Prove or disprove: Between any n-dimensional vector space V and Rn there is exactly one isomorphism...

Prove or disprove: Between any n-dimensional vector space V and Rⁿ there is exactly one isomorphism T : V → Rⁿ .

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Expert Solution

Suppose be a n-dimensional vector space which has a basis containing n elements given by .

The standred basis of is given by , .

As is a basis of so for all each there exist scalaers such that , .

is defined by , is an isomorphism as it is an linear transformation and T is bijective .

Also is defined by , is an isomorphism as it is an linear transformation and T is bijective .

But .

Hence there are more than one isomorphism .

Colby Messinger answered 10 months ago

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