Question 1179680
Absolutely! Let's break down how to represent Min Lee's weighted average as a dot product and how to calculate class averages using matrix multiplication.

**1. Min Lee's Weighted Average as a Dot Product:**

* **Min's Test Scores:** [84, 78, 91, 80, 72, 88, 83]
* **Weights:** The first six tests have a weight of 1, and the last test has a weight of 2. So the weights are [1, 1, 1, 1, 1, 1, 2].
* **Total Weight:** 1 + 1 + 1 + 1 + 1 + 1 + 2 = 8

To calculate Min's weighted average as a dot product, we need to normalize the weights by dividing each weight by the total weight:

* **Normalized Weights:** [1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 2/8] = [0.125, 0.125, 0.125, 0.125, 0.125, 0.125, 0.25]

Now, we can calculate the dot product of Min's test scores and the normalized weights:

* Weighted Average = (84 * 0.125) + (78 * 0.125) + (91 * 0.125) + (80 * 0.125) + (72 * 0.125) + (88 * 0.125) + (83 * 0.25)
* Weighted Average = 10.5 + 9.75 + 11.375 + 10 + 9 + 11 + 20.75
* Weighted Average = 82.375

This is the same as:

```
import numpy as np

# Min's test scores
min_scores = np.array([84, 78, 91, 80, 72, 88, 83])

# Weights for each test
weights = np.array([1, 1, 1, 1, 1, 1, 2]) 

# Calculate Min's final average using dot product
min_average = np.dot(min_scores, weights) / sum(weights) 

print(f"Min's final average: {min_average}")
```

**2. Class List of Test Averages Using Matrix Multiplication:**

Let's say the teacher has a class of 'm' students and the same 'n' number of tests. We can represent the test scores in a matrix called 'Scores', where:

* Scores is an m x n matrix (m rows, n columns).
* Each row represents a student's test scores.
* Each column represents the scores for a particular test.

We can also represent the normalized weights in a column vector called 'Weights', where:

* Weights is an n x 1 matrix (n rows, 1 column).
* Each element represents the normalized weight for a test.

To get the class list of test averages, we can multiply the 'Scores' matrix by the 'Weights' vector:

* Averages = Scores * Weights

The result will be an m x 1 matrix (a column vector) where each element represents a student's weighted average.

Here's how it would look in Python using NumPy:

```python
import numpy as np

# Example: 3 students, 7 tests
scores = np.array([
    [84, 78, 91, 80, 72, 88, 83],  # Min Lee's scores
    [90, 85, 78, 92, 88, 95, 90],  # Another student's scores
    [75, 82, 70, 78, 80, 75, 81]   # Another student's scores
])

weights = np.array([1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 2/8])

class_averages = np.dot(scores, weights)

print("Class Averages:", class_averages)
```

In essence, matrix multiplication allows the teacher to efficiently calculate the weighted averages for all students in the class with a single operation.