A matrix moves the whole plane

Think of graph paper

Pin graph paper at the origin. You may stretch, turn, slant or flip it. You may not bend, fold or tear it. Every 2 by 2 matrix does some mix of these. A matrix whose columns lie on one line (dependent columns, lesson 4) can also squash the paper flat onto a line, or onto the origin.

Why no bending? Because A(cv + dw) = cAv + dAw. This property is what “linear” means. Each entry of Ax is a row times the vector (lesson 1, slide 3), and row·(cv + dw) = c(row·v) + d(row·w). So evenly spaced points on a line stay evenly spaced on a line:

Two landmarks decide everything

Every point is xy = x·10 + y·01. Since A respects combinations,

Know where 10 and 01 go, and you know where every point goes. To build a matrix, choose where they land and write those as the columns.

Building a rotation

To turn the plane counterclockwise by θ, move both landmarks around the unit circle. 10 is at 0°, so it lands at θ: cos θsin θ. 01 is at 90°, a quarter turn ahead of 10, so it lands a quarter turn ahead of cos θsin θ. A quarter turn counterclockwise sends any xy to −yx (check: 10 → 01 and 01 → −10). Apply it to cos θsin θ: 01 lands at −sin θcos θ. Those are the columns:

A new number: the determinant

When the plane moves, shapes can grow or shrink. Take the little square with corners 00, 10, 01 and 11. Its area is 1. After the move it becomes a slanted box, a parallelogram. The area of that box tells you how much the matrix stretches every area. That stretch factor has a name: the determinant, written det A.

For a 2 by 2 matrix A = abcd, it is ad − bc. You first met this number in lesson 3: the test for whether a 2 by 2 matrix has an inverse. Now you can see what it measures:

• det A = 3: every area becomes 3 times bigger.
• det A = 1: areas stay the same, like a turn.
• det A = 0: the square is squashed flat onto a line, so its area is 0. That is the case of dependent columns (lesson 4).

A negative determinant means the plane was also flipped over, like turning a page face down. Slide 2 builds such a flip. A rotation has det Q = cos²θ + sin²θ = 1: it turns without stretching. The rest of this lesson (Strang's chapter 5) covers determinants fully.