Let $F$ and $K$ be fields such that $K \subset F$. We can consider the tensor product $F\, \otimes \, K^n$ as $F-$vector space with the operation:

$$ \lambda (a \otimes v) = (\alpha a \otimes v), \, \forall a \in F, \, \forall a\otimes v \in F\,\otimes \, K^n. $$

How one can show that $F^n$ and $F \,\otimes K^n$ are isomorphic as $F-$vector spaces?

I've tried the canonic way $\lambda \otimes (x_1,...,x_n) \in F \, \otimes \, K^n \mapsto (\lambda x_1,...,\lambda x_n) \in F^n$. However, I couldn't proof that this transform is isomorphism.



Let $x_i$ be a bsais of $V=K^n$. Then the elements of $F\otimes V$ are of the form $$\sum \lambda_i\otimes x_i.$$ Mapping this to $(\lambda_1,\cdots ,\lambda_n)$ is an isomorphism.


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