Proof of Theorem LIVHS

(⇐) Suppose that $\homosystem{A}$ has a unique solution. Since it is a homogeneous system, this solution must be the trivial solution $\vect{x}=\zerovector$. By Theorem SLSLC, this means that the only relation of linear dependence on $S$ is the trivial one. So $S$ is linearly independent.

(⇒) We will prove the contrapositive. Suppose that $\linearsystem{A}{\zerovector}$ does not have a unique solution. Since it is a homogeneous system, it is consistent (Theorem HSC), and so must have infinitely many solutions (Theorem PSSLS). One of these infinitely many solutions must be nontrivial (in fact, almost all of them are), so choose one. By Theorem SLSLC this nontrivial solution will give a nontrivial relation of linear dependence on $S$, so we can conclude that $S$ is a linearly dependent set.