Onto matrices

A matrix $$A \in \mathbb{R}^{m\times n}$$ is called onto if $$\mathcal{R}(A) = \mathbb{R}^m$$, which happens if and only if :
 * $$Ax=y$$ can be solved in $$x$$ for any $$y$$
 * $$A$$ has full row rank
 * columns of $$A$$ span $$\mathbb{R}^m$$
 * $$A$$ has a right inverse, i.e. $$\exist B\in \mathbb{R}^{n\times m}$$ s.t. $$AB = I$$
 * rows of $$A$$ are independent
 * $$\mathcal{N}(A^T) = \{0\}$$
 * $$\text{det}(A A^T) \neq 0$$