If two vector spaces are isomorphic, does that guarantee that there's a linear transformation $T$ that is an isomorphism?

Question

If $T: V \longrightarrow W$ and $V$ and $W$ are isomorphic vector spaces, does there always exist a $T$ that is an isomorphism?

Answer 1

Let $\mathbb{V}$ and $\mathbb{W}$ linear spaces over the same field $\mathbb{K}$.
If $\mathbb{V}$ and $\mathbb{W}$ are isomorphics, so exists a linear transformation $T:\mathbb{V}\longrightarrow \mathbb{W}$ such that $T$ is injective and surjective.

You do not assume that the $T$ function exists. This follows from the fact that $\mathbb{V}$ and $\mathbb{W}$ are isomorphic.