Linear transformation (complex)

A map $w=az+b$ ( $a\neq 0$ ) built from a rotation, a scaling, and a translation of the plane.

A complex linear transformation is a map of the form $w=az+b$ with $a,b\in\mathbb{C}$ and $a\neq 0$ . Writing $a=re^{i\theta}$ shows what it does geometrically -- it rotates the plane by $\theta$ , scales all lengths by the factor $r=|a|$ , and then translates by $b$ . Every such map sends lines to lines and circles to circles, and it is the simplest member of the larger family of Möbius transformations, arising when a linear fractional transformation's denominator is the constant $1$ .

Examples

$w=2iz+1$ rotates by $90^\circ$ , doubles all lengths, and shifts the result one unit to the right.
The map $w=z+(3-i)$ is a pure translation, sliding every point by the vector $3-i$ .