What is the Difference Between Kernel and Null Space?

Contextual Differences in Terminology

In linear algebra, students often get confused by the terms Kernel and Null Space. They seem to represent the same concept—finding vectors that map to zero—yet they are used in different sections of textbooks. Why do we have two words for the same thing?

The distinction is not mathematical, but rather a matter of perspective and notation.

The Abstract Perspective: Kernel

The term Kernel (often written as ker(T)) comes from abstract algebra and group theory. It is defined in the context of a linear transformation T: V → W between two vector spaces V and W:

ker(T) = { v ∈ V | T(v) = 0 }

This definition is highly abstract. The vector spaces V and W do not have to be spaces of coordinate columns; they could be spaces of continuous functions, polynomials, or infinite sequences. The Kernel is the set of all inputs that map to the additive identity (zero) of the target space.

The Matrix Perspective: Null Space

The term Null Space (often written as Null(A) or N(A)) is concrete and defined specifically for a matrix A of size m × n:

Null(A) = { x ∈ Rⁿ | Ax = 0 }

This is a specific algebraic system. The Null Space is the solution set to the homogeneous system of linear equations represented by Ax = 0. We calculate a basis for the null space by converting the matrix to RREF and writing the solution in vector parametric form.

How They Connect

Every linear transformation T between finite-dimensional spaces can be represented as a matrix A once we choose bases for the spaces. Under this basis choice:

The Kernel of T is isomorphic to the Null Space of A.

Essentially, they are the same mathematical structure. You use "Kernel" when talking about functions and transformations, and "Null Space" when working directly with columns of numbers in matrices.