Suppose $A$ is an $m \times n$ matrix and $0 \leq r \leq $ min($m,n$) is an integer. The following are equivalent:
(a) Rank($A$) = $r$
(b) Im($A$) = Span($A$) is an $r$-dimensional subspace of $\mathbb{R}^{m}$
(c) The dimension of Null($A$) is $n-r$.
(d) The solution set of $Ax = b$ has dimension $n-r$
(e) $A$ has an $r$-minor of nonzero determinant, but every $(r+1)$-minor has determinant zero
(f) $A$ has an invertible $r$-minor, but every $(r+1)$-minor is singular
(g) Rank($A^T$) = $r$
(h) $A$ has exactly $r$ pivot positions
(i) $A$ has exactly $r$ linearly independent columns
(j) There are exactly $n-r$ columns of $A$ that is a linear combination of other columns of $A$.
Reminds you of the Invertible matrix theorem? I thought that the invertible matrix theorem is a somewhat special case of a more general theorem. That is the case when $M$ is an $m \times n$ matrix of rank $r$. This is my question: Can you think of points that should be added to the list?