Null Space of a Matrix

Definition

The null space (or kernel) of an $m \times n$ matrix $A$ is $\operatorname{Null}(A) = \{ \mathbf{x} \in \mathbb{R}^n : A \mathbf{x} = \mathbf{0}\}$

It's a subspace of $\mathbb{R}^n$: closed under addition and scalar multiplication, contains $\mathbf{0}$.

The dimension of the null space is called the nullity; by rank–nullity, $\operatorname{nullity}(A) = n - \operatorname{rank}(A)$.

Step-by-step

$A = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 4 & 2 \end{pmatrix}$

Step 1 — Row reduce.

$R_2 \to R_2 - 2R_1$: $(2, 4, 2) - 2(1, 2, 1) = (0, 0, 0)$.

$\text{RREF}(A) = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 0 & 0 \end{pmatrix}$

Step 2 — Identify pivots and free variables.

Pivot column: column 1. Free columns: columns 2 and 3. Let $x_2 = s$, $x_3 = t$.

Step 3 — Solve for pivot variables.

From row 1: $x_1 + 2 x_2 + x_3 = 0 \implies x_1 = -2 s - t$.

Step 4 — Write the general solution.

$\mathbf{x} = \begin{pmatrix} -2s - t \\ s \\ t \end{pmatrix} = s \begin{pmatrix} -2 \\ 1 \\ 0 \end{pmatrix} + t \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}$

Step 5 — A basis for $\operatorname{Null}(A)$: $\left\{ \begin{pmatrix} -2 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix} \right\}, \quad \operatorname{nullity} = 2$

Verification:

• $A \cdot (-2, 1, 0)^T = (-2 + 2 + 0, -4 + 4 + 0) = (0, 0)$ ✓
• $A \cdot (-1, 0, 1)^T = (-1 + 0 + 1, -2 + 0 + 2) = (0, 0)$ ✓

Why there are two free variables

$A$ is $2 \times 3$, rank 1 (one pivot). Rank-nullity: $1 + \operatorname{nullity} = 3$, so $\operatorname{nullity} = 2$. Our answer has a 2-parameter family of solutions, matching.

Structure of solutions to $A \mathbf{x} = \mathbf{b}$

Solutions to $A \mathbf{x} = \mathbf{b}$ (if any exist) have the form $\mathbf{x} = \mathbf{x}_p + \mathbf{x}_h$

where $\mathbf{x}_p$ is a particular solution and $\mathbf{x}_h$ ranges over the null space. So the null space describes the freedom in any linear solution.

• If $\operatorname{Null}(A) = \{\mathbf{0}\}$, the solution (if it exists) is unique.
• If $\operatorname{Null}(A)$ is larger, the solution is a translated copy of the null space.

Where null spaces show up

• Homogeneous ODEs: the space of solutions is the null space of a differential operator.
• Linear regression: redundant features span a non-trivial null space of the design matrix.
• Computer graphics: rotations have trivial null space; projections onto lower-dimensional subspaces have non-trivial null spaces.

Common mistakes

• Including the zero solution as a basis vector. The zero vector is always in the null space, but a basis consists of non-zero independent vectors.
• Mixing up "null space" and "zero matrix." Null space is a subspace of $\mathbb{R}^n$, not a matrix.
• Solving $A \mathbf{x} = \mathbf{b}$ and forgetting to translate by $\mathbf{x}_p$. For non-homogeneous systems, you add a particular solution to a null-space element.

Try it in the visualization

In 3D, the null-space vectors span a plane through the origin. The plane tilts as you edit $A$'s entries; its dimension updates from 0 (only origin) up to 3.