Let $S$ be a linearly dependent set. Then, for each $x$ in $S$, is it true that $x$ is in $\operatorname{span}(S\setminus \{x\})$?

Question

My attempt :

Let $S =$ {$v_1, v_2, v_3, ..., v_n$} ($S$ is finite)

Then there exist $a_i$ which are not all zero such that

$a_1v_1 + a_2v_2 ... + a_nv_n = 0$

Let $v_k$ be an element in $S$ such that $a_k = 0$ in this equation. If no such element exists, then the claim is true. Else ...?

I don't know how to continue, and I don't even think it is true if $a_k = 0$.

Answer 1

No, it is not true. Take $S=\{(1,0),(0,1),(0,-1)\}$. It is linearly dependent, but$$(1,0)\notin\bigl\langle(0,1),(0,-1)\bigr\rangle.$$