Null Space Calculator

Find a basis for the null space of a matrix and review the rank-nullity relationship beside the result.

Last updated: March 2026 | By ForgeCalc Engineering

Null Space Solver

Linear algebra

Rows

Columns

Null Space Steps

1.Form the matrix A with dimensions 3 by 3.

2.Solve the homogeneous system Ax = 0.

3.The null space is the set of all solutions x.

4.The dimension of the null space is 3.

Null Space Basis

Dimension (nullity): 3

The null space of a matrix is the set of all vectors x such that Ax = 0. It is also called the kernel and is always a subspace of the domain.

rank(A) + nullity(A) = number of columns

For a matrix with full column rank, the null space is just the zero vector.

nullity = 0

What does nullity mean?

Nullity is the dimension of the null space.

What if the null space is {0}?

That means the only solution to Ax = 0 is the zero vector.

Is the null space always a subspace?

Yes. It is always a subspace of R^n for an m by n matrix.

Does the calculator accept decimals?

Yes. Any finite real matrix entries are accepted.

Related Tools

Calculate adjoint matrix.

Find angle between vectors.

Find characteristic polynomial.

Cholesky factorization.

Expand determinant by cofactors.

Calculate cofactor matrix.