One-to-one

A (skinny or square) matrix $$A$$ is called one-to-one if 0 is the only element of its nullspace: $$\mathcal{N}(A) = \{0\}$$, which is possible if and only if:
 * $$x$$ can always be uniquely determined from $$y=Ax$$
 * mapping from $$x$$ to $$Ax$$ is one-to-one: different $$x$$'s map to different $$y$$'s
 * Columns of $$A$$ are independent (hence, a basis for their span)
 * $$A$$ has a left inverse
 * $$A$$ has full column rank
 * $$\det(A^T A) \neq 0$$