Linear Transformations: Scale and Rotate

When the drone clips a gull, it tumbles through the air. Each frame of that tumble turns the sprite a little further, and each turn is a matrix. That sounds like a formula to memorise, but there's a way to build any such matrix from scratch in seconds: watch where two small arrows go.

The rule

A matrix's columns are where the x and y arrows land.

Multiply any matrix by $(1, 0)$ and out comes its first column; multiply by $(0, 1)$ and out comes its second. Every other vector is a mix of those two, so knowing where they land tells you everything. A transformation that works like this is linear: it keeps the origin fixed, keeps straight lines straight, and keeps evenly spaced grid lines evenly spaced.

Scale. Stretch x by $s_x$ and y by $s_y$: $(1, 0)$ lands on $(s_x, 0)$ and $(0, 1)$ on $(0, s_y)$, so the matrix is $\begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix}$. The landing squash from the last lesson is one of these.

Rotate by an angle $\theta$, counter-clockwise. The x arrow swings to $(\cos\theta, \sin\theta)$ and the y arrow to $(-\sin\theta, \cos\theta)$, so

$R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$

For a quarter-turn, 90°, that is $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$, which sends $(x, y)$ to $(-y, x)$.

Combine two transformations by multiplying their matrices. "First A, then B" is the product $BA$, read right to left, because the vector meets A first. The order matters. Stretch x by 2 and then turn 90°: $(1, 0) \to (2, 0) \to (0, 2)$. Turn first and then stretch x: $(1, 0) \to (0, 1) \to (0, 1)$. Two different answers.

The shortcut

Don't memorise matrices — build them. Ask where $(1, 0)$ lands and write it down as the first column; ask where $(0, 1)$ lands and write it as the second.

The trap is writing where the arrows land as rows instead of columns. For a rotation that quietly gives you the rotation the other way round.

Drills

Answer fast — trust the rule.

Remember

• A matrix's columns are where (1, 0) and (0, 1) land.
• Rotation by θ has columns (cos θ, sin θ) and (−sin θ, cos θ); a quarter-turn sends (x, y) to (−y, x).
• "First A, then B" is BA: order matters, and writing rows for columns reverses a rotation.