The Plane Moves, All at Once

You'll learnto watch a matrix move the whole plane by moving its two basis vectors.

A 2 by 2 matrix is not a grid of numbers waiting to be multiplied. It is a machine that picks up the entire plane and sets it down somewhere else, and its instruction set is written in plain sight: the two columns are exactly where the basis vectors land. Watch the grid follow them.

9 STEPS · 6 QUICK CHECKS · COLUMNS · GRID · DETERMINANT · COMPOSITION · v2

Unit 4 is not assessed on the AP exam; schools include it where local requirements ask.

columns are destinations

Before you start

What you're looking at

The two basis vectors alone, about to travel to the matrix's columns while the grid follows.

The question

Where do (1,0) and (0,1) land, and what does the rest of the plane do?

Watch for

The first column as the destination of (1,0), the second as the destination of (0,1).

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