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Null Space — Definition, Formula & Examples

The null space of a matrix AA is the set of all vectors x\mathbf{x} that produce the zero vector when multiplied by AA. In other words, it collects every solution to the homogeneous equation Ax=0A\mathbf{x} = \mathbf{0}.

Given an m×nm \times n matrix AA, the null space (or kernel) of AA is the subspace Nul(A)={x∈Rn∣Ax=0}\text{Nul}(A) = \{\mathbf{x} \in \mathbb{R}^n \mid A\mathbf{x} = \mathbf{0}\}. It is a subspace of Rn\mathbb{R}^n and always contains at least the zero vector.

Key Formula

Nul(A)={x∈Rn∣Ax=0}\text{Nul}(A) = \{\mathbf{x} \in \mathbb{R}^n \mid A\mathbf{x} = \mathbf{0}\}
Where:
  • AA = An m × n matrix
  • x\mathbf{x} = A vector in ℝⁿ
  • 0\mathbf{0} = The zero vector in ℝᵐ

How It Works

To find the null space, set up the equation Ax=0A\mathbf{x} = \mathbf{0} and row-reduce the augmented matrix [A∣0][A \mid \mathbf{0}] to reduced row echelon form. Identify the free variables (columns without pivots) and express each pivot variable in terms of the free variables. Write the general solution as a linear combination of vectors, each scaled by one free variable. Those vectors form a basis for the null space. If there are no free variables, the null space contains only the zero vector, meaning Ax=0A\mathbf{x} = \mathbf{0} has only the trivial solution.

Worked Example

Problem: Find the null space of the matrix A = [[1, 2, 1], [2, 4, 0]].
Set up the system: Write the augmented matrix for Ax = 0 and row-reduce.
[12102400]→R2−2R1[121000−20]\left[\begin{array}{ccc|c} 1 & 2 & 1 & 0 \\ 2 & 4 & 0 & 0 \end{array}\right] \xrightarrow{R_2 - 2R_1} \left[\begin{array}{ccc|c} 1 & 2 & 1 & 0 \\ 0 & 0 & -2 & 0 \end{array}\right]
Reach reduced row echelon form: Divide R₂ by −2, then eliminate above the pivot in column 3.
[12000010]\left[\begin{array}{ccc|c} 1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]
Express the general solution: Pivots are in columns 1 and 3, so x₂ is free. Let x₂ = t. Then x₁ = −2t and x₃ = 0.
x=t[−210],t∈R\mathbf{x} = t\begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix}, \quad t \in \mathbb{R}
Answer: The null space of A is spanned by the vector (−2, 1, 0), forming a one-dimensional subspace of ℝ³.

Why It Matters

The dimension of the null space (called the nullity) directly determines how many free variables a linear system has. In differential equations and data science, the null space reveals redundancies among variables. The rank-nullity theorem ties it to the column space, making it essential for understanding when systems have unique versus infinitely many solutions.

Common Mistakes

Mistake: Confusing the null space with the set {0} whenever a matrix is square.
Correction: A square matrix can still have a nontrivial null space if it is singular (determinant = 0). Only invertible matrices have a null space consisting of just the zero vector.