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1.30 These prove that isomorphism is an equivalence relation. (a) Show that the identity map id: V → V is an isomorphism. Thus, any vector space is isomorphic to itself. (b) Show that if f: V → W is an isomorphism then so is its inverse f-¹: W → V. Thus, if V is isomorphic to W then also W is isomorphic to V. (c) Show that a composition of isomorphisms is an isomorphism: if f: V → W is an isomorphism and g: W → U is an isomorphism then so also is go f: V → U. Thus, if V is isomorphic to W and W is isomorphic to U, then also V is isomorphic to U.

1.30 These prove that isomorphism is an equivalence relation. (a) Show that the identity map id: V → V is an isomorphism. Thus, any vector space is isomorphic to itself. (b) Show that if f: V → W is an isomorphism then so is its inverse f-¹: W → V. Thus, if V is isomorphic to W then also W is isomorphic to V. (c) Show that a composition of isomorphisms is an isomorphism: if f: V → W is an isomorphism and g: W → U is an isomorphism then so also is go f: V → U. Thus, if V is isomorphic to W and W is isomorphic to U, then also V is isomorphic to U.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do part A,B,C and please show step by step and explain

1.30 These prove that isomorphism is an equivalence relation. (a) Show that the identity map id: V → V is an isomorphism. Thus, any vector space is isomorphic to itself. (b) Show that if f: V → W is an isomorphism then so is its inverse f-¹: W → V. Thus, if V is isomorphic to W then also W is isomorphic to V. (c) Show that a composition of isomorphisms is an isomorphism: if f: V → W is an isomorphism and g: W → U is an isomorphism then so also is go f: V → U. Thus, if V is isomorphic to W and W is isomorphic to U, then also V is isomorphic to U.
Transcribed Image Text:1.30 These prove that isomorphism is an equivalence relation. (a) Show that the identity map id: V → V is an isomorphism. Thus, any vector space is isomorphic to itself. (b) Show that if f: V → W is an isomorphism then so is its inverse f-¹: W → V. Thus, if V is isomorphic to W then also W is isomorphic to V. (c) Show that a composition of isomorphisms is an isomorphism: if f: V → W is an isomorphism and g: W → U is an isomorphism then so also is go f: V → U. Thus, if V is isomorphic to W and W is isomorphic to U, then also V is isomorphic to U.
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