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.