# Thread: Linear independence and span

1. ## Linear independence and span

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 ?

2. ## Re: Linear independence and span

Originally Posted by mrmaaza123
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 ?

Well, you can think of $\displaystyle S$ as a basis for the subspace of $\displaystyle V$ generated by $\displaystyle S$ itself, so...
Let S be the set of vectors $\displaystyle \{v_1, v_2, ..., v_n\}$ and T the subset $\displaystyle \{v_1, v_2, ..., v_m\}$ with m< n, of course. Obviously, $\displaystyle v_n$ is in Span(S) so if Span(S)= Span(T) then we must have $\displaystyle v_n= a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_mv_m$. From that it follows that $\displaystyle a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_mv_m- v_m= 0$ contradicting the fact that the vectors in S were independent.