Let $u, v, w \in V $ be three nonzero vectors in a vector space over $C$ such that any list of two of the three vectors is linearly independent. With these conditions, prove or give a counterexample that $u, v, w$ is linearly independent.
My guess is that $u,v,w$ cannot be assumed to be linearly independent with the given conditions that each list of two vectors is linearly independent.
For $u,v,w$ to be linearly independent, there must exist some $a,b,c$ such that
which means that $a=b=c=0$.
Can I assume that since each set of 2 vectors are linearly independant, that these $a,b,c$ are automatically $0$ for $u,v,w$?