- #1
caffeinemachine
Gold Member
MHB
- 816
- 15
I am trying to prove the following.
Let be finite dimensional vector spaces over a field .
There is a natural isomorphism between and .
Define a map as
for all .
It can be seen that is a multilinear map.
By the universal property of tensor product, there exists a unique linear map such that .
We also know that
Thus we just need to show that to show that and are isomorphic.
My Problem: I want to show the triviality of the kernel in a basis free manner. But here I am stuck.
Can anybody help?
Let
There is a natural isomorphism between
Define a map
for all
It can be seen that
By the universal property of tensor product, there exists a unique linear map
We also know that
Thus we just need to show that
My Problem: I want to show the triviality of the kernel in a basis free manner. But here I am stuck.
Can anybody help?