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**Pinkk** Write the following isometries in terms of a complex variable $\displaystyle z = x + iy$.

1. $\displaystyle \tau_{a}(x) = \left[$\begin{array}{c}

x_{1}\\

x_{2}\end{array}$]\right + \left[$\begin{array}{c}

a_{1}\\

a_{2}\end{array}$]\right$

2. $\displaystyle \rho_{\theta}(x) = \left[$\begin{array}{cc}

\cos ( \theta) & -\sin (\theta) \\

\sin (\theta) & \cos (\theta) \\ \end{array}$]\right \left[$\begin{array}{c}

x_{1}\\

x_{2}\end{array}$]\right$

3. $\displaystyle r(x) = \left[$\begin{array}{cc}

1 & 0 \\

0 & -1 \\ \end{array}$]\right \left[$\begin{array}{c}

x_{1}\\

x_{2}\end{array}$]\right$

I'm not sure what the problem is asking for. Does it mean to plug in a complex variable $\displaystyle z$ instead of the $\displaystyle x$? Does it mean to change the matrices themselves or what? Wouldn't these matrices look exactly the same in the complex plane? Any help would be appreciated.