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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.

An $m\times n$ matrix has full row rank when its matrix rank is $m$, equivalently when its rows are linearly independent vectors. This requires $m\le n$. Its column space is all of ${\mathbb{F}}^m$, so the represented linear map is surjective and every linear system $Ax=b$ is solvable. If $n>m$, its nullity is $n-m>0$, so solutions are not unique.^[ref]

For example, the matrix transpose of the matrix in full column rank isIt has full row rank two. Every output $(p,q{)}^T$ has solutions $(p-t,q-t,t{)}^T$, with $t\in \mathbb{R}$. A square matrix of size $n$ has full row rank and full column rank together exactly when its matrix rank is $n$; then it is an invertible matrix. A nonsquare matrix cannot have both properties, and has no two-sided inverse: such an inverse would make its linear map a linear isomorphism between vector spaces of different dimensions.