I am new to advanced linear algebra so please excuse the problem goes:
Let S be a subset of a vector space V , such that S is also contained in a subspace W of V . If S is linearly independent as a subset of W, can it be dependent as a subset of V ?
My thoughts are that it does not matter if you are in a subspace or the vector space itself, the set will be linearly dependent or independent regardless. I can't think of any example that would say otherwise... Hints are greatly appreciated.