Take X1 = 0 vector; and all other constants besides C1 to be zero.
(C1)(0) + (0)(X2) + ... +(0)(Xn) = 0
In this case, C1 does not have to be equal to 0, it could be any number. In order for them to be independent, c1=c2=cn=0; since c1 does not equal 0 therefore they're dependent.
This is the way my professor described it to me, hopefully this helps.
It is well known that a vector space always admits an algebraic (Hamel) basis. This is a theorem that follows from Zorn's lemma based on the Axiom of Choice (AC).
Now consider any specific instance of vector space. Since the AC axiom may or may not be included in the underlying set theory, might there be examples of vector spaces in which an Hamel basis actually doesn't exist ?