Vector space/Isomorphic iff equal dimension/Fact

Let ${}K$ denote a field and let ${}V$ and ${}W$ denote finite-dimensional ${}K$-vector spaces.

Then

${}V$ and ${}W$ are isomorphic to each other if and only if their dimension

coincides.

In particular, an ${}n$-dimensional ${}K$-vector space is isomorphic to ${}K^{n}$.

Write a proof