Matrices as linear transformations in two and three dimensions (rotations, reflections, enlargements, stretches and shears), composition by multiplication, invariant points and lines, and the determinant as an area or volume scale factor.

What this dot point is asking

OCR wants you to read a matrix as a linear transformation: know the standard $2 \times 2$ matrices for rotations, reflections, enlargements, stretches and shears (and the $3 \times 3$ matrices for rotations and reflections in space), compose transformations by multiplying their matrices in the right order, find invariant points and invariant lines, and use the determinant as the area or volume scale factor.

Matrices as transformations and the column rule

A linear transformation fixes the origin and sends straight lines to straight lines. The key insight is that the columns of the matrix are the images of the standard basis vectors: the first column is where $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ goes, the second column where $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ goes. This lets you build a matrix from a described transformation, or read off a transformation from a matrix.

Standard transformations in two and three dimensions

Besides rotations and reflections, you should recognise a stretch parallel to an axis (for example $\begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix}$ stretches in the $x$-direction by factor $a$) and a shear (for example $\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$ shears parallel to the $x$-axis). In three dimensions, rotations about a coordinate axis and reflections in a coordinate plane have $3 \times 3$ matrices; for instance a rotation about the $z$-axis keeps the $z$-coordinate fixed and rotates the $x$ and $y$ coordinates.

Composing transformations

To apply transformation $P$ then transformation $Q$ to a point, you compute $\mathbf{Q}(\mathbf{P}\mathbf{x}) = (\mathbf{Q}\mathbf{P})\mathbf{x}$. The combined matrix is $\mathbf{Q}\mathbf{P}$, with the first transformation on the right. Order matters because matrix multiplication is not commutative.

Invariant points and invariant lines

An invariant point is fixed by the transformation: $\mathbf{M}\mathbf{x} = \mathbf{x}$. For any transformation that fixes the origin, the origin itself is invariant; other invariant points satisfy $(\mathbf{M} - \mathbf{I})\mathbf{x} = \mathbf{0}$. An invariant line is a line that maps onto itself (though individual points on it may move along it). To find invariant lines $y = mx$, apply $\mathbf{M}$ to a general point on the line and require the image to satisfy the same equation.

The determinant as a scale factor

The determinant of a transformation matrix is the factor by which it multiplies area (in two dimensions) or volume (in three). A determinant of $1$ preserves area, a negative determinant reverses orientation (a reflection is present), and a zero determinant collapses the figure onto a line or plane, so the transformation cannot be undone.

Matrices as transformations connect to complex numbers, where multiplication by $r e^{i\theta}$ is exactly a rotation by $\theta$ and an enlargement by $r$.

Try this

Q1. Write the matrix for a reflection in the $y$-axis. [1 mark]

• Cue. The image of $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ is $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$ and of $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ is itself: $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$.

Q2. State the area scale factor of the transformation $\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$. [2 marks]

• Cue. $|\det| = |12 - 2| = 10$, so areas are multiplied by $10$.