Hmmm..... What about
?
This set is clearly linearly dependent. But I cant express
each element of S as a linear combination of the others. For example, (1,1) cannot be expressed as a linear combination of the others, yet S is linearly dependent.
Considering your knowledge, perhaps you meant:
Technically this statement is wrong too, unless you
define it that way. A set S is
defined to be linearly dependent if it is not linearly independent. So technically(formally, logically...) you have to negate the definition of linear independence to get the mathematical formulation for linear dependence. And that formulation is the one you dont like....
I think this dilemma of implication and equivalence is pretty common. Generally if you have a set of vectors S where one is able to express some element of S as a linear combination of the others, then the set is definitely linearly dependent. However the other way round is not necessarily true. And your question works as wonderful illustration to this fact.
These are my thoughts and it could be unconvincing. Perhaps some MHF Algebraist can explain it better.