Four Subspaces - Spaces | Linear Algebra

The Four Fundamental Subspaces

Every $m \times n$ matrix $A$ has four fundamental subspaces:

Subspace	Of	Dimension	Basis
Column space $C(A)$	$\mathbb{R}^m$	$r$	Pivot columns of $A$
Null space $N(A)$	$\mathbb{R}^n$	$n - r$	Special solutions
Row space $C(A^T)$	$\mathbb{R}^n$	$r$	Nonzero rows of $U$
Left null space $N(A^T)$	$\mathbb{R}^m$	$m - r$	From $E$ in $EA = U$

Where $r = \text{rank}(A)$ .

\mathbb{R}^n = C(A^T) \oplus N(A)

\mathbb{R}^m = C(A) \oplus N(A^T)

Row space and null space are orthogonal complements in $\mathbb{R}^n$
Column space and left null space are orthogonal complements in $\mathbb{R}^m$

The rank $r$ is the number of pivots. It equals: