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
.