Matrices as Linear Transformations: Rotation Matrix Intuition

A matrix also moves the whole plane.

Think of A as a machine. A vector v goes in, Av comes out. Do that to every point at once and the whole plane moves.

A 10 = column 1 A 01 = column 2

Think: a sheet of graph paper

Pin a sheet of graph paper at the origin. You may stretch it, turn it, slant it or flip it over. You may not bend it, fold it or tear it.

That is everything a 2 by 2 matrix can do. Straight lines stay straight. Evenly spaced lines stay evenly spaced. The pin never moves.

Two landmarks are enough

If you know where two landmarks moved, you can work out where everything else went. The landmarks are (1, 0) and (0, 1).

Every point is a combination of them: (x, y) = x·(1, 0) + y·(0, 1). The machine keeps combinations intact, so A(x, y) = x·(column 1) + y·(column 2). It is slide 2 again.

To build any matrix, decide where the two landmarks should land and write those as the columns.

Worked example: rotation

Turning by θ sends (1, 0) to (cos θ, sin θ) and (0, 1) to (−sin θ, cos θ). Those are the columns:

Q = cos θ−sin θsin θcos θ

For 90°: cos = 0 and sin = 1, so Q = 0-110. It sends (1, 0) to (0, 1), a quarter turn.

Area: the determinant

Watch the shaded unit square. After the move it is a parallelogram with area |det A| (§5.3). Every other shape scales by the same factor.

A rotation has det = cos²θ + sin²θ = 1. It turns the paper without stretching it.

Where you meet this

An IMU measures in the drone's own axes. A 3 by 3 rotation matrix turns those readings into the world's axes. That is what an attitude estimator keeps updating.

Worked examples

Example 1 · Stretch sideways

Triple every x-coordinate, keep y: (1, 0) → (3, 0) and (0, 1) → (0, 1).

A = 3001 det A = 3

(1, 1) → (3, 1). Every area becomes 3 times larger

Example 2 · Half a turn

Rotate by 180°: cos 180° = −1, sin 180° = 0.

Q = −100−1 det Q = 1

(2, 1) → (−2, −1): every vector points the opposite way

Problems

Try each one before you open the solution.

Find the matrix that sends (1, 0) to (0, 2) and (0, 1) to (1, 0). What does det say?

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The landing spots are the columns: A = 0120. det A = 0 − 2 = −2: areas double and the plane is flipped.

Write the rotation by 60°. Where does (1, 0) go?

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cos 60° = ½ and sin 60° = √3/2, so Q = ½−√3/2√3/2½. (1, 0) goes to (½, √3/2).

Where does the shear 1021 send (1, 0) and (0, 1)? What happens to areas?

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Read the columns: (1, 0) → (1, 2) and (0, 1) → (0, 1). det = 1, so areas do not change.