Full column rank ($\mathrm{rank}A=n$)

An $m\times n$ matrix has full column rank when its matrix rank is $n$, equivalently when its columns are linearly independent vectors. This requires $n\le m$. Its kernel is $\{0\}$, so the represented linear map is injective and every solvable linear system $Ax=b$ has a unique solution. It need not reach every output.^[ref]

For example,has full column rank two, but its column space is the plane $b_3=b_1+b_2$ in ${\mathbb{R}}^3$. An attainable output fixes the input uniquely as $(b_1,b_2{)}^T$; an output outside this plane has no preimage.