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Lesson 8: Matrix-Vector Multiplication & Linear Transformations

1. Matrix × Vector Multiplication

To multiply matrix A (m×n) by vector v (n×1), result is m×1 vector.

Rule: each entry of result is dot product of row of A with vector v.

Example:

\[ \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix} \begin{pmatrix} 7 \\ 8 \end{pmatrix} = \begin{pmatrix} 1·7 + 2·8 \\ 3·7 + 4·8 \\ 5·7 + 6·8 \end{pmatrix} = \begin{pmatrix} 23 \\ 53 \\ 83 \end{pmatrix} \]

Exercise 1

Compute:
\[ \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 4 \\ 5 \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} \] x = , y =

2. Linear Transformations

A function T that maps vectors to vectors is linear if it preserves addition and scalar multiplication:

Every linear transformation from \( \mathbb{R}^{n} \) to \( \mathbb{R}^{m} \) can be represented by an \( m \times n \) matrix.

Examples:

Exercise 2

Which transformation is represented by the matrix \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\)?

3. Matrix Multiplication (brief intro)

To compose two transformations, multiply their matrices.

Order matters: AB ≠ BA in general (transformations don’t commute).

Example: rotation then scaling ≠ scaling then rotation.

Exercise 3

Which properties does matrix multiplication have? (Select all that apply)
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