Minor updates

Author: HengchengZhang

lectures/linear_equations.md

@@ -13,9 +13,6 @@ kernelspec:
 
 # Linear Equations and Matrix Algebra
 
-```{index} single: Linear Equations and Matrix Algebra
-```
-
 ## Overview
 
 Many problems in economics and finance require solving linear equations.
@@ -127,10 +124,7 @@ basics of vectors and matrices, in both theory and computation.
 
 
 
-## {index}`Vectors <single: Vectors>`
-
-```{index} single: Linear Algebra; Vectors
-```
+## Vectors
 
 A **vector** of length $n$ is just a sequence (or array, or tuple) of $n$ numbers, which we write as $x = (x_1, \ldots, x_n)$ or $x = \begin{bmatrix}x_1, \ldots, x_n\end{bmatrix}$.
 
@@ -175,9 +169,6 @@ plt.show()
 
 ### Vector operations
 
-```{index} single: Vectors; Operations
-```
-
 Sometimes we want to modify vectors.
 
 The two most common operators on vectors are addition and scalar
@@ -355,18 +346,12 @@ y = np.array((2, 4, 6))   # Converts tuple (2, 4, 6) into a NumPy array
 x + y                     # Add (element-by-element)
 ```
 
-```{code-cell} ipython3
+```{code-cell}
 4 * x                     # Scalar multiply
 ```
 
 ### Inner product and norm
 
-```{index} single: Vectors; Inner Product
-```
-
-```{index} single: Vectors; Norm
-```
-
 The **inner product** of vectors $x,y \in \mathbb R^n$ is defined as
 
 $$
@@ -413,9 +398,6 @@ np.linalg.norm(x)      # Norm of x, method two
 
 ## Matrix operations
 
-```{index} single: Matrix; Operations
-```
-
 When we discussed linear price systems, we mentioned using matrix algebra.
 
 Matrix algebra is similar to algebra for numbers.
@@ -617,9 +599,6 @@ It is a useful exercise to check the following:
 
 ### Matrices in NumPy
 
-```{index} single: Matrix; Numpy
-```
-
 NumPy arrays are also used as matrices, and have fast, efficient functions and methods for all the standard matrix operations.
 
 You can create them manually from tuples of tuples (or lists of lists) as follows
@@ -860,9 +839,6 @@ the following questions
 
 ## Solving systems of equations
 
-```{index} single: Matrix; Solving Systems of Equations
-```
-
 Recall again the system of equations {eq}`la_se`, which we write here again as
 
 ```{math}
@@ -1021,9 +997,6 @@ This is the solution to $Ax = b$ --- the solution we are looking for.
 
 ### Linear equations with NumPy
 
-```{index} single: Linear Algebra; SciPy
-```
-
 In the two good example we obtained the matrix equation,
 
 $$