Complex Mapping Linearity Test

[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 .

 

[blue_raver22]

Hello,

Thank you for sharing your question about complex mapping linearity test. It is important to understand that the definition of linearity for complex linear mappings is different from that of real linear mappings. In the case of complex linear mappings, the rule T(a * z1 + z2) = a * T(z1) + T(z2) only applies if both a and z1 are complex numbers.

In your case, A is a real number and B is a complex number. This means that T(z) = Az + B is a complex linear mapping, but it does not follow the rule mentioned above. This is because a * z1 would be a complex number, while T(z1) would be a complex number multiplied by a real number (A). Therefore, they cannot be combined in the same way as in the real linear mapping rule.

In conclusion, the mapping T(z) = Az + B is still a linear mapping, but it follows a different rule for linearity due to the presence of a real number in the equation. I hope this explanation helps clarify any confusion. Thank you.