Null Space and Kernel Solver

Math Linear Algebra • Vector Spaces and Linear Independence

Written by STEM Calculators Team Published February 22, 2026 Updated February 24, 2026

Find the null space / kernel of a matrix \(A\): all vectors \(x\) such that \(Ax=0\). The calculator computes RREF, identifies free variables, builds a null-space basis, and reports the nullity \(= n-\mathrm{rank}(A)\).

Rows \(m\)

Columns \(n\)

Tolerance

Display decimals

Steps

Show row-ops log Show pivot equations Simplify basis to integers (when possible) Show 2D plot when \(n=2\)

Inputs accept -3.5, 2e-4, fractions like 7/3, and constants pi, e.

Matrix \(A\)

A is 2×2

Kernel / null space: \(N(A)=\{x:\ Ax=0\}\).

Results

Rank

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Nullity \(=n-\mathrm{rank}\)

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Pivot columns

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Free columns

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Null space basis

A basis \(\{v^{(1)},\dots,v^{(k)}\}\) such that every solution is \(x=t_1v^{(1)}+\cdots+t_kv^{(k)}\).

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Ready

RREF

RREF shows pivot columns and free variables used to build a null-space basis.

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Enter \(A\) and click “Calculate”.

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Theory

Null space / kernel

For a matrix \(A\in\mathbb{R}^{m\times n}\), the null space (also called the kernel) is \[ N(A)=\{x\in\mathbb{R}^n:\ Ax=0\}. \] It is always a subspace of \(\mathbb{R}^n\). Every solution to the homogeneous system \(Ax=0\) lies in the null space.

If \(A\) represents a linear map \(T:\mathbb{R}^n\to\mathbb{R}^m\), then \(N(A)=\ker(T)\), the set of inputs mapped to the zero vector.

Rank–nullity theorem and nullity

The rank of \(A\) is the dimension of its column space (equivalently, the number of pivot columns in RREF). The nullity of \(A\) is the dimension of the null space: \[ \mathrm{nullity}(A)=\dim(N(A)). \]

The rank–nullity theorem states: \[ \mathrm{rank}(A)+\mathrm{nullity}(A)=n, \] where \(n\) is the number of columns of \(A\). So once you know the rank, you immediately know the nullity: \[ \mathrm{nullity}(A)=n-\mathrm{rank}(A). \]

How RREF gives a null-space basis

Compute the reduced row echelon form \(R=\mathrm{rref}(A)\). The pivot columns correspond to pivot variables, and the remaining columns are free variables.

In RREF, each pivot variable can be written as a linear expression in the free variables. Setting one free variable to \(1\) and the others to \(0\) produces one nonzero null-space vector. Doing this for each free variable yields a set of vectors that forms a basis of \(N(A)\).

If there are no free variables (nullity \(=0\)), then \(N(A)=\{0\}\) (only the zero vector solves \(Ax=0\)).

Geometric intuition

The null space lives in \(\mathbb{R}^n\) (the space of input vectors). When \(n=2\):

nullity \(=0\): kernel is just the origin \(\{0\}\)
nullity \(=1\): kernel is a line through the origin
nullity \(=2\): kernel is all of \(\mathbb{R}^2\) (happens when \(A=0\))

For larger \(n\), the null space is a higher-dimensional subspace (a plane through the origin, a 3D subspace, etc.).

University note: kernel–image connection

In linear algebra, the kernel (null space) and the image (column space) are tied together by rank–nullity: the “lost dimensions” of the map are exactly the dimensions of the kernel. This idea generalizes beyond matrices to linear transformations on abstract vector spaces.

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