Rank–Nullity Theorem

Statement

Let $f : V \to W$ be a linear map between finite-dimensional vector spaces over a field $K$. Then:

Equivalently, $\operatorname{rank}(f) + \operatorname{nullity}(f) = \dim V$.

Visualization

Consider the linear map $f : \mathbb{R}^3 \to \mathbb{R}^3$ given by the matrix:

Step 1 — Row reduce to find the image (column space).

Two pivot columns $\Rightarrow$ $\operatorname{rank}(A) = 2$.

Image basis: columns 1 and 2 of $A$, i.e., $\{(1,2,0)^T,\; (2,4,1)^T\}$.

Step 2 — Find the null space.

Free variable: $x_3 = t$. Back-substituting: $x_2 = -t$, $x_1 = -2x_2 - 3x_3 = 2t - 3t = -t$.

Dimension table:

	Dimension
$\dim(\operatorname{im} A)$	2
$\dim(\ker A)$	1
$\dim(\mathbb{R}^3)$	3
rank + nullity	$2 + 1 = 3$ ✓

Proof Sketch

Choose a basis $\{k_1, \ldots, k_m\}$ for $\ker f$ and extend it to a basis $\{k_1, \ldots, k_m, v_1, \ldots, v_r\}$ of $V$. One shows that $\{f(v_1), \ldots, f(v_r)\}$ is a basis for $\operatorname{im} f$: they span $\operatorname{im} f$ (since $f(k_i) = 0$) and are linearly independent (any linear dependence $\sum c_j f(v_j) = 0$ implies $\sum c_j v_j \in \ker f$, forcing all $c_j = 0$). Hence $\dim(\operatorname{im} f) = r$ and $\dim V = m + r$.

Connections

• Cayley–Hamilton TheoremCayley–Hamilton Theorem$$p_A(A) = 0 \;\text{where}\; p_A(\lambda) = \det(\lambda I - A)$$Every square matrix satisfies its own characteristic polynomialRead more → — the minimal polynomial degree and kernel dimensions are intertwined.
• Spectral TheoremSpectral Theorem$$A = Q \Lambda Q^T \;\text{(real symmetric case)}$$Every real symmetric matrix is orthogonally diagonalizableRead more → — the nullity of $A - \lambda I$ is the geometric multiplicity of eigenvalue $\lambda$.
• Cramer's RuleCramer's Rule$$x_i = \frac{\det(A_i)}{\det(A)}$$Explicit determinant formula for the solution of a linear systemRead more → — Cramer's rule applies precisely when $\ker A = \{0\}$, i.e., $\operatorname{nullity}(A) = 0$.
• Determinant MultiplicativityDeterminant Multiplicativity$$\det(AB) = \det(A)\,\det(B)$$The determinant of a product equals the product of determinantsRead more → — $\det(A) \neq 0$ if and only if $\operatorname{nullity}(A) = 0$ if and only if $\operatorname{rank}(A) = n$.

/-- Rank–nullity theorem: for a linear map f : V → W between finite-dimensional
    K-vector spaces, the finrank of the image plus the finrank of the kernel
    equals the finrank of the domain. -/
theorem rank_nullity {K V W : Type*}
    [DivisionRing K] [AddCommGroup V] [Module K V]
    [AddCommGroup W] [Module K W]
    [FiniteDimensional K V] (f : V →ₗ[K] W) :
    Module.finrank K (LinearMap.range f) +
    Module.finrank K (LinearMap.ker f) =
    Module.finrank K V :=
  LinearMap.finrank_range_add_finrank_ker f