times m$, then to multiply $A$ and $B$
 we require $k = j$, and the resulting matrix $A B$ is $n \times m$.
 
 ```{prf:example}
+:label:ex_2dmul
+
 Here's an example of a $2 \times 2$ matrix multiplied by a $2 \times 1$ vector.
 
 $$
@@ -843,6 +855,7 @@ In matrix form, the system {eq}`la_se` becomes
 ```
 
 ```{prf:example}
+:label:ex_gls
 For example, {eq}`n_eq_sys_la` has this form with 
 
 $$ 

From b50aa75bbbc7bd2a2aebde7e0ea55e9a52b2da24 Mon Sep 17 00:00:00 2001
From: Longye Tian <longye.tian@outlook.com>
Date: Tue, 23 Jul 2024 18:40:13 +1000
Subject: [PATCH 4/6] update example

---
 lectures/linear_equations.md | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/lectures/linear_equations.md b/lectures/linear_equations.md
index c862d16f3..19fff2f0c 100644
--- a/lectures/linear_equations.md
+++ b/lectures/linear_equations.md
@@ -142,7 +142,7 @@ column vectors.
 The set of all $n$-vectors is denoted by $\mathbb R^n$.
 
 ```{prf:example}
-:label:ex_dim
+:label: ex_dim
 
 * $\mathbb R^2$ is the plane --- the set of pairs $(x_1, x_2)$.
 * $\mathbb R^3$ is 3 dimensional space --- the set of vectors $(x_1, x_2, x_3)$.
@@ -187,7 +187,7 @@ multiplication, which we now describe.
 When we add two vectors, we add them element-by-element.
 
 ```{prf:example}
-:label:ex_add
+:label: ex_add
 
 $$
 \begin{bmatrix}
@@ -277,7 +277,7 @@ plt.show()
 Scalar multiplication is an operation that multiplies a vector $x$ with a scalar elementwise.
 
 ```{prf:example}
-:label:ex_mul
+:label: ex_mul
 
 $$
 -2
@@ -435,7 +435,7 @@ matrices.
 Scalar multiplication and addition are generalizations of the vector case:
 
 ```{prf:example}
-:label:ex_asm
+:label: ex_asm
 
 $$
 3
@@ -469,7 +469,7 @@ $$
 $$
 
 ```{prf:example}
-:label:ex_ma
+:label: ex_ma
 
 Consider this example of matrix addition,
 
@@ -530,7 +530,7 @@ If $A$ is $n \times k$ and $B$ is $j \times m$, then to multiply $A$ and $B$
 we require $k = j$, and the resulting matrix $A B$ is $n \times m$.
 
 ```{prf:example}
-:label:ex_2dmul
+:label: ex_2dmul
 
 Here's an example of a $2 \times 2$ matrix multiplied by a $2 \times 1$ vector.
 
@@ -855,7 +855,7 @@ In matrix form, the system {eq}`la_se` becomes
 ```
 
 ```{prf:example}
-:label:ex_gls
+:label: ex_gls
 For example, {eq}`n_eq_sys_la` has this form with 
 
 $$ 

From bd962da3f665c844f48da247b19ae419694ccaa5 Mon Sep 17 00:00:00 2001
From: Matt McKay <mmcky@users.noreply.github.com>
Date: Tue, 23 Jul 2024 20:52:37 +1000
Subject: [PATCH 5/6] Update lectures/linear_equations.md

---
 lectures/linear_equations.md | 1 +
 1 file changed, 1 insertion(+)

diff --git a/lectures/linear_equations.md b/lectures/linear_equations.md
index 19fff2f0c..6b50e3ca6 100644
--- a/lectures/linear_equations.md
+++ b/lectures/linear_equations.md
@@ -147,6 +147,7 @@ The set of all $n$-vectors is denoted by $\mathbb R^n$.
 * $\mathbb R^2$ is the plane --- the set of pairs $(x_1, x_2)$.
 * $\mathbb R^3$ is 3 dimensional space --- the set of vectors $(x_1, x_2, x_3)$.
 ```
+
 Often vectors are represented visually as arrows from the origin to the point.
 
 Here's a visualization.

From d4ec437c01d249f1d3dd6ed1d6267d684b2efa1d Mon Sep 17 00:00:00 2001
From: Longye Tian <longye.tian@outlook.com>
Date: Wed, 24 Jul 2024 10:12:25 +1000
Subject: [PATCH 6/6] add lecture label

---
 lectures/linear_equations.md | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/lectures/linear_equations.md b/lectures/linear_equations.md
index 6b50e3ca6..4b9ccb0fb 100644
--- a/lectures/linear_equations.md
+++ b/lectures/linear_equations.md
@@ -142,7 +142,7 @@ column vectors.
 The set of all $n$-vectors is denoted by $\mathbb R^n$.
 
 ```{prf:example}
-:label: ex_dim
+:label: le_ex_dim
 
 * $\mathbb R^2$ is the plane --- the set of pairs $(x_1, x_2)$.
 * $\mathbb R^3$ is 3 dimensional space --- the set of vectors $(x_1, x_2, x_3)$.
@@ -188,7 +188,7 @@ multiplication, which we now describe.
 When we add two vectors, we add them element-by-element.
 
 ```{prf:example}
-:label: ex_add
+:label: le_ex_add
 
 $$
 \begin{bmatrix}
@@ -278,7 +278,7 @@ plt.show()
 Scalar multiplication is an operation that multiplies a vector $x$ with a scalar elementwise.
 
 ```{prf:example}
-:label: ex_mul
+:label: le_ex_mul
 
 $$
 -2
@@ -436,7 +436,7 @@ matrices.
 Scalar multiplication and addition are generalizations of the vector case:
 
 ```{prf:example}
-:label: ex_asm
+:label: le_ex_asm
 
 $$
 3
@@ -470,7 +470,7 @@ $$
 $$
 
 ```{prf:example}
-:label: ex_ma
+:label: le_ex_ma
 
 Consider this example of matrix addition,
 
@@ -531,7 +531,7 @@ If $A$ is $n \times k$ and $B$ is $j \times m$, then to multiply $A$ and $B$
 we require $k = j$, and the resulting matrix $A B$ is $n \times m$.
 
 ```{prf:example}
-:label: ex_2dmul
+:label: le_ex_2dmul
 
 Here's an example of a $2 \times 2$ matrix multiplied by a $2 \times 1$ vector.
 
@@ -856,7 +856,7 @@ In matrix form, the system {eq}`la_se` becomes
 ```
 
 ```{prf:example}
-:label: ex_gls
+:label: le_ex_gls
 For example, {eq}`n_eq_sys_la` has this form with 
 
 $$