Solution to Exercise 3.64.
Thought makes the whole dignity of man; therefore, endeavor to think well, that is the only morality.
—Blaise Pascal (1623–1662)

Exercise 3.64. Prove the set of affine translations of the Euclidean plane is a group under matrix multiplication.

Proof. First, we show the set of affine translations of the Euclidean plane is closed under matrix multiplication. Let T_PQ and T_RS be matrices of affine translations of the Euclidean plane determined by vectors PQ = (a, b, 0) and RS = (c, d, 0). Then

 

which is the matrix of an affine translation of the Euclidean plane determined by vector (c + a, d + b, 0). Hence, the set of affine translations of the Euclidean plane is closed under matrix multiplication.

        Next, we show that the inverse of an affine translation of the Euclidean plane is an affine translation of the Euclidean plane. Let T_PQ be the matrix of an affine translation of the Euclidean plane determined by vector PQ = (a, b, 0). By Exercise 3.63, the form of the inverse of T_PQ is

 

which is the matrix of an affine translation of the Euclidean plane determined by the vector (–a, –b, 0). Thus, the every affine translation of the Euclidean plane has an inverse that is an affine translation of the Euclidean plane.

        Therefore, the set of affine translations of the Euclidean plane is a group under matrix multiplication.//
 

Solutions for Chapter 3

© Copyright 2005, 2006 - Timothy Peil