4.5 Rank

Rank

The rank of a matrix is the dimension of its range (or equivalently, its column space). It tells us how many linearly independent columns (or rows) exist in the matrix. Formally, for a general matrix $A\in {\mathbb{R}}^{m\times n}$,
$\mathrm{Rank}(A)\le \min (m,n),$
where if $\mathrm{Rank}(A)=\min (m,n)$, then $A$ is said to be in full rank.

Conversely, the nullity of a matrix $A$, denoted as $\mathrm{Nullity}(A)$ is the dimension of its null space It represents the number of independent solutions to the homogeneous equation $Ax=0$. Per the rank-nullity theorem,
$\mathrm{Rank}(A)+\mathrm{Nullity}(A)=n.$

Properties

(Determinant Relation) $\mathrm{Rank}(A)=\min (m,n)$ if and only if $\det (A)\ne 0$.