Theorem: Two finite-dimensional vector spaces are isomorphic if and only if they have the same dimension.
I can understand how to prove that if they are isomorphic then they have the same dimension. Yet for the other direction I cannot totally understand.
To quote Axler's Linear Algebra Done Right, 2nd edition page 55:
To prove the other direction, suppose $V$ and $W$ are finite-dimensional vector spaces with the same dimension. Let $(v_1,...,v_n)$ be a basis of $V$ and $(w_1,...w_n)$ be a basis of $W$. Let $T$ be the linear map from $V$ to $W$ defined by $$T(a_1v_1+...+a_nv_n)=a_1w_1+...+a_nw_n \ (*)$$ Then $T$ is surjective because $(w_1,...,w_n)$ spans $W$, and $T$ is injective because $(w_1,...,w_n)$ is linearly independent. Because $T$ is injective and surjective, $T$ is invertible.
I cannot understand how on earth can we define $T$ that satisfy (*) above. I don't think Axler has given a proof that this can be defined. Could somebody help me on this please?