Math 482
Take-Home Exam 1 Solutions
In these problems, suppose that G is the symmetry group of a wallpaper pattern, and
that the translation subgroup has the form T = f(I; nt1 + mt2 ) : n; m 2 Zg for some vectors
t1 ; t2 . Let V = fnt1 + mt2 : n; m 2 Zg,
Problem 1. Let r; s be rotation about the origin.
1. (a) Prove that r s = s r.
(b) Let (r; v ) and (s; w) be rotations (so r; s are both rotations about the origin).
Determine a condition on v; w in terms of r; s so that the two rotations have the
same ce
Homework #1 Solutions
Math 482/526
Problem 1. From the formula given above, show that a re
ection across a line passing
through the origin is both a linear transformation and an isometry.
Proof. Let w be a vector on a line ` passing through the origin. If
Math 526
Take-Home Exam 1 Solutions
In these problems, suppose that G is the symmetry group of a wallpaper pattern, and
that the translation subgroup has the form T = f(I; nt1 + mt2 ) : n; m 2 Zg for some vectors
t1 ; t2 . Let V = fnt1 + mt2 : n; m 2 Zg,
Problem 1. Let ft1 ; t2 g be an integral basis for V . Prove that ft1 ; t1 + t2 g is another
integral basis for V .
Solution. Because the set ft1 ; t1 + t2 g has two elements, if it spans V , then it is linearly
independent, and so is a basis. Let t 2 V .