Why Does Complex Mapping T(z) = Az + B Fail Linearity?

[seminum]

Hello,

Given the complex linear mapping: T(z) = Az + B where A is real and B is complex. However trying to show that T(a * z1 + z2) = a * T(z1) + T(z2) does not work which implies the mapping is not linear? Why does not this rule apply here?

Thanks.

 

[micromass]

The map you posted is not a linear map. It's only linear if $B=0$. The map you posted is called an affine map.

That said, I can probably imagine some books which define "linear map" somewhat different than usual.

 

[seminum]

I see now. Well I found this in the K. Stroud's Advanced Engineering Mathematics on complex analysis. The same transformation when applied to a line, the image is another line in the w-plane, but not sure if the same applies to any other shape.

 

[WWGD]

Note that while a Real-valued linear map y=kx only scales , a complex-valued linear map both scales and rotates (e.g., multiply using polars); Complex lines, being 2-dimensional, are Real planes: notice that , for fixed Complex a, the set { ${az: z \in \mathbb C}$} is the entire complex plane .