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