It means that your vectors that span
a subspace of V does not span all of V, since is a vector that cannot "be reached" by v1,....,vn.
It says that is in V, meaning that is a vector in V.
The line through the origin and w is a subspace of V, that is, all scalar multiples of is in the subspace spanned by .
I am not sure either how to do this proof correct (I am myself currently studying linear algebra). But maybe something like this:
(If mr_fantastic or anybody wants to correct, plz do!)
Giving all coefficients equaling zero.