Isomorphism of orientation preserving rigid motions

[kp266]

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

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]

Moderator's note:

Homework assignments or any textbook style exercises for which you are seeking assistance are to be posted in the appropriate forum in our https://www.physicsforums.com/forumdisplay.php?f=152" area. This should be done whether the problem is part of one's assigned coursework or just independent study.

 

[wofsy]

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

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

goes to az + b.