Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is one-to-one/injective.

Theorem: An $n \times n$ matrix $A$ is nonsingular iff $T(x) = Ax$ is one-to-one (injective). $T$ is injective iff $\ker(T) = \{0\}$, which means $Ax = 0$ has only the trivial solution.