Isomorphism of orientation preserving rigid motions

[kp266]

Find an isomorphism from the subgroup of GL2(C) of the form
<br /> \begin{pmatrix}<br /> a &amp; b\\ <br /> 0 &amp; 1<br /> \end{pmatrix}<br /> <br /> ,\left | a \right |=1<br /> to the group of orientation preserving rigid motions.
*The problem is from Artin's Algebra Chapter5

 

[torquil]

Find an isomorphism from the subgroup of GL2(C) of the form
<br /> \begin{pmatrix}<br /> a &amp; b\\ <br /> 0 &amp; 1<br /> \end{pmatrix}<br /> <br /> ,\left | a \right |=1<br />

to the group of orientation preserving rigid motions.

*The problem is from Artin's Algebra Chapter5

How about using the argument (angle) of a to represent rotation of the rigid object around the z-axis, and use the real and imaginary parts of b to represent an arbitrary translation along the (x,y) plane? These are orientation preserving motions of rigid objects in R^3.

Torquil

 

[Redbelly98]

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[wofsy]

Find an isomorphism from the subgroup of GL2(C) of the form
<br /> \begin{pmatrix}<br /> a &amp; b\\ <br /> 0 &amp; 1<br /> \end{pmatrix}<br /> <br /> ,\left | a \right |=1<br /> to the group of orientation preserving rigid motions.
*The problem is from Artin's Algebra Chapter5

<br /> \begin{pmatrix}<br /> a &amp; b\\ <br /> 0 &amp; 1<br /> \end{pmatrix}<br /> <br /> ,\left | a \right |=1<br />

goes to az + b.