Math 331 – Homework Assignment

April 19, 2002

Reading in the Text

§§ 10.4 – 10.5

Commutators

• Definition. If f and g are transformations of a set X, the commutator of f and g is the transformation fgf^{−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.

Similarities.

• 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 Ux + b ,

where r is a positive scalar, U is an n × n orthogonal matrix, and b is a point of R^{n}.

Assignment for Monday, April 22

1. Prove the proposition stated above.

2. Prove the theorem stated above.

3. Show that the commutator of two affine transformations of R^{n} must always be orientationpreserving and volume-preserving but need not be an isometry.

4. Show that the commutator of any two similarities of R^{n} must be an isometry.

5. Show that for r > 0 and c a given point of R^{n} the formula

f(x) = (1 − r)c + rx

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.)

6. 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

σ_{c} ∘ σ_{b} ∘ σ_{a} = σ_{m} ∘ h_{p} ,

where σ_{m} is reflection in the line m and h_{p} is the half turn about the point p.