Vector Space and Linear Transformation

Let V and W be vector spaces over the ﬁeld F ; suppose that $\displaystyle

\{\vec v_i : i \in I \} $ is a basis for V , listed without redundancy.

Also suppose that for each $\displaystyle

i \in I $, there exists $\displaystyle

\{\vec w_i \in W \} $ (The list $\displaystyle

\{\vec w_i : i \in I \} $ might have redundancy; there’s no assumption on these, except that they are in W .) Show that there is a linear transformation $\displaystyle T : V \rightarrow W$ such that $\displaystyle T\vec v_i = \vec w_i $ for every $\displaystyle i \in I $ .

I was wondering if there was a less trivial way then to just say that there can be $\displaystyle T : V \rightarrow W$ such that $\displaystyle T\vec v_i = \vec w_i $ for every $\displaystyle i \in I $ (which is reasonable in itself). And then to show that it respects linear transformation definition because W is a vector space.