Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is onto/surjective.

Theorem: An $n \times n$ matrix $A$ is nonsingular iff $T(x) = Ax$ is onto (surjective). $T$ is surjective iff the column space of $A$ is all of $\mathbb{R}^n$, i.e., $\text{rank}(A) = n$.