Q: If S is a linearly independent subset of a vector space V, then is it true in general that for no proper subset T of S, span(T) = span(S) ?
The result should be true if S is a basis. But in the above question one can't make out whether or not S is a basis, right ? So what should be the answer ?
Let S be the set of vectors and T the subset with m< n, of course. Obviously, is in Span(S) so if Span(S)= Span(T) then we must have . From that it follows that contradicting the fact that the vectors in S were independent.