Rank and Null Space

For a matrix $A$ ($m \times n$):

Rank $= \text{rank}(A)$ — the number of linearly independent rows (or equivalently, columns). Also equals the number of pivots in Gaussian Elimination.

Four fundamental subspaces:

• Column space $\text{Col}(A)$ — span of columns of $A$; all possible outputs $A\mathbf{x}$. Dimension = rank
• Row space $\text{Row}(A)$ — span of rows of $A$. Dimension = rank
• Null space $\text{Null}(A)$ — all $\mathbf{x}$ such that $A\mathbf{x} = \mathbf{0}$. Dimension = $n - \text{rank}$ (nullity)
• Left null space $\text{Null}(A^\top)$ — dimension = $m - \text{rank}$

Rank-Nullity Theorem: $\text{rank}(A) + \text{nullity}(A) = n$ (number of columns).

Implications:

• $A\mathbf{x} = \mathbf{b}$ has a solution iff $\mathbf{b} \in \text{Col}(A)$
• Solution is unique iff $\text{Null}(A) = \{\mathbf{0}\}$ (nullity = 0)
• Full rank ($\text{rank} = \min(m,n)$) means maximum information, no redundancy
• Rank deficiency → Singular Value Decomposition (SVD) has zero singular values

See also: Gaussian Elimination, Matrix Multiplication