A Matrix Moves the Grid

Lesson 7 of 12 · 7 min · Free, no account needed

What you will learn

A linear transformation moves the whole plane at once.
Its rule: grid lines stay straight, parallel and evenly spaced, and (0, 0) stays put.
Turns, flips, stretches and skews are linear. Swirls, bulges and slides are not.

Photo filters turn, stretch and skew a picture. One of these four breaks the rule the others follow.
Tap the odd one out.
Draw a little picture of your own.
Tap or drag over the cells. Five cells or more.
Run your picture through each filter, slowly.
Pick a filter and slide how much. Watch the grid lines and the orange dot at (0, 0).
Moves that follow the rule have a name.
A linear transformation moves every point of the plane at once.
Grid lines stay straight, and stay parallel and evenly spaced.
The origin (0, 0) stays put.
Linear or not?
Tap a filter card, then its bin.
Linear needs both: straight, even grid lines and (0, 0) fixed. Fisheye fails the first, slide fails the second.
This turn filter is broken: it also slides the picture.
Drag the orange dot back home to (0, 0) to fix it.
Game: filter match. Six pictures, against the clock.
Tap the filter that made each grid.
A computer image is a grid of pixels. Turning it moves every pixel the same way.
The image turns a quarter turn left around (0, 0). Where does the orange pixel at (2, 1) land?
Turning 90 degrees counterclockwise sends (x, y) to (-y, x), so (2, 1) goes to (-1, 2).
This filter keeps every line straight but moves (0, 0) to (1, 0). Is it linear?
Linear transformations keep (0, 0) fixed. This filter sends (0, 0) to (1, 0).
A linear move keeps the grid straight, even and anchored at (0, 0).
Your picture, before and after. Tap filters to make your card.

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