Null Space Calculator

The null space, also called the kernel, of a matrix A is the set of all column vectors x that satisfy Ax = 0. It is always a subspace, so it contains the zero vector and is closed under addition and scaling. This calculator reports a basis for that subspace and its dimension, the nullity. If the only solution is x = 0, the null space is trivial.

Reduce the matrix to reduced row echelon form (RREF) using Gauss-Jordan elimination, then identify the pivot columns and the free columns. For each free column you build one basis vector by setting that free variable to 1 and the other free variables to 0, then solving the pivot rows for the basic variables. This calculator does each step and lists the resulting basis vectors.

The nullity is the dimension of the null space, which equals the number of free columns. The rank-nullity theorem states that rank plus nullity equals the number of columns n, so nullity = n minus rank. For example, a 3 by 4 matrix of rank 2 has nullity 4 minus 2, which is 2, meaning two basis vectors span its null space.

If every column is a pivot column, the matrix has full column rank and the only solution of Ax = 0 is x = 0. The nullity is then 0 and the basis is empty. This happens exactly when the columns of A are linearly independent, which for a square matrix is the same as the determinant being nonzero.

The basis comes directly from the reduced row echelon form, so the components are exact ratios of the matrix entries. When a pivot does not divide evenly, those ratios show up as decimals, for example a coefficient of -0.5. You can scale any basis vector by a nonzero number and it still spans the same null space, so you may clear fractions if you prefer integer entries.

You can enter any matrix from 1 by 1 up to 6 by 6, and the number of rows and columns can differ. The tool handles wide matrices (more columns than rows), tall matrices, and square matrices. Changing the size keeps the values you already typed where they still fit.