Python-Workshop-3-NumPy.md

@@ -231,9 +231,18 @@ Problem](https://www.cs.odu.edu/~tkennedy/cs417/s21/Public/approximationExample2
 
 The code snippets in this section are extracted from [least_squares.py](https://git-community.cs.odu.edu/tkennedy/python-workshop/-/blob/master/NumPy/least_squares.py).
 
+The first line is the NumPy import. By convention the `numpy` module is given
+the alias `np`.
+
 ```python
 import numpy as np
+```
+
+The first function (i.e., `print_matrices`) was debugging/instrumentation
+logic. It was used to display the `XTX` (X-Transpose-X) and `XTY`
+(X-Transpose-Y) matrices during development.
 
+```python
 def print_matrices(matrix_XTX, matrix_XTY):
     """
     Print the XTX and XTY matrices
@@ -245,8 +254,50 @@ def print_matrices(matrix_XTX, matrix_XTY):
     print()
     print("{:*^40}".format("XTY"))
     print(matrix_XTY)
+```
+
+Next... let us jump to the `main` function and...
+
+  1. set up the input data (points)
+
+    ```python
+    points = [(0., 0.), (1., 1.), (2., 4.)]
+    ```
+
+  2. create the `X` matrix
+    ```python
+    matrix_X = np.array([[1., 0., 0.],
+                         [1., 1., 1.],
+                         [1., 2., 4.]])
+    ```
+
+  3. create the `Y` matrix 
+
+    ```python
+    matrix_Y = np.array([0,
+                         1,
+                         4])
+    ```
+
+  4. compute X-transpose
+
+    ```python
+    matrix_XT = matrix_X.transpose()
+    ```
 
+After setting everything up... it is time for a couple matrix multiplications.
 
+```python
+    matrix_XTX = np.matmul(matrix_XT, matrix_X)
+    matrix_XTY = np.matmul(matrix_XT, matrix_Y)
+```
+
+The augmented `XTX|XTY` matrix is what we are going to solve. *Note that the
+`@` operator is often used instead of `np_matmul`.*
+
+---
+
+```python
 def _backsolve(matrix_XTX, matrix_XTY):
 
     num_rows, _ = matrix_XTX.shape