Matrix rank ($\mathrm{rank}A$)

The rank of a matrix is the dimension of its column space, equivalently the dimension of its row space.

Two rectangular matrices are related by invertible left and right multiplication exactly when their ranks agree. The rank-$r$ orbit in ${\mathrm{Mat}}_{b\times a}(k)$ has dimension $r(a+b-r)$: choose its image in a Grassmannian, then a surjective map to that image. These are the representation orbits of the one-arrow quiver.

The image of the linear map $A+B$ is contained in the sum of the images of $A$ and $B$. Since the dimension of a vector space of this sum is at most the sum of their dimensions, matrix rank is subadditive. Iteration shows that a sum of $r$ matrices of matrix rank at most one has matrix rank at most $r$, over any field.

An $m\times n$ matrix has full column rank when its rank is $n$, equivalently when its columns are linearly independent vectors.

An $m\times n$ matrix has full row rank when its rank is $m$, equivalently when its rows are linearly independent vectors.