**Exercise 8.2.8.**Let \( \varphi : \mathbb{C}^\times \rightarrow \mathbb{R}^\times \) be the map \( \varphi(z) = |z|^2 \) where \( |z| \) is the modulus of \( z \).1. Show \( \varphi \) is a homomorphism.2. Compute \( \ker \varphi \) and \( \varphi(\mathbb{C}^\times) \).3. Show \( \mathbb{C}^\times / \ker \varphi \cong \varphi(\mathbb{C}^\times) \).

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Answered: Exercise 8.2.8. Let q: CX → RX be the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Exercise 8.2.8. Let q: CX → RX be the map o(z) = |z|² where |z| is the modulus of z. (1) Show q is a homomorphism. (2) Compute ker q and q(CX). (3) Show CX/ker q = q (CX).