Linear Transformations

Complex Analysis · Axiom Academy

The simplest conformal maps: f(z) = az + b rotates, scales, and shifts the whole plane — and never bends an angle. A linear (affine) transformation of the complex plane is just f(z) = az + b with . It looks plain, but it carries the whole geometry of conformal maps in miniature: written as , it is nothing more than a scaling , a rotation , and a translation stitched together. Because everywhere, it preserves angles — it is conformal on all of . This lesson shows what that decomposition looks like, why the angle survives, and how just two point–image pairs lock the map down completely. 1. Three Moves in One: Scale, Rotate, Shift Split the coefficient as . Then f(z) = az + b is the same as doing three simple things in a row: scale by |a| , rotate by , then translate by b . Watch a square ride through all three beats of f(z) = (1+i)z + 2 — it lands tilted and enlarged, but it is still a square. scale by |a| , then rotate by , then add b example: ( , turn , shift +2 ) 2. Why It's Conformal: Angles Survive A map is conformal where its derivative is nonzero, and f'(z) = a is never zero (we required ). Geometrically, multiplying by a rotates every direction by the same angle and scales every length by the same factor |a| . So two curves crossing at some angle still cross at exactly that angle after the map — watch a right angle stay a right angle even as both arms swing. Every tangent vector is multiplied by a , so its angle increases by .

This is the written version of the interactive lesson above. See the full Complex Analysis course.