Proving Isomorphisms of Vector Spaces

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Homework Statement

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Homework Equations

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The Attempt at a Solution

isomoprism is bijective

i have no clue whatsoever..im going to research a little bit now
ill appreciate any help

 

[HallsofIvy]

You are given that F is an isomorphism from X to Y and G is an isomorphism from Y->Z. You are asked to show that G\circF is an isomorphism from X to Z. I can guess that X, Y, and Z are vector spaces since another part of the problem talks about a "spanning set" but you should have said that! Proving an isomorphism of groups, rings, fields, etc. is quite different. An isomorphis is, as you say, bijective: you must prove this is surjective: that if z is any member of Z, then there exist x in X such that G\circF(x)= z. Since G is an isomorphism from Y to Z it is surjective: what does that tell you? Once you have that, you know that F is surjective from X to Y. Use that.

An isomorphism must be injective also. If x₁ and x₂ are such that G\circF(x₁)= G\circ(x₂) then you must prove that x₁= x₂. Use can use the fact that F and G are each injective to prove that. Of course, a isomorphism must also "preserve" the operations. You need to show that G\circF(au+ bv)= aG\circF(u)+ bG\circF(v).

As for b) using the definition of "span" together with that last statement: G\circF(au+ bv)= aG\circF(u)+ bG\circF(v) should be enough.