# Casting out of linearly dependent vectors from a set

Lemma: Let $V$ be a vector space and let $S=\{v_1,\ldots,v_n\}$ be a linearly dependent subset of $V$ and let $U={\rm Span}(S)$. Then there exists an index $1\le h\le n$ such that if $T=S\,\backslash \{v_h\}$ then $U={\rm Span}(T)$.

I am trying to consider what happens when $n=1$, now we could say $S=\{v_1\}$ but this wouldn't be linearly dependent and if we include another element then the set $S$ could no longer a subset of $V$, if we remove the element then we have the empty set which is linearly independent, so does this lemma even work for $n=1$?

• $S = \{ \mathbf{0} \}$ is a linearly dependent set of vectors... – Michael Biro Apr 15 '15 at 18:12
• ah of course... thanks – user2850514 Apr 15 '15 at 18:14