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Thread: complex analysis: linear transformations

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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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    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 $\displaystyle a = re^{i \theta} $ and $\displaystyle z = be^{i \gamma} $. Then $\displaystyle az = rbe^{i(\theta + \gamma)} $. And let $\displaystyle b = b_{1}+ i b_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.
    You need to know that $\displaystyle a\not =0$!

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

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

    Therefore, given $\displaystyle f(z) = \alpha z + \beta$ where $\displaystyle f(z) = re^{i\theta} + \beta$. So first you send $\displaystyle z\mapsto rz$ then $\displaystyle z\mapsto e^{i\theta}z$ and then $\displaystyle z\mapsto z+\beta$. 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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