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Real-World Examples for Matrix-Vector Multiplication

  • 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

This forms the price vector:

\[ \mathbf{p} = \begin{bmatrix} 2.50 \\ 3.20 \\ 1.80 \end{bmatrix} \]

Five shoppers purchase different quantities:

\[ A = \begin{bmatrix} 2 & 1 & 4 \\ 1 & 2 & 3 \\ 3 & 0 & 2 \\ 2 & 3 & 1 \\ 1 & 1 & 5 \end{bmatrix} \quad \text{(Rows: Alice, Ben, Clara, David, Emma)} \]

Matrix-vector multiplication gives us:

\[ A \cdot \mathbf{p} = \begin{bmatrix} 15.10 \\ 14.30 \\ 11.10 \\ 15.70 \\ 15.10 \end{bmatrix} \]

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: Liam 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 Liam 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: \[ \mathbf{x} = \begin{bmatrix} 4 & 2 & 6 \end{bmatrix} \cdot \begin{bmatrix} 2.50 \\ 3.20 \\ 1.80 \end{bmatrix} = 27.20 \]
  • 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: \[ \mathbf{p}_{\text{new}} = \begin{bmatrix} 3.00 \\ 3.50 \\ 2.00 \end{bmatrix} \quad \Rightarrow \quad \text{Alice: } 21.50, \quad \text{Clara: } 13.00 \]
  • Answer 4: \[ \begin{bmatrix} 3 & 2 & 5 \\ 2 & 1 & 3 \\ 1 & 4 & 2 \end{bmatrix} \cdot \begin{bmatrix} 2.50 \\ 3.20 \\ 1.80 \end{bmatrix} = \begin{bmatrix} 26.60 \\ 17.10 \\ 19.60 \end{bmatrix} \]

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