MIT OpenCourseWare
49 minutes
Four Fundamental Subspaces in Linear Algebra
Every $m \times n$ matrix $A$ comes with four important subspaces. The column space and the left null space live in $\mathbb{R}^m$. The row space and the null space live in $\mathbb{R}^n$. These four spaces pair up. The row space and the null space are perpendicular to each other, and together they fill $\mathbb{R}^n$. The column space and the left null space are perpendicular to each other, and together they fill $\mathbb{R}^m$. If the rank of $A$ is $r$, then the column space and the row space both have dimension $r$, the null space has dimension $n - r$, and the left null space has dimension $m - r$. Knowing these four spaces tells you almost everything about what the matrix does to a vector.