**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.