3.9 Matrix transformationsIB Maths: Applications and Interpretation HL: Revision notes

A $2\times2$ matrix transforms a point by multiplying its position vector: $\begin{pmatrix}x' \\ y'\end{pmatrix}=\begin{pmatrix}a & b \\ c & d\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}.$ The columns of the matrix are the images of $(1,0)$ and $(0,1)$, so you can write a matrix straight from a picture. All these transformations fix the origin.

Standard matrices:

• Reflection in the $x$-axis: $\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$; in the $y$-axis: $\begin{pmatrix}-1 & 0 \\ 0 & 1\end{pmatrix}$
• Reflection 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: $\begin{pmatrix}k & 0 \\ 0 & k\end{pmatrix}$
• Horizontal stretch, factor $k$: $\begin{pmatrix}k & 0 \\ 0 & 1\end{pmatrix}$; vertical stretch, factor $k$: $\begin{pmatrix}1 & 0 \\ 0 & k\end{pmatrix}$
• Rotation through $\theta$ anticlockwise about the origin: $\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}$, so $90^\circ$ anticlockwise is $\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$

Example: $\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}\begin{pmatrix}3 \\ 2\end{pmatrix}=\begin{pmatrix}-2 \\ 3\end{pmatrix}$.

A translation adds a vector to every point, so it cannot be written as a $2\times2$ matrix multiplication by itself. A general transformation is $\begin{pmatrix}x' \\ y'\end{pmatrix}=\begin{pmatrix}a & b \\ c & d\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}+\begin{pmatrix}e \\ f\end{pmatrix}.$ Do the matrix multiplication first, then add the translation vector.

Example: $\begin{pmatrix}x \\ y\end{pmatrix}\to\begin{pmatrix}2 & 0 \\ 0 & 2\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}+\begin{pmatrix}1 \\ -3\end{pmatrix}$ sends $(4,2)$ to $(8,4)+(1,-3)=(9,1)$.

A translation does not change lengths, angles or area.

A composition applies one transformation after another. If $\mathbf{A}$ is applied first and then $\mathbf{B}$, the single matrix is $\mathbf{B}\mathbf{A}$ because $\mathbf{B}(\mathbf{A}\mathbf{x})=(\mathbf{B}\mathbf{A})\mathbf{x}$. The order matters, since matrix multiplication is not commutative.

Example: rotate $90^\circ$ anticlockwise, $\mathbf{M}=\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$, then reflect in the $x$-axis, $\mathbf{R}=\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$: $\mathbf{R}\mathbf{M}=\begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix},$ which is a reflection in $y=-x$. Doing it the other way round gives $\mathbf{M}\mathbf{R}=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$, a reflection in $y=x$.

For $\mathbf{A}=\begin{pmatrix}a & b \\ c & d\end{pmatrix}$, $\det\mathbf{A}=ad-bc$. The area scale factor of the transformation is $|\det\mathbf{A}|$: $\text{area of image}=|\det\mathbf{A}|\times\text{area of object}.$

• If $\det\mathbf{A}<0$, the image is also reflected (orientation reverses).
• If $\det\mathbf{A}=0$, the shape collapses to a line or a point.
• Translations do not change area.

Example: $\mathbf{N}=\begin{pmatrix}3 & 1 \\ 2 & 2\end{pmatrix}$ has $\det\mathbf{N}=6-2=4$, so a triangle of area $8$ becomes area $32$. For a composition, multiply the determinants: $\det(\mathbf{B}\mathbf{A})=\det\mathbf{B}\times\det\mathbf{A}$.

A fractal is built by applying transformations over and over. Each step uses matrix transformations of the form $\mathbf{x}\to\mathbf{M}\mathbf{x}+\mathbf{t}$.

Example (Sierpinski triangle): start with a filled triangle of area $A_0$. Apply three transformations, each an enlargement of scale factor $0.5$ followed by a different translation, and keep all three images. Each image has area $0.25A_0$ since $\det=0.25$. After $n$ iterations there are $3^n$ small triangles and the shaded area is $A_n=\left(\tfrac34\right)^nA_0,$ a geometric sequence that tends to 0 as $n$ increases. Sums of lengths or areas of repeated, scaled copies are infinite geometric series, with $S_\infty=\frac{a}{1-r}$ for $|r|<1$.

Repeatedly applying a matrix to a vector, $\mathbf{x}_{n+1}=\mathbf{A}\mathbf{x}_n$, links to Markov chains, where $\mathbf{A}$ is a transition matrix.