Q:

Proof of linear independence of non-empty subsets

The question states:

Show that if $S = \{v_1, v_2, \ldots , v_r\}$ is a linearly independent set of vectors, then so is every non-empty subset of $S$.

I understand that if $r>n$, $S$ is linearly dependent, but other than that, I'm not sure where to start with this proof. 

A:

I would be using proof by contradiction as mentioned in one of the answers. 
$$S=(v_1,v_2,........v_n)$$
Let $S_0$ be a subset of $S$ and assume that $S_0$ is linearly dependent 
$$S_0=(a_1,a_2,a_3.........,a_b)$$ where $b<n$. Since it is linearly dependent there exists scalars $(k_1,k_2,.......,k_b)$ such that $$a_1k_1+a_2k_2+.........+a_bk_b=0$$ but this can only happen when $b>n$ but in this case we had written that $b<n$. So therefore we arrive at a contradiction and our assumption is false. 
Hence proved.  Q.E.D

