Base of vector space from a finite set

Question

Let $V$ be a vector space of finite dimension. $S=\{v_1,...,v_r\} \subset V$ and $Span(S)=V$. For each $v_i\in S$ there is a linear combination from $S\setminus \{v_i\}$. How can I show that for each $1\leq i \leq r$, $S\setminus \{v_i\}$ is a base for V? I tried displaying all vectors in S as a linear combination of themeselves with $S\setminus \{v_i\}$ but I got a really big useless experssion. What is the right way to go here?

Answer 1

The claim is not true. Consider the following example $V = \mathbb{R}^2, S= \{e_1, e_2, e_1 + e_2, e_1 - e_2\}$ where $e_1 = (1, 0), e_2 = (0, 1).$ Then every element of this set can be written as a linear combination of the others. But deleting one of them from the list won't give you a basis. It will be a generating set, but not basis.