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.