Null Space Calculator

The null space (also called the kernel) of a matrix A is the set of all vectors x such that Ax = 0. It is a subspace of R^n whose dimension equals the nullity of A, which is n minus the rank.

If the matrix has full rank (rank = n), the null space contains only the zero vector — the equation Ax = 0 has only the trivial solution.

If the matrix is rank-deficient, the null space has one or more basis vectors, each representing a free direction in the solution space. Any linear combination of these basis vectors is also a solution to Ax = 0.

This calculator finds the null space via Gaussian elimination (RREF). After row reduction, columns without pivot entries correspond to free variables. For each free variable, set that variable to 1 and all other free variables to 0, then solve for the pivot variables by back-substitution. Each such assignment gives one null space basis vector.

Example: A = [[1,2,3],[4,5,6],[7,8,9]]. RREF reduces this to a rank-2 matrix. One free variable exists (nullity = 1). The null space basis is the vector (1, -2, 1), meaning x = t*(1,-2,1) satisfies Ax = 0 for any scalar t.

Enter the matrix as rows separated by semicolons: “1,2,3;4,5,6;7,8,9” for 3x3. The calculator shows the RREF, the rank, and each basis vector of the null space.

The null space matters in linear systems, differential equations, and least-squares problems. When fitting a model with collinear predictors, the null space of the design matrix reveals which parameter combinations are unidentifiable.