[linear_equations]Update the example environment

lectures/linear_equations.md

@@ -141,11 +141,12 @@ column vectors.
 
 The set of all $n$-vectors is denoted by $\mathbb R^n$.
 
-For example, 
+```{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 $(x_1, x_2, x_3)$.
-
+```
 Often vectors are represented visually as arrows from the origin to the point.
 
 Here's a visualization.
@@ -185,7 +186,8 @@ multiplication, which we now describe.
 
 When we add two vectors, we add them element-by-element.
 
-For example,
+```{prf:example}
+:label: ex_add
 
 $$
 \begin{bmatrix}
@@ -208,6 +210,7 @@ $$
  1
 \end{bmatrix}.
 $$
+```
 
 In general,
 
@@ -273,7 +276,8 @@ plt.show()
 
 Scalar multiplication is an operation that multiplies a vector $x$ with a scalar elementwise.
 
-For example,
+```{prf:example}
+:label: ex_mul
 
 $$
 -2
@@ -292,6 +296,7 @@ $$
  14
 \end{bmatrix}.
 $$
+```
 
 More generally, it takes a number $\gamma$ and a vector $x$ and produces
 
@@ -429,7 +434,8 @@ matrices.
 
 Scalar multiplication and addition are generalizations of the vector case:
 
-Here is an example of scalar multiplication
+```{prf:example}
+:label: ex_asm
 
 $$
 3
@@ -443,6 +449,7 @@ $$
  0 & 15
 \end{bmatrix}.
 $$
+```
 
 In general for a number $\gamma$ and any matrix $A$,
 
@@ -461,6 +468,9 @@ $$
 \end{bmatrix}.
 $$
 
+```{prf:example}
+:label: ex_ma
+
 Consider this example of matrix addition,
 
 $$
@@ -479,6 +489,7 @@ $$
  7 & 12
 \end{bmatrix}.
 $$
+```
 
 In general,
 
@@ -518,6 +529,9 @@ $j$-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}
+:label: ex_2dmul
+
 Here's an example of a $2 \times 2$ matrix multiplied by a $2 \times 1$ vector.
 
 $$
@@ -536,6 +550,7 @@ Ax =
  a_{21}x_1 + a_{22}x_2
 \end{bmatrix}
 $$
+```
 
 As an important special case, consider multiplying $n \times k$
 matrix $A$ and $k \times 1$ column vector $x$.
@@ -839,6 +854,8 @@ In matrix form, the system {eq}`la_se` becomes
  \end{bmatrix}.
 ```
 
+```{prf:example}
+:label: ex_gls
 For example, {eq}`n_eq_sys_la` has this form with 
 
 $$ 
@@ -848,7 +865,7 @@ $$
  \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