Definition. A subset S of R^{n} is fixed by a transformation f if f(x) = x for each x in S. A subset S of R^{n} is invariant under f or is stabilized by f if both f and f^{-1} carry points of S to points of S.
Note. Terminology on these matters is not uniform. Some use the term ``fixed'' for what is called ``invariant'' here and then would say ``pointwise-fixed'' to describe what is called ``fixed'' here.
Theorem. If f(x) = A x + b is an affine transformation of R^{n} and L is an invariant line of f, then L must be parallel to some eigenvector of the matrix A. (However, there may be lines parallel to an eigenvector of A that are not invariant lines of f. )
Recall that a ``half turn'' is a rotation of the plane (about any point) through the angle pi. Show that the invariant lines of a half turn are precisely the lines through its center.
Show that the rotation of the plane (about any point) through an angle that is not an integer multiple of pi has no invariant lines.
For what type of isometry is the set of invariant lines the family of all lines parallel to a given line?
What type of isometry of the plane has one and only one invariant line?
Show that if an affine transformation f of R^{n} carries each point of an affine subspace S of R^{n} to a point of S, then f^{-1} automatically does likewise. (Use a result from linear algebra.)