2.2.1 Transformations using a Matrix

Introduction to Matrix Transformations

Matrix transformations are a powerful way to represent geometric transformations in two dimensions. Each transformation can be expressed as a specific $2×2$ matrix, which, when multiplied by a coordinate vector, results in a new position vector. These transformations include reflection, rotation, enlargement, and stretch, all centered at the origin unless stated otherwise.

Key Transformations and Their Matrices

Reflections

Reflections flip points across a line. The matrix for a reflection depends on the axis or line of reflection:

Reflection in the x-axis:

$$\text{Matrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$

Reflection in the y-axis:

$$\text{Matrix} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$$

Reflection in the line $y = x$:

$$\text{Matrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$

Reflection in the line $y = -x$:

$$\text{Matrix} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$$

Rotations

Rotations turn points about the origin by a specified angle $\theta$

Rotation through angle $\theta$ (anticlockwise):

$$\text{Matrix} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$$

For specific angles:

90$$° anticlockwise rotation:

$$\text{Matrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$

$180°$ rotation:

$$\text{Matrix} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$$

$270°$ anticlockwise (or $90°$ clockwise):

$$\text{Matrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$

Stretches

Stretches scale distances from the origin along one axis.

Stretch parallel to the $x-axis$ by a factor $k$:

$$\text{Matrix} = \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix}$$

Stretch parallel to the $y-axis$ by a factor $k$:

$$\text{Matrix} = \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix}$$

Enlargement

Enlargement scales distances from the origin by a factor $k$.

Enlargement by scale factor $k$.

$$\text{Matrix} = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$$

Combining Transformations

When transformations are combined, the order of application matters. The resulting transformation matrix is obtained by multiplying the matrices of the individual transformations:

If a transformation represented by matrix $A$ is followed by another represented by matrix $B$, the combined transformation is represented by $BA$, not $AB$.

Worked Examples

Note Summary