Write down in the form ${Z}\rightarrow{AZ+B}$ the following transformations of the complex plane:
(a) translation in the direction $(2,-3)$
(b) rotation about (0,1) through $\pi/4$
I know from my student answer key that the answer for part (b) is ${Z}\rightarrow{AZ-iA+i}$
where $A=\frac{(1+i)}{\sqrt2}$
can someone explain the steps for part (b)? Part (a) is straightforward.
If you find part (a) easy, I guess you understand the geometry of complex addition and how complex numbers can be thought of as vectors.
If you can't do part (b), maybe you don't understand complex multiplication. If you have complex numbers $z$ and $z'$, then $zz'$ is $z$ with its length multiplied by the length of $z'$, and rotated by the angle $z'$ makes with the real axis.
So for instance, if you have a complex number whose vector is of length $5$ and makes an angle of $15°$ with the horizontal, then multiplying $z$ by that complex number will make $z$ five times longer and rotate it $15°$ in the positive direction.