Question

In: Advanced Math

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 .

Solutions

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 .


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