Homework5Solutions

Homework5Solutions - Homework 5 Solutions Math 332 Spring...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Homework 5 Solutions Math 332, Spring 2010 Problem 1. Proposition. Let G be the following subgroup of GL (2 , R ) : G = a- b b a : a,b ∈ R and ( a,b ) 6 = (0 , 0) . Then G is isomorphic to C # , the group of nonzero complex numbers under multiplication. Proof. Define ϕ : C # → G by ϕ ( a + bi ) = a- b b a . Clearly ϕ is a bijection. Furthermore, if a + bi,c + di ∈ C # , then ϕ ( ( a + bi )( c + di ) ) = ϕ ( ( ac- bd ) + ( ad + bc ) i ) = ac- bd- ad- bc ad + bc ac- bd = a- b b a c- d d c = ϕ ( a + bi ) ϕ ( c + di ) , and therefore ϕ is an isomorphism. Problem 2. Proposition. Let Aff ( R ) be the group of all functions f : R → R of the form f ( x ) = ax + b where a,b ∈ R and a 6 = 0 . Then Aff ( R ) is isomorphic to the group G = a b 0 1 : a,b ∈ R and a 6 = 0 . Proof. Let ϕ : Aff ( R ) → G be the function that maps f ( x ) = ax + b to the matrix a b 0 1 . Clearly ϕ is a bijection. To prove that ϕ is an isomorphism, let f,g ∈ Aff ( R ), where f ( x ) = ax + b and...
View Full Document

{[ snackBarMessage ]}

Page1 / 4

Homework5Solutions - Homework 5 Solutions Math 332 Spring...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online