Linear transformationsAQA A-Level Further Maths: Revision notes

A $2\times2$ matrix $\mathbf{M}$ acts on a position vector by multiplication, $\begin{pmatrix}x'\\ y'\end{pmatrix}=\mathbf{M}\begin{pmatrix}x\\ y\end{pmatrix}$, and so defines a linear transformation of the plane. The origin is always fixed. The key fact: the columns of $\mathbf{M}$ are the images of the unit vectors $\begin{pmatrix}1\\0\end{pmatrix}$ and $\begin{pmatrix}0\\1\end{pmatrix}$. If $(1,0)\mapsto(a,c)$ and $(0,1)\mapsto(b,d)$ then $\mathbf{M}=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$. This lets you write down any matrix from two images, with no formula to remember. Example: a transformation sends $(1,0)$ to $(2,-1)$ and $(0,1)$ to $(3,4)$, so $\mathbf{M}=\begin{pmatrix}2&3\\-1&4\end{pmatrix}$, and $(2,1)\mapsto(7,2)$.

Learn these matrices (angles measured anticlockwise about the origin):

• Rotation through $\theta$: $\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}$. For $90^\circ$ anticlockwise this is $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$; for $180^\circ$ it is $\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$.
• Reflection in the $x$-axis $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$; in the $y$-axis $\begin{pmatrix}-1&0\\0&1\end{pmatrix}$; in $y=x$ $\begin{pmatrix}0&1\\1&0\end{pmatrix}$; in $y=-x$ $\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$.
• Enlargement, scale factor $k$, centre the origin: $\begin{pmatrix}k&0\\0&k\end{pmatrix}$.
• Stretch parallel to the $x$-axis, factor $a$: $\begin{pmatrix}a&0\\0&1\end{pmatrix}$; parallel to the $y$-axis, factor $b$: $\begin{pmatrix}1&0\\0&b\end{pmatrix}$.
• Shear with the $x$-axis invariant: $\begin{pmatrix}1&\lambda\\0&1\end{pmatrix}$; with the $y$-axis invariant: $\begin{pmatrix}1&0\\ \lambda&1\end{pmatrix}$. To describe a transformation fully, name its type and give the key data: centre, angle and direction for a rotation; the mirror line for a reflection; the scale factor for an enlargement or stretch; the invariant line and where a point goes for a shear.

If transformation $A$ has matrix $\mathbf{A}$ and $B$ has matrix $\mathbf{B}$, then $A$ followed by $B$ has matrix $\mathbf{B}\mathbf{A}$. The matrix for the first transformation is on the right, next to the vector, because $\mathbf{B}(\mathbf{A}\mathbf{x})=(\mathbf{B}\mathbf{A})\mathbf{x}$. Matrix multiplication is not commutative, so the order matters. Example: $P$ a clockwise quarter turn, $\mathbf{P}=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$, and $Q$ a reflection in $y=x$, $\mathbf{Q}=\begin{pmatrix}0&1\\1&0\end{pmatrix}$. Then $P$ then $Q$ is $\mathbf{Q}\mathbf{P}=\begin{pmatrix}-1&0\\0&1\end{pmatrix}$ (reflection in the $y$-axis), but $Q$ then $P$ is $\mathbf{P}\mathbf{Q}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ (reflection in the $x$-axis). Repeating a transformation $n$ times gives $\mathbf{M}^n$. Two shears $\begin{pmatrix}1&2\\0&1\end{pmatrix}$ in a row give $\begin{pmatrix}1&4\\0&1\end{pmatrix}$.

In 3D a transformation is represented by a $3\times3$ matrix whose columns are the images of $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$. The specification confines you to two types. Reflections in the coordinate planes change the sign of one coordinate: in $x=0$, $\begin{pmatrix}-1&0&0\\0&1&0\\0&0&1\end{pmatrix}$; in $y=0$, $\begin{pmatrix}1&0&0\\0&-1&0\\0&0&1\end{pmatrix}$; in $z=0$, $\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}$. Rotations about a coordinate axis leave that axis fixed. A positive angle $\theta$ is anticlockwise when looking from the positive axis towards the origin (the right-hand rule):

• about $x$: $\begin{pmatrix}1&0&0\\0&\cos\theta&-\sin\theta\\0&\sin\theta&\cos\theta\end{pmatrix}$
• about $y$: $\begin{pmatrix}\cos\theta&0&\sin\theta\\0&1&0\\-\sin\theta&0&\cos\theta\end{pmatrix}$
• about $z$: $\begin{pmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\0&0&1\end{pmatrix}$ The $y$-rotation has its signs the opposite way round because $z\to x$ is the positive turn about $y$.

Reflection after rotation. In 3D, $R$ is a rotation through $90^\circ$ about the $x$-axis and $T$ is reflection in $z=0$. Then $R$ followed by $T$ has matrix $\mathbf{T}\mathbf{R}=\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}\begin{pmatrix}1&0&0\\0&0&-1\\0&1&0\end{pmatrix}=\begin{pmatrix}1&0&0\\0&0&-1\\0&-1&0\end{pmatrix}$, and $(2,3,4)\mapsto(2,-4,-3)$. Squaring a rotation. $\mathbf{U}=\begin{pmatrix}0&0&1\\0&1&0\\-1&0&0\end{pmatrix}$ is a $90^\circ$ rotation about the $y$-axis, and $\mathbf{U}^2=\begin{pmatrix}-1&0&0\\0&1&0\\0&0&-1\end{pmatrix}$ is a rotation through $180^\circ$ about the $y$-axis. Check habits: test a matrix on $(1,0)$ and $(0,1)$; confirm the order of multiplication by asking which transformation acts first; and keep exact values ($\cos90^\circ=0$) rather than decimals.