Recently, I am learning complex analysis using " Complex Analysis for mathematics and engineering" by John H. Mathews and Russell W. Howell. In Chapter 2, it says that "...because a linear transformation can be considered to be a composition of a rotation, a magnification, and a translation", where in the text a linear transformation is defined as w=L(z)=Az+B. This is really confusing, because in linear algebra, this doesn't make sense. So, is the definition of linear transformation in complex number different from that of linear algebra? Thank you!

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    $\begingroup$ This comes from the definition of a linear function. $+b$ is the translation. $a$ is a complex number and every complex number can be written in polar form which can be realized as a rescaling and rotation. $\endgroup$ Jun 16, 2018 at 1:43
  • $\begingroup$ Well, I think that I didn't express my question nicely! The question is edited. I am confused why the way of defining linear transformation in complex number can be so explicitly written in L(z)=Az+B, where A and B are complex numbers. $\endgroup$
    – Ivy
    Jun 16, 2018 at 1:52
  • $\begingroup$ @CameronWilliams I understood! Is it because it is a one-dimensional linear transformation over the field C? Thank you! $\endgroup$
    – Ivy
    Jun 16, 2018 at 1:54
  • $\begingroup$ I'm not sure what you mean by linear transformations not having such an explicit definitions: Linear maps from $\mathbb{R}$ to itself are given by $f(x)=ax$ for $a$ a real number. As mentioned, colloquially we say a funciton is linear if it is really any affine function, i.e. we allow for translations (basically we just want the graph to be a line, $f(x)=ax+b$) $\endgroup$ Jun 16, 2018 at 2:13
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    $\begingroup$ Affine is a fine word. $\endgroup$
    – copper.hat
    Jun 16, 2018 at 2:49

1 Answer 1


There is a very sad confusion that arises naturally as a result of poor choice of terminology in English mathematics, which is calling functions of the form $ f (x)=ax+b $ linear functions - $ x $ be a real or complex number, it doesn't matter.

The reason for this is that those are functions whose graph is a line (be it a real or complex line).

The reason it is unfortunate is that these functions are not necessarily linear transformations, since $ f (0)=b $ is not necessarily $0 $.

But note that they are however affine transformations, which means that if they don't necessarily respect linear combinations, they do respect affine combinations. Other languages would more appropriately call such functions affine functions.

  • $\begingroup$ Now that I see why I am confused! Thank you so much! $\endgroup$
    – Ivy
    Jun 16, 2018 at 2:57
  • $\begingroup$ @ivy You're welcome! $\endgroup$ Jun 16, 2018 at 5:22

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