Further Maths/Lessons/Linear Transformations/Reflections & Rotations
Core Pure · Linear Transformations · Year 1

Reflections & Rotations

Two of the most familiar transformations, both captured perfectly by a matrix — and a neat trigonometric derivation shows exactly where the rotation matrix comes from.

16 min read 4 worked examples

Reflections

Standard reflection matrices

Reflection in the $x$-axis is represented by the matrix:

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

Reflection in the $y$-axis is represented by the matrix:

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

Reflection in the line $y=x$ is represented by the matrix:

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

Reflection in the line $y=-x$ is represented by the matrix:

$$\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$$
A coordinate grid showing a triangle and its mirror image reflected across the line y equals x, with the line of reflection drawn as a dashed diagonal
Fig. 1 — reflecting a shape in the line y = x swaps the x and y coordinates of every point.
Example 1 · Deriving a reflection matrix from first principles
Find the matrix representing reflection in the line $y=x$.

Reflecting $(1,0)$ in $y=x$ swaps its coordinates: image is $(0,1)$. Reflecting $(0,1)$ gives $(1,0)$.

These images become the columns: $M=\begin{pmatrix}0&1\\1&0\end{pmatrix}$.

DerivationThe other three standard reflection matrices, from the unit vectors
1
Reflection in the $x$-axis: $\mathbf{i}=(1,0)$ lies on the mirror line so is unchanged; $\mathbf{j}=(0,1)$ flips to $(0,-1)$.
Points on the mirror line are invariant; points off it land the same distance on the opposite side.
2
Columns give $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ for reflection in the $x$-axis.
Definition 24.1: the images of $\mathbf{i}$ and $\mathbf{j}$ are the columns.
3
Reflection in the $y$-axis: $\mathbf{i}=(1,0)$ flips to $(-1,0)$; $\mathbf{j}=(0,1)$ lies on the mirror line so is unchanged.
Same reasoning, mirrored about the vertical axis this time.
4
Columns give $\begin{pmatrix}-1&0\\0&1\end{pmatrix}$ for reflection in the $y$-axis.
Definition 24.1 again.
5
Reflection in $y=-x$: $\mathbf{i}=(1,0)$ maps to $(0,-1)$; $\mathbf{j}=(0,1)$ maps to $(-1,0)$.
Reflecting in $y=-x$ swaps the coordinates and negates both — this can be checked directly by folding the plane along the line $y=-x$.
6
Columns give $\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$ for reflection in $y=-x$.
Definition 24.1 once more — all four standard reflections are built the same way.
∎
Invariant points and lines for reflections

For reflection in the mirror line $\ell$, every point on $\ell$ is invariant (mapped to itself). Any line perpendicular to $\ell$ is also an invariant line as a whole, even though the individual points on it move — they simply swap to the mirrored position on the same perpendicular line. For example, reflecting in the $x$-axis fixes every point on $y=0$, and maps each vertical line $x=k$ onto itself.

Rotations

Theorem 25.1 — rotation matrix

Rotation about the origin by angle $\theta$ anticlockwise is given by $$R_\theta=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$$

ProofDerivation of the rotation matrix
1
$\mathbf{i}=(1,0)$ rotates to $(\cos\theta,\sin\theta)$.
A point at angle $0$ and distance $1$ from the origin moves to angle $\theta$, distance unchanged.
2
$\mathbf{j}=(0,1)$, at angle $90°$, rotates to angle $90°+\theta$: $(\cos(90°+\theta),\sin(90°+\theta))=(-\sin\theta,\cos\theta)$.
Standard angle-addition identities: $\cos(90°+\theta)=-\sin\theta$, $\sin(90°+\theta)=\cos\theta$.
3
The images of $\mathbf{i}$ and $\mathbf{j}$ become the columns of $R_\theta$.
Definition 24.1.
∎
A coordinate grid showing a triangle rotated anticlockwise about the origin by an angle theta, with a curved arrow indicating the direction and size of rotation
Fig. 2 — rotating a shape anticlockwise by angle θ about the origin.
Standard angles as special cases

Substituting the common angles into $R_\theta$ gives matrices you should recognise on sight:

$R_{90°}=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, $\quad R_{180°}=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$, $\quad R_{270°}=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$.

The only invariant point for any rotation (other than $0°$ or $360°$) is the origin. The single exception is $\theta=180°$: because this sends every point to its negative, every line through the origin is invariant as a whole (though no point on it besides the origin is individually fixed) — for any other angle, there are no invariant lines at all.

Example 2 · Identifying a transformation from its matrix
Describe the transformation given by $M=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$.

Comparing with $R_\theta$: $\cos\theta=0$, $\sin\theta=1$, so $\theta=90°$.

$M$ represents a rotation of $90°$ anticlockwise about the origin.

Example 3 · Finding an exact rotation matrix
Find the exact matrix representing a rotation of $150°$ about the origin.

$\cos150°=-\dfrac{\sqrt3}{2}$, $\sin150°=\dfrac12$.

$$R_{150°}=\begin{pmatrix}-\frac{\sqrt3}{2}&-\frac12\\\frac12&-\frac{\sqrt3}{2}\end{pmatrix}$$
Example 4 · Combining knowledge to identify an unknown angle
Given that $M=\begin{pmatrix}-\tfrac{1}{\sqrt2}&-\tfrac1{\sqrt2}\\\tfrac1{\sqrt2}&-\tfrac1{\sqrt2}\end{pmatrix}$ represents a rotation, find the angle of rotation.

$\cos\theta=-\dfrac{1}{\sqrt2}$ and $\sin\theta=\dfrac{1}{\sqrt2}$: both conditions match $\theta=135°$ (second quadrant, since cosine negative and sine positive).

Common mistake

Confusing the reflection matrix for $y=x$ (swap coordinates) with a $90°$ rotation — although both matrices involve $0$s and $1$s in similar positions, $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ (reflection, $\det=-1$) and $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ (rotation, $\det=1$) are very different transformations. Checking the determinant is a quick way to distinguish: reflections always have $\det=-1$, rotations always have $\det=1$.

Exam-style questions

Question 1 5 marks

Find the matrix representing a rotation of $180°$ about the origin, and describe an alternative single transformation that has the same effect.

$R_{180°}=\begin{pmatrix}\cos180°&-\sin180°\\\sin180°&\cos180°\end{pmatrix}=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$ (M1, A1)

This is the same as an enlargement with scale factor $-1$ centred at the origin (a point reflection through the origin). (B1, B1, B1)

Question 2 5 marks

Describe fully the transformation represented by $M=\begin{pmatrix}-1&0\\0&1\end{pmatrix}$, and find the image of the point $(3,-4)$ under this transformation.

Reflection in the $y$-axis. (B1, B1)

$M\begin{pmatrix}3\\-4\end{pmatrix}=\begin{pmatrix}-3\\-4\end{pmatrix}$ (M1, A1, A1)

Question 3 6 marks

Find the exact matrix representing rotation by $210°$ about the origin, and use it to find the image of $(2,0)$.

$\cos210°=-\dfrac{\sqrt3}{2}$, $\sin210°=-\dfrac12$ (M1, A1)

$R_{210°}=\begin{pmatrix}-\frac{\sqrt3}{2}&\frac12\\-\frac12&-\frac{\sqrt3}{2}\end{pmatrix}$ (A1, A1)

$R_{210°}\begin{pmatrix}2\\0\end{pmatrix}=\begin{pmatrix}-\sqrt3\\-1\end{pmatrix}$ (M1, A1)

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