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Math Help - complex analysis: linear transformations

  1. #1
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    complex analysis: linear transformations

    A function of the form f(z)=az+b where a and b are complex constants is called a linear transformation. show that every linear transformation can be expressed as the composition of a magnification, a ratation, and a translation. deduce from this that a linear transformation maps lines to lines and circles to circles.

    Hint: write a in polar form.
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  2. #2
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    Quote Originally Posted by srw899 View Post
    A function of the form f(z)=az+b where a and b are complex constants is called a linear transformation. show that every linear transformation can be expressed as the composition of a magnification, a ratation, and a translation. deduce from this that a linear transformation maps lines to lines and circles to circles.

    Hint: write a in polar form.
    Let  a = re^{i \theta} and  z = be^{i \gamma} . Then  az = rbe^{i(\theta + \gamma)} . And let  b = b_{1}+ i b_2 .
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  3. #3
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    Quote Originally Posted by srw899 View Post
    A function of the form f(z)=az+b where a and b are complex constants is called a linear transformation. show that every linear transformation can be expressed as the composition of a magnification, a ratation, and a translation. deduce from this that a linear transformation maps lines to lines and circles to circles.

    Hint: write a in polar form.
    You need to know that a\not =0!

    If a\in \mathbb{R}^{\times} and a>1 then z\mapsto az is magnification by a. If 0<a<1 then z\to az is minification by a. If a<0 then a\to az is the same effect of a\to |a|z however the object is reflected through the origin. And of course if a=1 then nothing happens.

    Let \theta\in \mathbb{R} then z\mapsto e^{i\theta}z is a rotation counterclockwise by \theta. If if \alpha \in \mathbb{C} then z\mapsto z+\alpha is a translation in direction of \alpha (if you think of this complex number as a vector).

    Therefore, given f(z) = \alpha z + \beta where f(z) = re^{i\theta} + \beta. So first you send z\mapsto rz then z\mapsto e^{i\theta}z and then z\mapsto z+\beta. Thus, all we are doing are rotating, maginifying/minifying, and translating. Thus, lines and circles go to lines and circles, respectively.
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