Group Theory and Linear Algebra
Last updated: 24 September 2014
Lecture 28: The affine orthogonal group and isometries
The affine orthogonal group is
The orthogonal group is
The special orthogonal group is
A rotation is an element of and a reflection is an element of where and
For and let
The group acts on by
Note that, if then
is not a linear transformation, in particular
be the metric on given by
be the positive definite bilinear form given by
An isometry of is a function
Use (a) and (b) to show that if
is an isometry then satisfies
The group of isometries of is
with operation given by composition of functions.
is given by
Then is a group isomorphism.
Let Translation by is the function
Note that is an isometry since
Notes and References
These are a typed copy of Lecture 28 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on October 11, 2011.