- Understand matrix-vector multiplication through real-world examples.
- Explore the concept of linearity using grocery shopping data.
- Connect mathematical operations to practical decision-making.
- Build intuition for matrix transformations and scaling effects.
- Prepare for deeper study of matrix algebra and applications.
Grocery Shopping Example
Imagine a grocery store with 3 items:
- Bread: $2.50
- Milk: $3.20
- Apples: $1.80
Five shoppers purchase different quantities:
| Shopper | Bread | Milk | Apples |
|---|---|---|---|
| Alice | 2 | 1 | 4 |
| Ben | 1 | 2 | 3 |
| Clara | 3 | 0 | 2 |
| David | 2 | 3 | 1 |
| Emma | 1 | 1 | 5 |
Matrix-vector multiplication gives us:
\[ \begin{bmatrix} 2 & 1 & 4 \\ 1 & 2 & 3 \\ 3 & 0 & 2 \\ 2 & 3 & 1 \\ 1 & 1 & 5 \end{bmatrix} \cdot \begin{bmatrix} 2.50 \\ 3.20 \\ 1.80 \end{bmatrix} = \begin{bmatrix} 15.10 \\ 14.30 \\ 11.10 \\ 15.70 \\ 15.10 \end{bmatrix} \]
Each result is the total expense for one shopper. If prices double, the total paid also doubles—this is the essence of linearity.
Word Problems
- Problem 1: Sophia buys 4 bread, 2 milk, and 6 apples. What is her total expense?
- Problem 2: Ben buys 2 bread, 3 milk, and 1 apple. Olivia buys 1 bread, 2 milk, and 4 apples. Calculate their expenses.
- Problem 3: If prices change to Bread = $3.00, Milk = $3.50, Apples = $2.00, compute new expenses for Alice and Clara.
- Problem 4: Noah (3 bread, 2 milk, 5 apples), Ava (2 bread, 1 milk, 3 apples), Ethan (1 bread, 4 milk, 2 apples). Find their total expenses.
Exercises
- Question 1: Represent Sophia’s purchase as a vector and compute her total expense.
- Question 2: Create a matrix for Ben and Olivia’s purchases and multiply by the price vector.
- Question 3: Use the new price vector to compute updated totals for Alice and Clara.
- Question 4: Multiply the matrix of Noah, Ava, and Ethan’s purchases by the original price vector.
- Answer 1: \[ \begin{bmatrix} 4 & 2 & 6 \end{bmatrix} \cdot \begin{bmatrix} 2.50 \\ 3.20 \\ 1.80 \end{bmatrix} = 27.20 \]
- Answer 2: \[ \begin{bmatrix} 2 & 3 & 1 \\ 1 & 2 & 4 \end{bmatrix} \cdot \begin{bmatrix} 2.50 \\ 3.20 \\ 1.80 \end{bmatrix} = \begin{bmatrix} 17.90 \\ 17.70 \end{bmatrix} \]
- Answer 3: \[ \begin{bmatrix} 3.00 \\ 3.50 \\ 2.00 \end{bmatrix} \quad \Rightarrow \quad \text{Alice: } 21.50, \quad \text{Clara: } 13.00 \]
- Answer 4: \[ \begin{bmatrix} 3 & 2 & 5 \\ 2 & 1 & 3 \\ 1 & 4 & 2 \end{bmatrix} \cdot \begin{bmatrix} 2.50 \\ 3.20 \\ 1.80 \end{bmatrix} = \begin{bmatrix} 26.60 \\ 17.10 \\ 19.60 \end{bmatrix} \]