IB Mathematics Applications & Interpretation AHL 3.9—matrix Transformations Notes

Why Matrix Transformations Exist: One Multiplication Moves Every Point

1. Every image on a screen is really a set of points, so moving, turning, or resizing a shape means moving all of its points at once.
2. A $2\times2$ matrix packages a geometric transformation into one object, so a single matrix multiplication sends a point to its new position.
   1. This is exactly how computer graphics, animation, and game engines rotate and scale shapes: they multiply coordinates by a matrix.
3. Because AI is a technology course, your GDC does the matrix arithmetic on every paper, so your real job is to set up the correct matrix and read off the image.
4. The whole topic rests on one idea: the columns of the matrix are the images of the base vectors, so once you know where $\mathbf{i}$ and $\mathbf{j}$ go, you know the matrix.

The Key Idea: A Matrix's Columns Are The Images Of i And j

1. Write a point as a column vector and transform it by multiplying on the left by a matrix $M$, which works out entry by entry: $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax+by \\ cx+dy \end{pmatrix}$$
2. The base vectors are $\mathbf{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\mathbf{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, pointing one unit along the $x$-axis and the $y$-axis.
3. Multiplying $M$ by $\mathbf{i}$ picks out the first column and multiplying by $\mathbf{j}$ picks out the second: $$M\mathbf{i} = \begin{pmatrix} a \\ c \end{pmatrix}, \qquad M\mathbf{j} = \begin{pmatrix} b \\ d \end{pmatrix}$$
   1. So to build any transformation matrix, find where $\mathbf{i}$ and $\mathbf{j}$ land and write those images as the two columns.
4. Every one of these matrices transforms about the origin, which stays fixed because $M\begin{pmatrix} 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$.

Reflections: Flipping Across A Mirror Line

1. A reflection flips every point across a mirror line through the origin, and you can build its matrix from the images of $\mathbf{i}$ and $\mathbf{j}$.
2. Reflecting in the $x$-axis fixes $\mathbf{i}$ and sends $\mathbf{j}$ to $\begin{pmatrix} 0 \\ -1 \end{pmatrix}$, so its matrix is $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.
3. The AI HL formula booklet gives the general reflection in the line $y = (\tan\theta)x$, so the four standard reflections come from choosing the right angle: $$\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}$$

One-Way Stretches: Scaling Along A Single Axis

1. A one-way stretch scales one direction and leaves the other unchanged, which turns a square into a rectangle.
2. A horizontal stretch of factor $k$ sends $\mathbf{i}$ to $\begin{pmatrix} k \\ 0 \end{pmatrix}$ and fixes $\mathbf{j}$, giving $\begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix}$.
3. A vertical stretch of factor $k$ fixes $\mathbf{i}$ and sends $\mathbf{j}$ to $\begin{pmatrix} 0 \\ k \end{pmatrix}$, giving $\begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix}$.
4. These stretch matrices are not in the formula booklet, so build them from the base vectors each time.

Enlargements: Scaling Equally In Every Direction

1. An enlargement with scale factor $k$ centred on the origin multiplies every position vector by $k$.
2. Because both base vectors scale by $k$, $\mathbf{i}$ goes to $\begin{pmatrix} k \\ 0 \end{pmatrix}$ and $\mathbf{j}$ goes to $\begin{pmatrix} 0 \\ k \end{pmatrix}$.
3. The matrix is therefore $k$ times the identity, written $kI$: $$\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} = k\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = kI$$
4. A factor $k>1$ makes the shape bigger, $0<k<1$ shrinks it, and $k<0$ also turns it by $180^\circ$.
5. Like the stretch matrices, $kI$ is not given in the formula booklet, so construct it from the scale factor.

Rotations About The Origin: Turning Through An Angle

1. A rotation turns every point about the origin through an angle $\theta$, with anticlockwise taken as positive.
2. Derive the matrix from the base vectors: $\mathbf{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ turns to the unit-circle point $\begin{pmatrix} \cos\theta \\ \sin\theta \end{pmatrix}$.
3. The vector $\mathbf{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ starts at $90^\circ$ and turns to angle $90^\circ + \theta$, landing at $\begin{pmatrix} -\sin\theta \\ \cos\theta \end{pmatrix}$.
4. Writing these two images as columns gives the rotation matrix: $$\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$
5. This anticlockwise rotation matrix, together with its clockwise version, is given in the AI HL formula booklet, so you select it rather than memorise it.

Translations: The One That Is Not A Matrix Product

1. A translation slides every point by a fixed vector, and this cannot be done by a $2\times2$ matrix multiplication about the origin.
   1. Any $2\times2$ matrix keeps the origin fixed, but a translation moves the origin, so no such matrix can exist for it.
2. Instead you add the translation vector to the position vector: $$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix}$$
3. A transformation with both a turn and a slide is a matrix multiplication followed by a vector addition, $\mathbf{x}' = M\mathbf{x} + \mathbf{t}$.

Composition: Combining Transformations By Multiplying Matrices

1. Applying one transformation and then another is a composition, found by multiplying their matrices.
2. To apply $A$ first and then $B$, the combined matrix is $BA$, because $B(A\mathbf{x}) = (BA)\mathbf{x}$.
   1. The matrix nearest the point acts first, so read a product from right to left.
3. Order matters because matrix multiplication is not commutative, so $BA$ and $AB$ are usually different transformations.

The Determinant: How A Transformation Scales Area

1. The determinant of $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is $\det M = ad - bc$, which is given in the formula booklet.
2. Its absolute value is the area scale factor: the image of any region has area $|\det M|$ times the original area. $$\text{area of image} = |\det M| \times \text{area of object}$$
3. The sign of the determinant reports orientation, so a negative determinant means the shape has been flipped over.
   1. Reflections have $\det M = -1$, while rotations and enlargements keep orientation with a positive determinant.
4. On the GDC, store $M$ and use the determinant command instead of expanding by hand, especially for a larger set-up.

Reference Table: The Standard 2D Transformation Matrices

Only the rotation and general reflection matrices are given in the formula booklet; build the rest from the images of the base vectors.