Theorem describing the dimension of spaces associated to the block matrices of the extended reduced row echelon form of a matrix

Theorem (Dimensions from extended RREF): For an $m \times n$ matrix $A$ of rank $r$, the dimensions of the four fundamental subspaces are:

Subspace Dimension Row space (of $A$) $r$ Null space (of $A$) $n - r$ Column space (of $A$) $r$ Left null space (of $A^T$) $m - r$

Rank-Nullity Theorem: $\dim(\text{Row space}) + \dim(\text{Null space}) = n$, i.e., $r + (n - r) = n$.

Dual statement: $\dim(\text{Column space}) + \dim(\text{Left null space}) = m$, i.e., $r + (m - r) = m$.

Geometric interpretation: The rank $r$ is the dimension of the image of the linear transformation $T(x) = Ax$. The nullity $n - r$ is the dimension of the kernel.