2.27 Does every correspondence between bases, when extended to the spaces, give an isomorphism? That is, suppose that V is a vector space with basis B = (ß₁, ..., ßn) and that f: B → W is a correspondence such that D = (f(ß₁),…,f(ßn)) is basis for W. Must f: V→ W sending v=C₁ B₁ +…+cnßn to f(v) = c₁f(ß₁)+…+cnf(ßn) be an isomorphism?
2.27 Does every correspondence between bases, when extended to the spaces, give an isomorphism? That is, suppose that V is a vector space with basis B = (ß₁, ..., ßn) and that f: B → W is a correspondence such that D = (f(ß₁),…,f(ßn)) is basis for W. Must f: V→ W sending v=C₁ B₁ +…+cnßn to f(v) = c₁f(ß₁)+…+cnf(ßn) be an isomorphism?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:✓2.27 Does every correspondence between bases, when extended to the spaces, give an
isomorphism? That is, suppose that V is a vector space with basis B = (B₁,..., B₁)
and that f: B → W is a correspondence such that D = (f(ß₁),…,f(³₁)) is basis
for W. Must f: V → W sending v=c₁B₁+...+CnBn to f(v) = C₁ f(B₁)+···+Cnf (Bn)
be an isomorphism?
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In this question, we need to establish that whether or not every correspondence between bases, when extended to the space gives an isomorphism?
We know that a set is a basis of a vector space , if the set is linearly independent and .
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