Question

Let z c. Consider the map fz : c c defined by fz(w) = zw. Give a brief description of this map. Show that fz is a linear map and compute the matrix of this map with respect to the usual basis of c considered as a vector space over r.

Answer 1

Answer:

(i) The map $f_z$ is the "multiplication by z" map.  Geometrically, multiplication by z rotates by the angle θ = arg z and dilates by the factor |z|.

(ii) Linearity:

$f_z(au + bv) = z(au+bv)=z(au) + z(bv) = a(zu) + b(zv) = af_z(u) + bf_z(v)$

(iii) Matrix:

In view of the description as a rotation and a dilation, the matrix for the map is

$\left(\begin{array}{cc}|z|\cos\theta &-|z|\sin\theta\\ |z|\sin\theta &|z|\cos\theta\end{array}\right)$

where θ = arg z.