Hi could you provide a different or same but more thorough explanation for this problem? I don't understand what is going on this solution... Please give me better explanation. Let V be an n-dimensional vector space over a field F, where n ≥ 1. Prove that L(V, F)is isomorphic to F. You must explicitly write down an isomorphism (and prove that it is an isomorphism). Ans!-Given thatlet "V" be an N-dimensional vectar spaceover a field F.wheren21.To prove that / (V₁F) is isomorphic to 'F' is :-letus consider that13 = { 0₁ 0₂ 03₁ -- -- en 3ез,Now to write thatL (V₁F) = { collection of all linear transformationfrom v to E' }we tale thatf-> (f(e.) jf(²₂), f(es), _. f (en)}"& (f+g) =((f+g)lei), (fig)(e₁),,___.(f+g) (en))(f(e₁), f(l₂), f(en)) + (g(er), g(e.) _- -- g(en))= § (f) + $(9)=be a vender bases of 'V"-Q$ (f + g)Now let cys consider thatas theirg (df).for d EF..• (Af(e₁), Af(e)..· if(en) )= A (f(e₁), f(C₁).f(en))keng = {fEL (V₁F) / $$) = (0₁0.-- 0) }&where f(ei) =0 + 1 = 1, 2, 3 ~~.n.

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Answered: Let V be an n-dimensional vector space…

Let V be an n-dimensional vector space over a field F, where n ≥ 1. Prove that L(V, F) is isomorphic to Fn. You must explicitly write down an isomorphism (and prove that it is an isomorphism).

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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