Solution to Exercise 3.40.
What did the acorn say when it finally grew up?
Solution: Ge-om-e-try

Exercise 3.40. Prove Proposition 3.8.
Proposition 3.8. The product of the matrices of two affine direct or two affine indirect isometries of the Euclidean plane is the matrix of an affine direct isometry. Further, the product of an affine direct and an affine indirect isometry of the Euclidean plane is an affine indirect isometry of the Euclidean plane.

 

Proof. Let A₁ and A₂ be matrices of affine direct isometries of the Euclidean plane.

 

Hence, the product of two matrices of affine direct isometries of the Euclidean plane is the matrix of an affine direct isometry of the Euclidean plane.

        Let A₁ and A₂ be matrices of affine indirect isometries of the Euclidean plane.

 

Hence, the product of two matrices of affine indirect isometries of the Euclidean plane is the matrix of an affine direct isometry of the Euclidean plane.

        Let A₁ be the matrix of an affine direct isometry of the Euclidean plane and A₂ be the matrix of an affine indirect isometry of the Euclidean plane.

 

The product A₂A₁ has a similar result. Hence, the product of a matrix of an affine direct isometry and an affine indirect isometry of the Euclidean plane is the matrix of an affine indirect isometry of the Euclidean plane.//
 

Solutions for Chapter 3

© Copyright 2005, 2006 - Timothy Peil