The Transformation Zoo

You've built the vocabulary. Now let's meet the whole family. There are a handful of named transformations that appear over and over — in graphics, physics, machine learning, and geometry.

The named transformations

Rotation

Spins every point around the origin by angle $\theta$. Lengths and angles are perfectly preserved. The determinant is always 1.

Scaling

Uniform scaling zooms everything in or out. Non-uniform scaling stretches one axis more than another — turning circles into ellipses.

Shear

Tilts one axis while leaving the other in place. Think of slanting a stack of cards — each card slides a little more than the one below. Parallel lines stay parallel; right angles don't stay right.

Reflection

Flips space across a line through the origin. The determinant is −1: area is preserved but orientation flips (clockwise becomes counter-clockwise).

Projection

Squashes everything onto a line. Every point is dropped perpendicularly onto that line. The determinant is 0 — information is permanently lost. You can't "un-project" back to where you came from.

Combining them

Real-world transformations are usually combinations: a rotation followed by a non-uniform scale, or a shear followed by a reflection. Any combination of linear transformations is still linear — and can always be described by a single matrix.