Kernel — Definition, Formula & Examples

The kernel of a linear transformation is the set of all input vectors that get mapped to the zero vector. For a matrix

A

, the kernel is the set of all vectors

\mathbf{x}

satisfying

A\mathbf{x} = \mathbf{0}

Given a linear transformation $T: V \to W$ between vector spaces, the kernel of $T$ is defined as $\ker(T) = \{\mathbf{v} \in V \mid T(\mathbf{v}) = \mathbf{0}\}$ . For an $m \times n$ matrix $A$ , this coincides with the null space $\text{Null}(A) = \{\mathbf{x} \in \mathbb{R}^n \mid A\mathbf{x} = \mathbf{0}\}$ , which is always a subspace of $\mathbb{R}^n$ .

Key Formula

\ker(A) = \{\mathbf{x} \in \mathbb{R}^n \mid A\mathbf{x} = \mathbf{0}\}

Where:

$A$ = An m × n matrix
$\mathbf{x}$ = A vector in ℝⁿ
$\mathbf{0}$ = The zero vector in ℝᵐ

How It Works

To find the kernel of a matrix

A

, set up the homogeneous system

A\mathbf{x} = \mathbf{0}

and row-reduce the augmented matrix

[A \mid \mathbf{0}]

to reduced row echelon form. Free variables in the solution correspond to basis vectors of the kernel. The dimension of the kernel is called the nullity of

A

. By the Rank-Nullity Theorem,

\text{rank}(A) + \text{nullity}(A) = n

, where

n

is the number of columns.

Worked Example

Problem: Find the kernel of the matrix

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

Step 1: Set up the homogeneous system and row-reduce.

\begin{pmatrix} 1 & 2 & 1 \\ 2 & 4 & 2 \end{pmatrix} \xrightarrow{R_2 - 2R_1} \begin{pmatrix} 1 & 2 & 1 \\ 0 & 0 & 0 \end{pmatrix}

Step 2: Read off the solution. With

x_2 = s

and

x_3 = t

as free variables, the first equation gives

x_1 = -2s - t

\mathbf{x} = s\begin{pmatrix} -2 \\ 1 \\ 0 \end{pmatrix} + t\begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}, \quad s, t \in \mathbb{R}

Answer: The kernel is a 2-dimensional subspace of

\mathbb{R}^3

spanned by

\{(-2, 1, 0)^T,\, (-1, 0, 1)^T\}

. The nullity is 2.

Why It Matters

The kernel tells you exactly when a linear system has non-unique solutions and measures how much information a transformation loses. In differential equations, the kernel of a differential operator gives the space of homogeneous solutions. Understanding the kernel is also essential in data science applications like principal component analysis and least-squares fitting.

Common Mistakes

Mistake: Confusing the kernel with the set of outputs (image/column space) of the transformation.

Correction: The kernel lives in the domain (input space) and consists of vectors mapped to zero. The image lives in the codomain and consists of all possible outputs.