Vector Space and Linear Transformation

Let V and W be vector spaces over the ﬁeld F ; suppose that is a basis for V , listed without redundancy.

Also suppose that for each , there exists (The list might have redundancy; there’s no assumption on these, except that they are in W .) Show that there is a linear transformation such that for every .

I was wondering if there was a less trivial way then to just say that there can be such that for every (which is reasonable in itself). And then to show that it respects linear transformation definition because W is a vector space.