Transform a plane systematically

A 2 × 2 matrix maps a column vector to another column vector. Its columns are the images of the two standard basis vectors, so two simple tests reveal the whole linear transformation.

To apply A and then B, multiply BA. A determinant describes signed area scaling: a negative sign reverses orientation, while zero collapses the plane onto a line or point.

\displaystyle R_\theta=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}

WORKED EXAMPLE

Rotate (2, 1) by 90° anticlockwise about the origin.

At 90°, the rotation matrix has rows (0, −1) and (1, 0).
Multiplying by (2, 1)ᵀ gives (−1, 2)ᵀ.

Assumed knowledge

Multiply matrices by column vectors and by other matrices, and calculate a 2 × 2 determinant. Recognise Cartesian rotations, reflections and dilations.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Transform a plane systematically

Rotate (2, 1) by 90° anticlockwise about the origin.

Put the idea to work.