Abstract Algebra Theory and Applications - Computer Science ...

178 CHAPTER 10 MATRIX GROUPS AND SYMMETRY(a, b)(a, –b)(sin θ, – cos θ)(cos θ, sin θ)θFigure 10.2. O(2) acting on R 2whereas a rotation by an angle θ in a counterclockwise direction must comefrom a matrix of the form( )cos θ sin θ.sin θ − cos θIf det A = −1, then A gives a reflection.Two of the other matrix or matrix-related groups that we will considerare the special orthogonal group and the group of Euclidean motions. Thespecial orthogonal group, SO(n), is just the intersection of O(n) andSL n (R); that is, those elements in O(n) with determinant one. The Euclideangroup, E(n), can be written as ordered pairs (A, x), where A is inO(n) and x is in R n . We define multiplication by(A, x)(B, y) = (AB, Ay + x).The identity of the group is (I, 0); the inverse of (A, x) is (A −1 , −A −1 x). InExercise 6, you are asked to check that E(n) is indeed a group under thisoperation.xx + yFigure 10.3. Translations in R 2