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Complex Mapping Linearity Test

  1. Jul 13, 2014 #1

    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???

  2. jcsd
  3. Jul 13, 2014 #2
    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.
  4. Jul 13, 2014 #3
    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.
  5. Jul 14, 2014 #4


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    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 .
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