Proof of Theorem NMTNS

The null space of a square matrix, $A$, is equal to the set of solutions to the homogeneous system, $\homosystem{A}$. A matrix is nonsingular if and only if the set of solutions to the homogeneous system, $\linearsystem{A}{\zerovector}$, has only a trivial solution. These two observations may be chained together to construct the two proofs necessary for each half of this theorem.