Prove that if U, V and W are vector spaces such that U and V are...

Prove that if U, V and W are vector spaces such that U and V are isomorphic and V and W are isomorphic, then U and W are isomorphic.

Expert Solution

Defination : Two vector space V₁ and V₁ are isomorphic if there exist a linear map T : V₁V₂ which is bijective .

Now given U and V are isomorphic ,then there exist linear map T₁ : UV , which is bijective .

Also given V and W are isomorphic ,then there exist linear map T₂ : VW ,which is bijective .

Now T₁ : UV and T₂ : VW T₂ T₁ is also a linear map and composition two bijective map is bijective .

T₂ T₁ is a linear bijective map from U and W .

i.e., there exist a linear bijective map from U to W .

Hence U and W are isomorphic .

.

If you have any doubt or need more clarification at any step please comment .

Colby Messinger answered 1 year ago

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