Math 331 - Homework Assignment

§§ 10.4 - 10.5

Definition. If f and g are transformations of a set X, the commutator of f and g is the transformation f g f^{-1} g^{-1}. Sometimes the commutator of f and g is denoted by [f, g].
While [f, g]^{-1} = [g, f] and, therefore, the inverse of any commutator is always a commutator, it is not necessarily true that the product of two commutators is a commutator.

Definition. A transformation f of R^{n} is called a similarity if there is a positive scalar r such that for all points x, y in R^{n} one has the distance relation
d(f(x), f(y)) = r d(x, y) .
The number r is called the scaling factor of f.
Proposition. When two similarities are composed, the scaling factor of the composite similarity is the product of the scaling factors of the two original similarities.
Theorem. A transformation f of R^{n} is a similarity if and only if in a rectangular coordinate system it is given by the formula
f(x) = r U x + b ,
where r is a positive scalar, U is an n \times n orthogonal matrix, and b is a point of R^{n}.

Prove the proposition stated above.
Prove the theorem stated above.
Show that the commutator of two affine transformations of R^{n} must always be orientation-preserving and volume-preserving but need not be an isometry.
Show that the commutator of any two similarities of R^{n} must be an isometry.
Show that for r > 0 and c a given point of R^{n} the formula
f(x) = (1 - r) c + r x
defines a similarity f with scaling factor r that has c as a fixed point and that commutes with every affine transformation of R^{n} that has c as a fixed point. This type of similarity is called a dilatation. (Note that inasmuch as this definition of f involves a barycentric combination of c and x, the transformation f thereby defined does not depend on the choice of a rectangular coordinate system. Use this observation to simplify your argument.)
Given three lines a, b, and c in the plane that intersect in pairs so as to delimit a triangle, construct a point p and a line m so that the product of the reflections in the three given lines satisfies the equation
sigma_{c} \circ sigma_{b} \circ sigma_{a} = sigma_{m} \circ h_{p} ,
where sigma_{m} is reflection in the line m and h_{p} is the half turn about the point p.