MIT OpenCourseWare
48 minutes
Rectangular Matrix Subspaces and Rank in Linear Algebra
An $m \times n$ matrix $A$ has four fundamental subspaces: the row space, the column space, the null space, and the left null space. They come in two perpendicular pairs. The row space and the null space are orthogonal complements inside $\mathbb{R}^n$: they meet only at the zero vector, and their dimensions add up to $n$. That is the rank-nullity theorem, $r + \dim(\text{Nul } A) = n$. The column space and the left null space are orthogonal complements inside $\mathbb{R}^m$, and their dimensions add up to $m$. These spaces explain when $A\mathbf{x} = \mathbf{b}$ can be solved ($\mathbf{b}$ must be in the column space) and what every solution looks like: one particular solution plus any vector from the null space, written $\mathbf{x}_p + \text{Nul } A$.