If vectors in a set is linearly independent to each other, does it make the set an linearly independent set? Why or why not?

Question

A set S has vectors $a_1$,$a_2$,$a_3$,.....,$a_n$.
$a_1$ is linearly independent to $a_2$,$a_3$,......,$a_n$.
$a_2$ is linearly independent to $a_1$, $a_3$,......,$a_n$.
$a_3$ is linearly independent to $a_1$, $a_2$,......,$a_n$.
......
$a_n$ is linearly independent to $a_1$, $a_2$,$a_3$,......,$a_(n-1)$.
does this make the set S a linearly independent set? Why or why not?

Answer 1

Let $S=\{v_1,\dots,v_n\}$ be the set you describe in the question. Let $a_1,\dots,a_n$ be scalars such that $$a_1v_1+\dots+a_nv_n=0$$

This implies that the $$a_1=\dots=a_n=0$$ as if that were not the case, then there would exist a vector that could be written as a linear combination of the other vectors in $S$, contradicting the hypothesis that each vector is linearly independent to all the other vectors.

Trivial point but worth mentioning: None of the vectors can be the zero vector as then that vector would not be linearly independent to the other vectors.