Two isometries f and g of R^{n} are said to be conjugate if there is an is an isometry k such that g = k \circ f \circ k^{-1}. In this case g is called the conjugate of f by k.
Two affine transformations f and g are said to be conjugate if there is an affine transformation k such that g = k \circ f \circ k^{-1}. Also in this case g may be called the conjugate of f by k.
Proposition. If f and g are conjugate affine transformations, respectively, conjugate isometries, and g and h are conjugate affine transformations, respectively, conjugate isometries, then f and h are conjugate affine transformations, respectively, conjugate isometries.
Proposition. If two affine transformations f and g are conjugate, then they must have the same parity, i.e., f and g are either both orientation-preserving or both orientation-reversing.
Proposition. If a point p is a fixed point of an affine transformation f and h is any affine transformation, then the point q = h(p) is a fixed point of the conjugate of f by h.
Theorem. If two isometries of the plane are conjugate, then they must share the same of the four basic types of isometry of the plane.
Proposition. The conjugate of a translation of R^{n} by any affine transformation must be a translation.
Show that if a translation T_{a} of R^{n} is conjugated by an affine transformation f, where
then the result is the translation T_{Aa}.
What is the result of conjugating the translation T_{a} by the translation T_{b} ?
Show that conjugate rotations of the plane must involve the same angle or its negative.
Let p and q be given points of the plane, and let theta be a given angle. Let a = q - p. Let rho be the rotation about p through the angle theta. What is the isometry
If an isometry of the plane is conjugated by an affine transformation, must the result be an isometry? Either prove ``yes'', or provide a counter-example.
Explain why the theorem stated above is a consequence of the propositions that precede it.
Are any of the four propositions stated above difficult to prove?
Why is exercise 1 theoretically important? (Difficult)