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Range, Null Space and Rank–Nullity

Source lecture(s): PHY621 Ch. 2

Intuition

Every linear map \(A\) does two things: it sends some directions somewhere (the range), and it annihilates others (the null space). The rank–nullity theorem says these two are complementary — what the map destroys and what it preserves must add up to everything it was given.

\[\boxed{\;\dim(\text{null space}) + \operatorname{rank} = n\;}\]

for \(A\) acting on an \(n\)-dimensional space.

The definitions

  • Range (column space): \(\{A\mathbf{x}\}\) — everything reachable. Its dimension is the rank.
  • Null space (kernel): \(\{\mathbf{x} : A\mathbf{x} = 0\}\) — everything crushed to zero. Its dimension is the nullity.

Rank is also the number of linearly independent columns, and the number of independent rows — those are equal, which is not obvious and is worth remembering.

The three cases of \(A\mathbf{x} = \mathbf{b}\)

Rank–nullity settles the solvability question completely:

Condition Solutions
\(\mathbf{b}\in\) range, nullity \(= 0\) exactly one
\(\mathbf{b}\in\) range, nullity \(= k > 0\) infinitely many, a \(k\)-parameter family
\(\mathbf{b}\notin\) range none

The middle row is the structurally important one: the general solution is one particular solution plus any element of the null space. That sentence is also the structure of the general solution of an inhomogeneous linear ODE — particular plus complementary function — and it is the same theorem. See ODEs.

Where physicists meet it

  • Zero modes. A null-space vector of the stiffness matrix is a motion that costs no energy. In the normal modes lab the free triatomic has exactly one: uniform translation, \(\omega = 0\). Counting zero modes is counting the null space, and by Noether's logic each corresponds to a symmetry.
  • Conservation laws and constraints. A constraint matrix's null space is the space of allowed motions.
  • Gauge freedom. In electromagnetism the map \(\mathbf{A}\mapsto\mathbf{B} = \nabla\times\mathbf{A}\) has a null space — gradients — and that null space is gauge freedom. Fixing a gauge means choosing one representative from each null-space coset.
  • Ill-conditioned inversions. A matrix with a nearly zero singular value has an approximate null space; inverting it amplifies noise catastrophically. Every regularisation scheme in data analysis is a way of handling that.

Worked example

Find the rank, nullity and general solution structure for $\(A = \begin{pmatrix}1 & 2 & 3\\ 2 & 4 & 6\\ 1 & 1 & 1\end{pmatrix}\)$

Row 2 is twice row 1, so at most two independent rows; rows 1 and 3 are independent, so \(\operatorname{rank} = 2\). With \(n = 3\), rank–nullity gives nullity \(= 1\).

The null space: solving \(A\mathbf{x} = 0\) gives \(x + 2y + 3z = 0\) and \(x + y + z = 0\), so \(y = -2z\) and \(x = z\); the null space is spanned by \((1,-2,1)^{\mathsf T}\).

Consequently \(A\mathbf{x} = \mathbf{b}\) has either no solution (if \(\mathbf{b}\) is outside the 2-dimensional range) or a one-parameter family \(\mathbf{x}_p + t(1,-2,1)^{\mathsf T}\). Never a unique one.

Common mistakes

  • Assuming a square matrix is invertible. It is invertible exactly when its nullity is zero, equivalently when \(\det \neq 0\), equivalently when it has full rank. A square matrix with a zero eigenvalue is singular.
  • Forgetting the null space when solving. Reporting one particular solution as "the" solution hides an entire family — and in physics that family is usually the interesting part (zero modes, gauge freedom).
  • Treating a small singular value as zero. Numerically it behaves like a null direction, but the distinction matters for whether your inverse is meaningful.

Knowledge graph position

Prerequisites: vector spaces, matrices. Leads to: eigenproblems, solvability conditions, gauge freedom, least-squares and regularisation.

Quiz

Q1 (conceptual). A \(5\times5\) matrix has rank 3. How many solutions does \(A\mathbf{x} = \mathbf{b}\) have?

Answer

Nullity \(= 5 - 3 = 2\). If \(\mathbf{b}\) lies in the 3-dimensional range there is a two-parameter family of solutions; if not, there are none. There is never a unique solution.

Q2 (conceptual). What does the null space of a stiffness matrix mean physically?

Answer

Displacements that cost no potential energy — zero modes. For an unconstrained system these are the rigid-body motions (translations and rotations), and they appear as \(\omega = 0\) eigenvalues. Their count is a good check that the matrix was assembled correctly.

Q3 (MCQ). Gauge freedom in electromagnetism corresponds to:

  • (a) the range of the curl operator
  • (b) the null space of the curl operator acting on the vector potential
  • (c) the rank of Maxwell's equations
  • (d) an eigenvalue of the Laplacian
Answer

(b). Adding any gradient to \(\mathbf{A}\) leaves \(\mathbf{B} = \nabla\times\mathbf{A}\) unchanged, because gradients are annihilated by the curl. Gauge fixing selects one representative from each equivalence class.