3. Transformations of the Plane

Video coming soon2 worked examples

Theory

A $2\times 2$ matrix can act as a linear transformation of the plane. To transform a point, write it as a column vector and pre-multiply by the matrix. The standard transformation matrices are:

Anticlockwise rotation by $\theta$ about the origin: $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$
Reflection in the $x$ -axis: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ ; in the line $y=x$ : $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
Scaling (dilatation) by factor $k$ : $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$

The Golden Rule: to combine transformations, multiply their matrices — but the transformation applied first goes on the right. “Do $A$ then $B$ ” has combined matrix $BA$ .

⚠️ Common Examiner Traps

Wrong order for a composition: “A then B” is $BA$ , not $AB$ — the second transformation sits on the left.
Rotation direction: the standard matrix is anticlockwise (positive $\theta$ ); a clockwise rotation uses $-\theta$ .
Row instead of column: the point must be a column vector so the multiplication is defined.

Worked examples

Example 1

Find the image of the point $(3, 1)$ under an anticlockwise rotation of $90^\circ$ about the origin.

Step 1: With $\theta = 90^\circ$ , $\cos 90^\circ = 0$ and $\sin 90^\circ = 1$ , so the rotation matrix is:

\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}

Step 2: Multiply the matrix by the point written as a column vector:

\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} (0)(3)+(-1)(1) \\ (1)(3)+(0)(1) \end{pmatrix} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}

The image is $(-1, 3)$ .

Example 2

A point is first reflected in the $x$ -axis, then rotated $90^\circ$ anticlockwise about the origin. Find the single matrix representing this combined transformation, and describe it.

Step 1: Let $R = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ (reflection, applied first) and $T = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ (rotation, applied second). “Reflect then rotate” is $TR$ .

Step 2: Multiply, with the rotation on the left:

TR = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

Step 3: The result $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is the matrix for a reflection in the line $y = x$ .